1 Introduction↩︎

In this paper, we address the question of the variation of Picard ranks in families of K3 surfaces over a base in positive characteristic \(p > 2\). This question is very well understood in characteristic zero. Indeed, let \(\mathscr{X}\rightarrow \Delta\) be a holomorphic family of K3 surfaces. Define the Noether–Lefschetz locus (succinctly, the NL locus) to be the set of points \(s\in \Delta\) such that the Picard rank of \(\mathscr{X}_s\) is greater than the generic Picard rank of \(\mathscr{X}\). Work of Oguiso ([1]) and Green ([2]) shows that if the family is not isotrivial, the NL locus is dense in \(\Delta\), and in particular is infinite.

The NL locus is can be defined for families of K3 surfaces over more general bases. Specifically, let \(\mathscr{X}/C\) be an algebraic family of K3 surfaces over some irreducible scheme \(C\) with generic point \(\eta\). Define the NL locus to be the set of points \(s\in C\) such that the Picard rank of \(\mathscr{X}_{\bar{s}}\) is greater than that of \(\mathscr{X}_{\bar{\eta}}\). The works of Green and Oguiso shows that if \(C\) is a positive dimensional scheme over \(\mathbb{Q}\) and \(\mathscr{X}\) is isotrivial, then the exceptional locus is Zariski-dense in \(C\). Studying the NL locus becomes significantly more challenging when the \(C\) is a scheme over an arithmetic base. For instance, let \(\mathscr{X}/\mathcal{O}_K[1/N]\) be a K3 surface, where \(\mathcal{O}_K\) is the ring of integers of a number field \(K\). In this setting, the NL locus is a union of points of the form \(\mathop{\mathrm{Spec}}\mathbb{F}_q\). The series of works [3], [4], [5] and [6] shows that the NL locus is infinite, and therefore Zariski dense in \(C\).

1.1 Exceptional families: counterexamples in positive characteristic↩︎

The setting where \(C\) is defined over a field of positive characteristic is more subtle, and even the answer depends on characteristic \(p\) invariants that do not exist in characteristic zero. Note that the question reduces to the case where \(C\) is a curve. So in what follows, we will always assume that \(C\) is a curve. Moreover, up to replacing \(C\) by a finite cover, we assume that line bundles on the geometric generic fiber are already defined on the generic fiber.

The work [7] (along with [6] to treat the case of non-projective \(C\)) shows that the NL locus is Zariski dense if the generic K3 surface parameterized by \(C\) is ordinary. But as noted in [7], the ordinariness condition is very important. For example, the generic Picard rank and the Picard rank of every specialization is 22 for a family of supersingular K3 surfaces, and so the NL locus is empty. This is not the only obstruction to the exceptional locus not being dense. Consider any non-isotrivial family of elliptic curves \(\mathscr{E}/C\), and let \(E\) be a supersingular elliptic curve. Then, consider the Kummer family \(\mathcal{K}/C\) associated to the abelian surface \(\mathscr{E}\times E /C\). It is easy to check that the Picard rank of \(\mathcal{K}_{\bar{s}}\) is greater than the generic Picard rank if and only if \(s\in C\) is a point of supersingular specialization of \(\mathscr{E}\). As there are only finitely many such points in \(C\), it follows that the NL locus is finite. The above example is not an isolated one, and perists even for families of K3 surfaces with smaller Picard rank. We note that being non-ordinary alone is not sufficient for the NL locus to be finite. For instance, [8] shows that families of non-ordinary abelian surfaces with trivial endomorphism ring are split at infinitely many fibers, a question that is intimately related to the distribution of the NL locus.

To explain the phenomenon underlying these examples, we will need to work with moduli spaces of lattice-polarized K3 surfaces. These spaces are (open parts of) GSpin Shimura varieties \(S_L\) associated to a quadratic lattice \(L\) having signature \((b,2)\) with (\(b\leq 19\)). These Shimura varieties admit families of special divisors which are themselves GSpin Shimura varieties associated to sublattices of \(L\) having signature \((b-1,2)\). We make the standing assumption that the discriminant of \(L\) (or equivalently the discriminant of \(\mathrm{Pic}(\mathscr{X}_\eta)\)) is prime to \(p\). By [9], this implies the existence of smooth integral models \(\mathscr{S}_L\) of \(S_L\) at \(p\). Under this assumption, there is a minimum-rank lattice \(L\) and a map \(C\rightarrow \mathscr{S}_L\) that induces \(\mathscr{X}\), and the assumption on the Picard lattice of the generic fiber of \(\mathscr{X}\) implies that \(L\) has prime-to-\(p\) discriminant. The mod \(p\) special fiber of \(\mathscr{S}_L\) admits a Newton stratification, with the open stratum being the ordinary locus and the smallest stratum being the supersingular locus (\(\mathscr{S}_{\textrm{ss}}\)). We consider the case when \(b\) is even – indeed, the NL locus in the odd case necessarily consists of every \(\overline{\mathbb{F}}_p\)-point as the the Picard rank of a K3 surface over a finite field is necessarily even. This is also true in the more general setting of \(\textrm{GSpin}\) Shimura varieties and is a consequence of the group-theoretic fact that the rank of \(\textrm{SO}_{2n-1}\) equals the rank of \(\textrm{SO}_{2n}\). The geometry and combinatorics of the Newton strata only depend on the isomorphism class of \(L\otimes \mathbb{Z}_p\). There are two cases – when \(L\otimes \mathbb{Z}_p\) is a direct sum of hyperbolic planes, and when it isn’t. In the first case, the dimension of \(\mathscr{S}_{\textrm{ss}}\) is smaller than in the second case, and has dimension \(b/2 -1\) (whereas, in the second case, \(\mathscr{S}_{\textrm{ss}}\) has dimension \(b/2\)). In the first case, the fact that \(\mathscr{S}_{\textrm{ss}}\) is smaller than expected has the crucial consequence that every special divisor contains the \(\mathscr{S}_{\textrm{ss}}\). Further, the unique “almost supersingular” Newton stratum, which we call \(\mathscr{S}_{\textrm{as}}\) (whose closure \(\overline{\mathscr{S}_{\textrm{as}}}\) equals \(\mathscr{S}_{\textrm{as}} \cup \mathscr{S}_{\textrm{ss}}\)), does not intersect any special divisor! In other words, the intersection of \(\overline{\mathscr{S}_{\textrm{as}}}\) with every special divisor is contained in \(\mathscr{S}_{\textrm{ss}}\). In particular, if \(\mathscr{X}/C\) is induced by a map \(C\rightarrow \overline{\mathscr{S}_{\textrm{as}}}\), then the NL locus of \(C\) is necessarily contained in the supersingular locus of \(C\), and thus must be finite. It is exactly this phenomenon that shows up in the Kummer example.

1.2 Statement of results in the setting of K3 surfaces↩︎

Our first result is in the setting of non-isotrivial families \(\mathscr{X}/C\) which are not exceptional, and which satisfy a monodromy result.

We will elaborate on the big monodromy assumption – indeed, there are non-exceptional examples where the NL locus is provably finite when the family has smaller monodromy, which we describe in Section 1.3. But first, we state a stronger theorem that implies the above result.

The notion of enlarged formal Brauer group traces back several years; cf. [11] and [12], where the group scheme is defined at least for non-supersingular K3 surfaces over finite fields. We will define the enlarged formal Brauer group for \(\mathscr{X}_\eta\) in Section 8. Roughly speaking, it is a \(p\)-divisible group characterized by the property that (up to a Tate twist) its Dieudonné crystal equals the quotient of the K3 crystal of \(\mathscr{X}_\eta\) by the subcrystal of the smallest slope. Note that when \(\mathscr{X}\) is generically ordinary, the connected-étale exact sequence of the enlarged formal Brauer group admits a global splitting if and only if the family \(\mathscr{X}\rightarrow C\) is isotrivial.

A consequence of the enlarged formal Brauer group admitting such a splitting is that the generic Kuga–Satake abelian scheme admits extra endomorphisms. The above result therefore has the following corollary.

1.3 Counterexamples with small monodromy↩︎

We note that the case of exceptional K3 surfaces are not the only counterexamples to the NL locus being dense. For example, let \(F/\mathbb{Q}\) be a real quadratic field, and let \(V'/F\) be a 4-dimensional vector space along with a bilinear form that is positive definite at one real place and has signature \((2,2)\) at the other. The Shimura variety \(S_{V'}\) associated to quadratic space is defined over \(F\), and embeds in \(S_L\) where \(L\) is a rank 8 quadratic \(\mathbb{Z}\)-lattice with signature \((6,2)\). By arguments of Geeman and Nikulin (see [15]), one can arrange that \(L\) embeds into the K3 lattice so that \(S_L\) parametrizes K3 surfaces with real multiplication by \(F\).

It is easy to see that there are infinitely many primes \(p\) such that \(p\) splits in \(F\) with \(v_1,v_2 \mid p\) the places above \(p\), and such that \(V'_{v_i}\) (for \(i = 1,2\)) is a direct sum of two hyperbolic planes over \(\mathbb{Q}_p\). Fix such a \(p\) and \(v = v_i\) for some \(i\in \{1,2 \}\). The mod \(v\) Shimura variety \(\mathscr{S}_{V',v}\) admits three Newton strata: the ordinary stratum, the supersingular stratum, and one intermediate stratum. This intermediate stratum does not intersect \(\mathscr{S}_{\textrm{as}}\subseteq \mathscr{S}_L\). However, it behaves like the 1-dimensional Newton stratum in the moduli space of a pair of elliptic curves in that the NL locus intersects the closure only along the supersingular locus. Therefore, the NL locus is always empty for a family of K3 surfaces pulled back from this intermediate Newton stratum. It is easy to construct similar examples where the K3 surface in question has smaller Picard ranks.

1.4 General GSpin Shimura varieties↩︎

In this section, we state the results in the general setting of GSpin Shimura varieties that go beyond the K3 case. We will always fix \(p\) to be an odd prime.

Let \(L\) be a lattice of signature \((b,2)\) which is self-dual at \(p\), and let \(\mathscr{S}_{L}\) be the canonical integral model of the associated GSpin Shimura variety at \(p\). Though \(\mathscr{S}_L\) does not parametrize K3 surfaces in general, one can still equip \(\mathscr{S}_L\) with a universal Kuga–Satake abelian scheme \(\mathcal{A}/\mathscr{S}_L\) via a suitable Hodge embedding. Similar to the K3 case, \(\mathscr{S}_L\) is equipped with a family of special divisors \(\{\mathcal{Z}(m)\}_{m\geq 1}\) which are Zariski closure in \(\mathscr{S}\) of GSpin Shimura subvarieties associated to sublattices of \(L\) having signature \((b-1,2)\). The special divisors parameterize points with so-called special endomorphisms (see Section 2.1 for definitions). The notion of formal Brauer groups also generalizes; cf, §8.

Similar to the case of K3 surfaces, when \(b\) is even, there is a dichotomy between \(L\otimes \mathbb{Z}_p\) being a direct sum of hyperbolic planes, and otherwise. In the former case, the supersingular locus has dimension \(b/2 - 1\) and \(\overline{\mathscr{S}_{\textrm{as}}} \cap \mathcal{Z}(m)\) is contained in the supersingular locus for every \(m\), and thus the NL locus cannot be dense for curves generically mapping to this locus. Now we let \(\iota:C\rightarrow \mathscr{S}_{L}\) be a nontrivial morphism, where the discriminant of \(L\) is prime to \(p\). We further assume that \(\mathscr{S}_L\) is the smallest \(\textrm{GSpin}\) Shimura varieties containing the image of \(\iota\), i.e. \(\iota\) does not factor through any \(\mathcal{Z}(m)\). Similar to the K3 case, we say that \(\iota\) is exceptional, if its image is supersingular, or \(L\otimes \mathbb{Z}_p\) is a direct sum of hyperbolic planes and \(\iota\) factors through \(\overline{\mathscr{S}_{\textrm{as}}}\subseteq \mathscr{S}_{L}\).

1.5 Outline of proof↩︎

The overarching outline follows [16], [17] and [7], and is a local-global argument. The idea (first used in [16]) is to use global inputs to estimate the global intersection \(C . \mathcal{Z}(m)\) as \(m\) varies, and to show that the local contribution \(i_P(C. \mathcal{Z}(m))\) (i.e., the intersection multiplicity of \(C\) with \(\mathcal{Z}(m)\) at \(P\)) from any one point \(P\in C\) is small relative to \(C.\mathcal{Z}(m)\). The difficulty, of course, is in proving good enough bounds for \(i_P(C . \mathcal{Z}(m))\). Our proof of this consists of three parts having distinct flavors. The first part is mostly global, and in particular shows that \(i_P(C.\mathcal{Z}(m))\) is small enough as long as \(P\) is not a supersingular point. For supersingular points, we reduce the main theorem to a purely local statement in the context of a formal curve \(\mathop{\mathrm{Spec}}\mathbb{F}[\![t]\!] \rightarrow \mathscr{S}\). This technique first appeared in [5] and [7], and the lack of ordinarity introduces no additional difficulty. Part 2 is purely local, and we establish a combinatorial characterization of “decaying” lattices of special endomorphisms, which is needed to control \(i_P(C.\mathcal{Z}(m))\) when \(P\) is a supersingular point. The fact that the curve avoids the ordinary locus introduces significant difficulties, which take new ideas to overcome. Part 3 is an algebraization result used to handle an extreme situation occurring in Part 2 where “maximal amount” of special endomorphisms resist decay (hence the local intersections can not be effectively controlled). When this happens, we show that the connected-étale exact sequence of the enlarged formal Brauer group over the generic point of \(C\) must split. This implies that the generic Kuga–Satake abelian scheme admits extra algebraic endomorphisms.

We now describe these three parts in more detail.

1.5.1 Local-global comparison↩︎

Borcherd’s theory tells us the asymptotic growth rate of the global intersection number \(C . Z(m)\) as \(m \to \infty\). More precisely, \(C . Z(m)\) is of the form \(|q_L(m)|(C \cdot \omega) + o(m^{b/2})\) and \(q_L(m) \asymp m^{(b + 2) / 2}\). Here, \(\omega\) is the Hodge bundle of \(\mathscr{S}\).

Let \(P\) be a \(\overline{\mathbb{F}}_p\)-point on \(C\) and let \(t\) be a uniformizer of \(C\) at \(P\), so that the formal completion of \(C\) at \(P\) is \(\mathop{\mathrm{Spf}}(\overline{\mathbb{F}}_p[\![t]\!])\). Let \(\mathcal{A}_r\) be the restriction of \(\mathcal{A}\) to \(\overline{\mathbb{F}}_p[t]/ (t^r)\), and let \(L_r := L(\mathcal{A}_r)\). Note that we have a nested sequence of lattices \(L(\mathcal{A}_P) = L_1 \supseteq L_2 \supseteq L_3 \supseteq \cdots\), such that a special endomorphism \(v \in L(\mathcal{A}_P)\) deforms to \(\overline{\mathbb{F}}_p[t]/ (t^r)\) if and only if \(v \in L_r\). By the moduli interpretation of \(\mathcal{Z}(m)\), the local intersection number \(i_P(C . \mathcal{Z}(m))\) is computed by \(\sum_{r = 1}^\infty \# \{ v \in L_r : v^2 = m \}\). It is proved in [7] (crucially using inputs from [5]) that if \(P\) is non-supersingular, then for large \(X\), \(\sum_{m \le 2X} i_P(C \cdot Z(m)) = O(X^{b/2} \log X)\), so the growth rate is not on the same order as the global intersection number. By the work of Tayou [6], a similar statement holds when \(P\) is a boundary point. When \(P\) is supersingular, we have \(i_P(C . \mathcal{Z}(m)) \asymp m^{(b + 2) / 2}\) as well, so we need to compare it with \(|q_L(m)|(C \cdot \omega)\) more carefully, and this is done in Part 2.

1.5.2 Supersingular points and decaying lattices of special endomorphisms↩︎

Now let \(P\) be supersingular. Roughly speaking, to bound \(i_P(C . \mathcal{Z}(m))\) we need \(L_r\) to become small quickly as \(r\) grows. This is done by examining for each \(n\), the “\(n\)-th order decay rate” \(d_n(v) = \max \{ r : p^{n - 1} v \in L_r \}\) for \(v \in L(\mathcal{A}_P)\). There are two kinds of results that we need:

1. There is a saturated sublattice (having large enough rank) of \(L(\mathcal{A}_P)\) consisting of vectors \(v\) that decays rapidly. See sections 5 and 7 for the definition of rapid decay, which roughly means that \(d_n(v)\)’s are subject to certain bounds.

2. The first time a primitive endomorphism \(v\) in this sublattice decays is sufficiently early (i.e., \(d_1(v)\) is sufficiently small). See section 6.

We need a lot more control for both steps as compared to the ordinary case. In particular, “large enough rank” just means rank 2 in the ordinary case, and we are able to establish this in a comparatively ad-hoc way in [7] and [17], whereas our treatment in the case of general Newton strata is (and needs to be) more systematic. Indeed, we are able to precisely relate the rates of higher-order decay to the degree of tangency of the curve to deeper Newton strata. This takes up all of Section 5 and part of 7.

When \(C\) generically maps into the ordinary stratum, one only requires one special endomorphism to decay “early” to first order, which can be deduced just from the fact that curve mapping non-constantly to \(\mathscr{S}\). However, the deeper the Newton stratum into which \(C\) generically maps, the more special endomorphisms are required to decay “sufficiently early” to first order. Even quantifying what sufficiently “early” (i.e., what bounds to put on \(d_n(v)\)’s) requires a significantly more involved setup. It turns out that the degree of tangency of the curve with the Artin strata is what governs the first order decay, and we end up proving that a large-rank isotropic space decays early to first order. Proving this takes up all of Section 6 and part of 7.

Taking both these steps together, it is possible to obtain a comprehensive description of the lattices \(\{ L_r \}_{r \ge 1}\). We find that the decay of special endomorphisms must follow some fascinating stringent patterns, which we were not able to find a satisfactory conceptual explanation. For curious reader, we provide a typical pattern of decay:

1.5.3 Algebraization↩︎

The precise description of decay in Part 2 allows us to find very tight bounds on the local intersection \(i_P(C. \mathcal{Z}(m))\). For most curves \(C\), this already yields the desired local-global comparison, hence the denseness of NL locus. The bound only fails when the curve is in an extreme position, where the \(p\)-divisible group \(\mathcal{A}_P[p^\infty]\) admits a “maximal amount” of formal special endomorphisms that do not decay (i.e., they deform all the way to the formal neighborhood \(C^{/P}\)), so \(i_P(C. \mathcal{Z}(m))\) is as large as possible. In fact, when this happens, the main term of the local intersections at supersingular points already equal the main term of the global intersection, hence the local–global comparison breaks down and one cannot conclude that NL locus is dense.

We note that this is not a byproduct of our decay results being insufficiently strong. Indeed, it is precisely this behaviour that gives rise to the counterexamples with small monodromy discussed in Section 1.3.

We overcome this structural difficulty by proving that in this extreme case, the connected-étale sequence \(\mathbf{\Psi}\) of the enlarged formal Brauer group over the generic point of \(C\) must split, cf. section 8. This is termed as algebraization in the following sense: let \(K(C)\) be the function field of \(C\). The existence of this “maximal amount of” formal special endomorphisms of \(\mathcal{A}_{C^{/P}}[p^\infty]\), which is a \(p\)-divisible group over a formal base, implies the splitting of \(\mathbf{\Psi}\) over \(K(C)\), which is a global and algebraic statement. It then follows that \(\mathcal{A}|_C\) has strictly more endomorphisms than \(\mathcal{A}\), i.e., \(\mathrm{End}(\mathcal{A}|_C) \otimes \mathbb{Q}\supsetneq \mathrm{End}(\mathcal{A}) \otimes \mathbb{Q}\).

Therefore, if \(\mathbf{\Psi}\) does not split over \(K(C)\), or if \(\mathrm{End}(\mathcal{A}|_C) \otimes \mathbb{Q}= \mathrm{End}(\mathcal{A}) \otimes \mathbb{Q}\), then we are not in the extreme situation, and the NL locus is dense as we have noted earlier.

1.6 Prior and related work↩︎

Our paper fits into the broader context of studying the distribution of algebraic cycles in a family. Over \(\mathbb{C}\), this reduces to studying the Hodge locus (at least when assuming the Hodge conjecture), and much recent progress has been made. In [18], [19], and [20], the authors thoroughly studied when the Hodge locus, or a distinguished part of it, is dense. From [20], we typically expect a theorem ensuring density to involve two parts in its assumptions.

1. The period dimension (i.e., the rank of the Kodaira-Spencer map at a generic point) is large enough for the Hodge locus to have non-negative expected dimension.

2. The monodromy group is not “too degenerate”.

Note that for a variation of Hodge structures, having infinite monodromy is equivalent to having positive period dimension — both are equivalent to non-isotriviality. In retrospect, it is somewhat a coincidence that when \(h^{2, 0} = 1\), non-isotriviality already ensures a dense NL locus (cf. [21]). In general, one would expect (I) and (II) to be separate (though related) conditions. In fact, this already happens in positive characteristic even when \(h^{2, 0} = 1\): the supersingular locus has positive period dimension and yet finite monodromy.

In analogy to the complex analytic situation, the hypotheses of our 3 consist of three parts: Non-isotriviality corresponds to (I). The non-exceptional-ness assumption excludes the obvious obstruction to a dense NL locus coming from Frobenius, while conditions (a), (b), (c) are analogues of (II). In particular, (a) implies the following: if the crystalline monodromy group, (cf. [22]) on the open Newton stratum of \(C\)¹ does not split as a product over \(W(\overline{\mathbb{F}}_p)[1/p]\), then the NL locus is dense. So (a) is indeed a condition on monodromy.

Another general theme which our results fit into is the intersection theory on Shimura varieties. During an AIM workshop, several participants (including the second named author) conjectured ([24]) that for an integral model \(\mathscr{S}\) of a Shimura variety of Hodge type and generically ordinary subvarieties \(X, Y \subset \mathscr{S}_{\mathbb{F}_p}\) of complementary dimension, the intersection of \(X\) with the union of prime-to-\(p\) Hecke translates of \(Y\) is Zariski dense in \(X\). The ordinary Hecke orbit (HO) conjecture ([25]) is the special case \(X = \mathscr{S}_{\mathbb{F}_p}\), \(Y\) a point; [7] treats the case where \(X\) is a curve in an orthogonal Shimura variety and \(Y\) a special divisor; and [26] treats the case when \(X,Y\) are ordinary curves inside a suitable Hilbert modular surface. These density results have some unexpected geometric applications, e.g., to a good reduction criterion for K3 surfaces ([27]). There is more known about this conjecture in characteristic zero - for example, the work [28] address this question in many cases, including when \(X\) is a curve and \(Y\) is a divisor, and ongoing work of Baldi-Urbanik addresses this question over the complex numbers whenever \(X,Y\) are Hodge-generic in the ambient Shimura variety.

Beyond the ordinary case, these questions become significantly harder, and even the correct formulation becomes mysterious. For \(X = \mathscr{S}_{\mathbb{F}_p}\) and \(Y\) a point, the full Hecke orbit conjecture (recently proved by van Hoften and D’Addezio [29]) roughly says that the obvious Frobenius constraint—that Hecke translates lie in the same central leaf—is the only one. However, as this paper illustrates, to extend [7] beyond the obvious Frobenius constraint, there are additional conditions whose failure may lead to counterexamples (e.g., §1.3). On the other hand, these additional conditions have certain parallels in complex Hodge theory. In [8], the first named author studied the almost ordinary locus when \(b = 3\) (i.e., for abelian surfaces). In that paper, the question considered isn’t a Picard-rank jumping question (which would be automatic as \(b\) is odd), but the more fine question of showing that the almost-ordinary curve \(C\) intersects \(\mathcal{Z}(m)\) where \(m\) varies over a fixed square-class. In this paper, we now provide a full generalization for all \(b\). In doing so, we hope to further illustrate how the Newton and Artin stratification governs the distribution of algebraic cycles in positive characteristic, and the parallel between the positive characteristic and the complex analytic situations.

1.7 Organization of the paper↩︎

In Section 2, we define the various objects in play, including \(\textrm{GSpin}\) Shimura varieties, special divisors and endomorphisms, and the Newton stratification. In Section 3, we setup the intersection-theoretic framework required to prove our main result, and using results from [5], [7] and [6], we reduce our main theorem to a local statement at supersingular points. In Section 4, we recall work of Kisin and Ogus to describe the universal K3 crystal at a formal neighbourhood of a supersingular point and use this to derive explicit equations for the Newton strata. Sections 5 and 6 are the technical heart of the paper where we bound local-intersection numbers at superspecial points under a certain local assumption. We generalize this to all supersingular points at 7. We finally prove the main theorem in Section 8, where we show that the local assumption always holds under the global hypotheses of our main result. We also have included a glossary with the notation that we use in our paper (especially for Sections 5, 6, and 7) for the reader’s convenience.

Acknowledgements↩︎

We are very grateful to Davesh Maulik, Arul Shankar, Yunqing Tang, and Salim Tayou for several helpful discussions over the past several years. We are especially grateful to Davesh and Yunqing for invaluable discussions at the start of this project. We also thank David Urbanik for helpful comments. A.S. was partially supported by the NSF grant DMS-2338942, the Institute for Advanced Studies (via the NSF grant DMS-2424441), and a Sloan research fellowship. Z.Y. was supported by the CUHK start up grant, and an ECS grant from the Hong Kong RGC (Project No. 24308225).

2 Preliminaries on orthogonal Shimura varieties↩︎

2.1 Basic setup↩︎

We will set up basic definitions and terminology for our Shimura varieties, special endomorphisms, and special divisors. At first, our exposition and setup will follow [7] and [5], and we will refer to [30] and [31]. We will then work modulo \(p\) and collect various results (mostly from [32], [33], and [7]) that together describe Newton strata.

Let \(L\) be a quadratic lattice with signature \((b,2)\) which is self-dual at \(p\). Write \(V\) for \(L_\mathbb{Q}\). We may also assume that \(L \subset V\) is maximal among those lattices contained in \(V\) on which which the bilinear form has integer values. Let \(\mathrm{Cl}(L)\) denote the Clifford algebra associated to \(L\). Note that we have a natural embedding of free \(\mathbb{Z}_{(p)}\)-modules \(L\otimes \mathbb{Z}_{(p)} \hookrightarrow \mathrm{Cl}(L)\otimes \mathbb{Z}_{(p)}\). Let \(G:= \mathrm{GSpin}(L\otimes \mathbb{Z}_{(p)})\) be the group of spinor similitudes. Note that \(G\) is a reductive group over \(\mathbb{Z}_{(p)}\) and is naturally a subgroup of \(\mathrm{Cl}(L \otimes \mathbb{Z}_{(p)})^{\times}\). Let \(\mathcal{D}\) be the Hermitian symmetric domain \(\{ z \in V_\mathbb{C}| \langle z, \overline{z} \rangle < 0 \}\). Then \((G, \mathcal{D})\) defines a spinor Shimura datum with reflex field \(\mathbb{Q}\). For every neat compact open subgroup \(\mathbb{K}\subseteq G(\mathbb{A}_f)\), we obtain a Shimura variety \(\mathrm{Sh}_{\mathbb{K}}(G, \mathcal{D})\) over \(\mathbb{Q}\).

Set \(H = \mathrm{Cl}(L)\), viewed as a \(\mathrm{Cl}(L)\)-bimodule. Note that \(H\) has a natural \(\mathbb{Z}/2 \mathbb{Z}\)-grading. Left multiplication of \(\mathrm{Cl}(L)\) on \(H\) the induces spin representation \(G \to \mathrm{GL}(H)\) and an embedding \(L \hookrightarrow\mathrm{Cl}(L) \hookrightarrow\mathrm{End}(H)\). Equip \(\mathrm{End}(H)\) with the pairing \((\alpha, \beta) := 2^{-\mathrm{rank\,} L} \mathrm{tr}(\alpha \circ \beta)\). Then the composite embedding \(L \hookrightarrow\mathrm{End}(H)\) is isometric and we can form the orthogonal projection \(\psi : \mathrm{End}(H) \to L\). The spinor group \(G\) can be conversely viewed as the stabilizer of the \(\mathbb{Z}/ 2 \mathbb{Z}\)-grading, the right \(\mathrm{Cl}(L\otimes \mathbb{Z}_{(p)})\)-action and the idempotent projector \(\psi\).

Suppose now that \(\mathbb{K}\) is of the form \(\mathbb{K}_p \mathbb{K}^p\) for \(\mathbb{K}^p \subseteq G(\mathbb{A}^p_f)\) and \(\mathbb{K}_p = G(\mathbb{Z}_p)\), i.e., is hyperspecial at \(p\). By [9] (cf. [31]) there is an canonical integral model \(\mathscr{S}_{\mathbb{K}}(G)\) over \(\mathbb{Z}_{(p)}\). We may endow \(H\) with a suitable symplectic form such that the left multiplication by \(G\) on \(H\) respects the form up to scaling, and indeed induces an embedding of Shimura data \((G, \mathcal{D}) \hookrightarrow(\mathrm{GSp}, \mathcal{H}^\pm)\), where \(\mathcal{H}^\pm\) is the associated Siegel half spaces. This Siegel embedding equips \(\mathscr{S}_{\mathbb{K}}(G)\) with a universal abelian scheme \(\mathcal{A}\), called the Kuga-Satake abelian scheme.² We use \(\mathcal{A}[p^\infty]\) to denote the \(p\)-divisible group associated to \(\mathcal{A}\). Throughout, we fix a choice of such level structure, and for brevity let \(\mathrm{Sh}\) (resp. \(\mathscr{S}\)) denote \(\mathrm{Sh}_{\mathbb{K}}(G,\mathcal{D})\) (resp. the integral model \(\mathscr{S}_{\mathbb{K}}(G,\mathcal{D})\)).

Define the sheaves \(\mathbb{H}_\mathrm{B}/\mathrm{Sh}_{\mathbb{C}}\), \(\mathbb{H}_\ell/\mathscr{S}\) (\(\ell \neq p\)), \(\mathbb{H}_p/\mathrm{Sh}\), \(\mathbb{H}_\mathrm{dR}/\mathscr{S}\) and \(\mathbb{H}_\mathrm{cris}/\mathscr{S}_{\mathbb{F}_p}\) denote the (first relative) Betti cohomology, \(\ell\)-adic etale cohomology, \(p\)-adic etale cohomology, de Rham cohomology, and the crystalline cohomology of the universal abelian scheme. The abelian scheme \(\mathcal{A}\) is equipped with a “CSpin-structure”: a \(\mathbb{Z}/ 2 \mathbb{Z}\)-grading, \(\mathrm{Cl}(L)\)-action and an idempotent projector \(\psi_? : \mathrm{End}(\mathbb{H}_?) \to \mathrm{End}(\mathbb{H}_?)\) for \(?\in \{B, \ell, p, \mathrm{dR}, \mathrm{cris}\}\) on (various applicable fibers of) \(\mathscr{S}\). We use \(\mathbb{L}_?\) to denote the images of \(\psi_?\).

2.2 Newton stratification↩︎

Let \(k\) be any perfect field of characteristic \(p \ge 5\) and let \(W = W(k)\). We describe the Newton stratification on \(\mathscr{S}_k\) following the scheme-theoretic approach of [33]. In order to extend the results there without referring to K3 surfaces, we can use the language of orthogonal \(F\)-zips. This extends the data of Hodge and conjugation filtrations, as well as the Frobenius action on the de Rham cohomology of K3 surfaces.

For any closed subscheme \(T\) of \(\mathscr{S}_k\), we write \(\widetilde{T}\) for the (\(p\)-complete) PD-envelope of \(T\) in \(\mathscr{S}_W\). When \(\widetilde{T}\) is \(p\)-torsion-free, we say that \(T\) is admissible, and the restriction \((\mathbb{L}_\mathrm{cris}|_T, p \mathop{\mathrm{Frob}})\) is a K3 crystal in the terminology of Nygaard–Ogus ([10]), ignoring the rank \(22\) condition. Recall that a local complete intersection is admissible ([34]). This condition allows us to view \(\mathbb{L}_\mathrm{cris}(\widetilde{T})\) as a submodule of \(\mathbb{L}_\mathrm{cris}(\widetilde{T})[1/p]\), on which \(\mathop{\mathrm{Frob}}\) natually acts.

The de Rham cohomology \(\mathbb{H}_\mathrm{dR}\) is equipped with a Hodge and conjugate filtration, these structures are inherited by the direct summand \(\mathbb{L}_\mathrm{dR}\) of \(\mathrm{End}(\mathbb{H}_\mathrm{dR})\). Note that there is a canonical (reduction mod \(p\)) map \(\mathbb{L}_\mathrm{cris}(\tilde{T}) \rightarrow \mathbb{L}_{\mathrm{dR},T}\), which we will denote in this section by \(\pi\). By Mazur–Ogus inequality [35], one can recover these structures using the Frobenius structure on \(\mathbb{L}_{\mathrm{cris}}|_T\): \[\begin{align} \mathop{\mathrm{Fil}}^i \mathbb{L}_{\mathrm{dR}, T} &= \pi\left(\mathop{\mathrm{Frob}}^{-1}(p^{i} \mathbb{L}_{\mathrm{cris}}(\widetilde{T}))\right) \tag{1} \\ \mathop{\mathrm{Fil}}_c^{i} \mathbb{L}_{\mathrm{dR},T} &= \pi\left(p^{-i} \left[\mathop{\mathrm{Frob}}(\mathbb{L}_\mathrm{cris}(\widetilde{T})) \cap p^i \mathbb{L}_\mathrm{cris}(\widetilde{T})\right]\right)\tag{2} \end{align}\] The first equation breaks into two \[\begin{align} \pi^{-1} (\mathop{\mathrm{Fil}}^0 \mathbb{L}_{\mathrm{dR}, T}) &= \{ \widetilde{x} \in \mathbb{L}_\mathrm{cris}(\widetilde{T}) \mid \mathop{\mathrm{Frob}}(\widetilde{x}) \in \mathbb{L}_\mathrm{cris}(\widetilde{T}) \} \\ \pi^{-1}(\mathop{\mathrm{Fil}}^1 \mathbb{L}_{\mathrm{dR}, T}) &\supsetneq \{ \widetilde{x} \in \mathbb{L}_\mathrm{cris}(\widetilde{T}) \mid \mathop{\mathrm{Frob}}(\widetilde{x}) \in p \mathbb{L}_\mathrm{cris}(\widetilde{T}) \}. \end{align}\] Note that the first is an equality because \(\mathop{\mathrm{Frob}}(\mathbb{L}_\mathrm{cris}) \in p^{-1} \mathbb{L}_\mathrm{cris}\), but the second is not. Technically, [35] applies to a smooth base. So in order to obtain (1 ) and (2 ) for arbitrary \(T\) (with \(p\)-torsion free \(\widetilde{T}\)), one can use that for any closed embedding \(T \hookrightarrow T'\), there are commutative diagrams \[\label{eqn:32embed32of32PD32envelopes} \begin{tikzcd} T & {T'} \\ {\widetilde{T}} & {\widetilde{T}'} \arrow[hook, from=1-1, to=1-2] \arrow[hook, from=1-1, to=2-1] \arrow[hook, from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \end{tikzcd} \Rightarrow \begin{tikzcd} {\mathbb{L}_\mathrm{cris}(\widetilde{T}')} & {\mathbb{L}_\mathrm{cris}(\widetilde{T})} \\ {\mathbb{L}_{\mathrm{dR}, T'}} & {\mathbb{L}_{\mathrm{dR}, T}} \arrow["{\otimes_{\mathcal{O}_{\widetilde{T}'}} \mathcal{O}_{\widetilde{T}}}", from=1-1, to=1-2] \arrow["\pi"', two heads, from=1-1, to=2-1] \arrow["\pi"', two heads, from=1-2, to=2-2] \arrow["{\otimes_{\mathcal{O}_{T'}} \mathcal{O}_T}", from=2-1, to=2-2] \end{tikzcd}\tag{3}\]

We will often implicitly make use of such diagrams. Over \(T = \mathscr{S}_k\) (and hence any \(T\)), the \(F\)-zip \(\mathbb{L}_{\mathrm{dR}, T}\) is equipped with two \(\mathcal{O}_T\)-linear isomorphisms \[\begin{align} \label{eqn:32gamma32-1} \gamma_{-1} : (F_T)^* (\mathrm{gr}^{-1} \mathbb{L}_{\mathrm{dR}, T}) \stackrel{\sim}{\to} \mathop{\mathrm{Fil}}^{-1}_c \mathbb{L}_{\mathrm{dR}, T} \text{ and } \gamma_0 : (F_T)^* (\mathrm{gr}^0 \mathbb{L}_{\mathrm{dR}, T}) \stackrel{\sim}{\to} \mathrm{gr}^{0}_c \mathbb{L}_{\mathrm{dR}, T}. \end{align}\tag{4}\] Also, note that \(\mathrm{gr}^{-1} \mathbb{L}_{\mathrm{dR}, T} = \pi(p \mathop{\mathrm{Frob}}(\mathbb{L}_\mathrm{cris}(\widetilde{T})))\).

Before we proceed, we briefly recall some basic facts abour K3 crystals over a point ([33]).

Our definition differs from that in [33] up to a Tate twist. Namely, we divide the Frobenius operator in the original definition by \(p\). Moreover, our conjugation filtration is ascending, which is compatible with the convention for F-zips but is opposite to the convention used in [33]. In particular, if \(\mathbb{L}_{\mathrm{dR}, T}\) comes from (the primitive part of) the de Rham cohomology of a K3 surface \(X\) over \(T\) as in [33], then our \(\mathop{\mathrm{Fil}}^{-1}_c \mathbb{L}_{\mathrm{dR}, T}\) corresponds to \(\mathop{\mathrm{Fil}}^2_c H^2_\mathrm{dR}(X/T)\).

Recall that the Artin invariant \(\sigma_0\) of the supersingular K3 crystal \((H, \mathop{\mathrm{Frob}})\) is the integer such that the Tate module \(H^{\mathop{\mathrm{Frob}}= 1}\) has discriminant isomorphic to \((\mathbb{Z}/ p \mathbb{Z})^{2 \sigma_0}\).

Now, following [33], we define a sequence of closed subschemes \(\mathcal{F}_1 = \mathscr{S}_k \supseteq \mathcal{F}_2 \supseteq\) of \(\mathscr{S}_k\) as follows:

Meanwhile, let us consider also the naive definition of Newton stratification which is customary in the literature on Shimura varieties. Namely, we simply define \(\mathscr{S}_h\) to be the closed subset consisting of points of height \(\ge h\), and then endow it with the reduced subscheme structure (see e.g., [36]). We write \(\mathscr{S}_{\infty}\) for the supersingular (or basic) locus on \(\mathscr{S}_k\). We recall the following dimension formula on \(\mathscr{S}_h\)’s from Howard–Pappas [37] (see also Shen–Zhang [36]).

Within \(\mathscr{S}_\infty\), let us write \(\mathscr{S}_{\infty, \sigma_0}\) for the closed subscheme of points \(s\) where \(\sigma_0(s) := \sigma_0(\mathbb{L}_{\mathrm{cris}, s}, p \mathop{\mathrm{Frob}}_s)\) has Artin-invariant \(\le \sigma_0\). We recall that:

Most of the arguments and results in [33] hold verbatim in the context of the general orthogonal Shimura varieties, except for Corollary 16 and Theorem 19, which only extend to the case (1). We shall need a statement for case (2). Therefore, below we rephrase and summarize the key inputs in [33] which hold regardless of the three cases in our general context.

The reader may be concerned that in 4, \(F_h\) is a priori only defined as a subfunctor for the functor of the points of \(\mathscr{S}_k\). That they are indeed representable by closed subschemes is justified by the surjectivity statement of (b) above (cf. [33]). Indeed, suppose that this is true for \(h\). Then \(\mathcal{F}_{h + 1}\) is the closed subscheme of \(\mathcal{F}_h\) defined by the ideal given by the image of the pairing \[\label{eqn:32ideal32of32F32h3243321} \mathop{\mathrm{Fil}}^1 \mathbb{L}_{\mathrm{dR}, \mathcal{F}_h} \otimes\left( E_h(\mathcal{F}_h) / E_{h - 1}(\mathcal{F}_h) \right) \to \mathcal{O}_{\mathcal{F}_h}.\tag{5}\] Suppose that \(\mathcal{F}'_{h + 1}\) is defined by this ideal. Then a \(\mathcal{F}_h\)-scheme \(T\) factors through \(\mathcal{F}'_{h + 1}\) if and only if \(E_h(\mathcal{F}_h) |_T \subseteq \mathop{\mathrm{Fil}}^0 \mathbb{L}_{\mathrm{dR}, T}\). But there is a commutative diagram \[\begin{tikzcd} {E_h(\mathcal{F}_h)|_T} & {E(T)} \\ {(\mathrm{gr}^{-1} \mathbb{L}_{\mathrm{dR}, \mathcal{F}_h})|_T} & {\mathrm{gr}^{-1} \mathbb{L}_{\mathrm{dR}, T}} \arrow[two heads, from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow[from=1-2, to=2-2] \arrow["\sim", from=2-1, to=2-2], \end{tikzcd}\] so the left vertical arrow vanishes if and only if the right one does. Hence \(\mathcal{F}'_{h + 1} = \mathcal{F}_{h + 1}\). In fact, as \(E_{h} / E_{h -1}\) is locally monogenic for \(\mathcal{F}_h\), (5 ) implies that \(\mathcal{F}_{h + 1}\) is locally cut out by a single equation from \(\mathcal{F}_h\). Next, we recall Ogus’ characterization of the tangent spaces of \(\mathcal{F}_h\).

Now we can extend [33] as follows.

In the next lemma, we extend [33] to a general base.³ Recall that \(\widetilde{T}\) denotes the PD-envelope in \(\mathscr{S}_W\) for a closed \(\mathscr{S}_k\)-scheme \(T\).

3 Arithmetic intersection theory↩︎

In this section, we will set up the intersection-theoretic framework required to prove Theorem 3. We will also reduce the proof of Theorem 3 to Theorem 7, which is a purely local statement at supersingular points.

3.1 Global intersection↩︎

We let \(L\) as in Section 2.

3.1.1 The global Eisenstein series↩︎

Let \(L\) be as in Section 2. Set \(U = \mathbb{C}[L^\vee / L]\). Let \(\mathrm{Mp}_2(\mathbb{Z})\) denote the metaplectic double cover of \(\mathrm{SL}_2(\mathbb{Z})\), which is equipped with a Weil representation \(\rho_L : \mathrm{Mp}_2(\mathbb{Z}) \to \mathrm{GL}(U)\). Let \(\kappa = 1 + \frac{b}{2}\) and \(M_{\kappa}\) be the space of vector-valued modular forms of \(\mathrm{Mp}_2(\mathbb{Z})\) of weight \(\kappa\) with respect to \(\rho_L\). Let \(\{ \mathfrak{e}_\mu : \mu \in L^\vee / L \}\) be the standard basis for \(U\). Let \(E_0(\tau)\in M_{\kappa}(\rho_L)\) be the vector valued Eisenstein series with constant term \(\mathfrak{e}_0\) as in [39]. Denote the Fourier expansion of \(E_0(\tau)\) as \[\label{eqn:32Fourier32expansion} E_0(\tau)= {\sum_{\mu \in L^\vee / L}} \sum_{m\geq 0,\mu\in \mathbb{Z}+Q(\mu)}q_L(m,\mu) q^m\mathfrak{e}_\mu ,\text{ where }q=e^{2\pi i\tau}.\tag{7}\] Let \(q_L(m) = q_L(m, 0)\). The theorem [40] provides an explicit expression of \(q_L(m)\). Before stating it, we first recall the notion of local density, which will be used throughout the paper:

By [40], the limit stablizes when \(a \ge 1 + 2 v_\ell(2m)\) (take \(d_\mu = 1\) as \(\mu = 0\)).

Given \(0\neq D\in\mathbb{Z}\) such that \(D\equiv 0,1\pmod{4}\), we use \(\chi_D\) to denote the Dirichlet character \(\chi_D(a)=\left(\frac{D}{a}\right)\), where \(\left(\frac{\cdot}{\cdot}\right)\) is the Kronecker symbol. For a Dirichlet character \(\chi\) we set \(\sigma_s(m,\chi)=\sum_{d\mid m}\chi(d)d^{s}\).

Here the last assertion is a direct consequence of the above explicit formulae and the fact \(\delta(\ell,L,m)\asymp 1\) (see also [7]). It plays an important role in arithmetic intersection theory.

3.1.2 Arithmetic intersection theory↩︎

Let \(k=\overline{\mathbb{F}}_p\). In the following we review the arithmetic intersection theory on \(\mathscr{S}^\mathrm{tor}_k\). Let the setup be as in §3.1.1. We will use the notation \(\overline{\mathcal{Z}(\mu, m)}\) to denote the closure of the special divisor \(\mathcal{Z}(\mu, m)\) in \(\mathscr{S}^{\text{tor}}\). To every \(\overline{\mathcal{Z}(\mu, m)}\), there is a carefully chosen associated divisor \[\mathcal{Z}^{\text{tor}}(\mu, m)= \overline{\mathcal{Z}(\mu, m)}+ \mathcal{B}\in \mathrm{Pic}(\mathscr{S}^\mathrm{tor})_{\mathbb{R}},\] where \(\mathcal{B}\) is a certain \(\mathbb{R}\)-linear combinations of boundary divisors; see [6]. Write \(Z^{\mathrm{tor}}(\mu,m)\) for the mod \(p\) reduction of \(\mathcal{Z}^{\text{tor}}(\mu, m)\), and let \(Z^\mathrm{tor}(m)\) be \(Z^{\mathrm{tor}}(0,m)\). Let \(\overline{Z(m)}\) be the Zariski closure of \(Z(m)\) in \(\mathscr{S}_k^\mathrm{tor}\). Consider the following generating series \[\Phi^{\text{tor}}_L:=\mathcal{L}^{-1}\mathfrak{e}_0+\sum_{\mu\in L^\vee/L}\sum_{m\in Q(\mu)+\mathbb{Z}}{Z}^{\text{tor}}(\mu, m) q^m\mathfrak{e}_\mu ,\text{ where }q=e^{2\pi i\tau}.\] It is known that \(\Phi^{\mathrm{tor}}_L\) is modular, i.e., \(\Phi^{\mathrm{tor}}_L\in M_{\kappa}(\rho_L)\otimes \mathrm{Pic}(\mathscr{S}_k^\mathrm{tor})_\mathbb{Q}\); cf. [6].

Let \(C\) be a proper smooth connected curve with a non-constant map \(C\rightarrow \mathscr{S}_{k}^\mathrm{tor}\) whose image lies generically in \(\mathscr{S}_{k}\). The modularity of \(\Phi^{\mathrm{tor}}_L\) enables us to relate the intersection number \((\overline{Z(m)}\cdot C)\) and the coefficients \(q_L(m)\) of the Eisenstein series \(E_0(\tau)\):

We note that \(q_L(m) \asymp m^{\frac{b+2}{2}}\). [7] and [6] control the local intersection numbers at non-supersingular points. For \(X\) some integer, let \(S_X\) denote the set of positive integers up to \(X\) which are relatively prime to \(p\).

In order to prove the main theorem, we will need the following result that treats supersingular points.

We now prove Theorem 3 assuming Theorem 6. As the deduction of Theorem 3 from the above stated local and global results follows the same contours as the proofs in [16], [3],[4], [17], [5], and [7], we will be content with a brief account.

In the next subsection, we will reduce Theorem 6 (using [7]) to purely local statement for formal curves passing through supersingular points.

3.2 The local and global intersection numbers at a supersingular point↩︎

Recall that \(a\) is the largest integer such that \(C\) maps to the locus \(\mathcal{F}_{a+1} \subseteq \mathscr{S}\). Set \(h_P = i_P(C\cdot\mathcal{F}_{a+2})\), the local intersection multiplicity of \(C\) and \(\mathcal{F}_{a + 2}\) at \(P\).

Since \(\mathcal{F}_{a+2}\) is the vanishing locus of a section of \(\omega^{p^{a+1}-1}|_{\mathcal{F}_{a+1}}\); cf. Lemma 3, we immediately get \[\label{eq:g95Pandglobal} (C\cdot\omega)|q_L(m)| = \sum_{P\in C\cap \mathcal{F}_{a+2}}g_P(m).\tag{8}\]

Now, let \(P\in C_{\mathrm{ss}}\) be a supersingular point, and let \(t\) be a local parameter at \(P\). Let \(\mathcal{A}_r := \mathcal{A}/\mathop{\mathrm{Spf}}k[\![t]\!]/(t^r)\) denote the pullback of \(\mathcal{A}\) to the \(r\)th infinitesimal neighbourhood of \(C\) at \(P\).

By construction, we have that \(L_{P,r} \subset L_{P,r-1}\). The equation below follows directly from the moduli interpretation of \(Z(m)\). \[\label{eq:local32intersection32defn} i_P(C\cdot Z(m)) = \sum_{r=1}^{\infty} \#\{v\in L_{P,r}: Q(v) = m \}.\tag{9}\]

For the rest of this section, we will reduce Theorem 6 to the purely local result stated just below:

4 Local structure of Shimura varieties at a superspecial point↩︎

4.1 Local structure in characteristic \(p\)↩︎

Let \(k\) be a perfect field of characterisic \(p\). We recall the group-theoretic description of [9] of the formal neighborhood of a closed point \(P\) in \(\mathscr{S}\), which we shall denote by \(\widehat{\mathscr{S}}_P\).

Let us write the K3 crystal \(\mathbb{L}_{\mathrm{cris}, P}\) simply as \(\mathbb{L}\). Then recall that \(\mathbb{L}\otimes_W k = \mathbb{L}_{\mathrm{dR}, P}\) is equipped with a Hodge filtration. Let \(\mu : \mathbb{G}_{m,W} \to \text{SO}(\mathbb{L})\) be a co-character (referred to as the “Hodge co-character") whose reduction modulo \(p\) splits the Hodge filtration. Define \(U\) to be the opposite unipotent subgroup in \(\mathrm{SO}(\mathbb{L})\) relative to \(\mu\), and let \(\mathop{\mathrm{Spf}}(R)\) denote the completion of \(U\) at the identity section. Let \(\sigma\) be a lift of the Frobenius endomorphism on \(R \otimes_W k\), and let \(u \in U(R)\) be the tautological point. Then, according to [9] (see also [41]), there exists an isomorphism \(\mathscr{S}^{/P}_W \simeq \mathop{\mathrm{Spf}}(R)\) through which \(\mathbb{L}_\mathrm{cris}(R)\) can be identified with \(\mathbb{L}\otimes_W R\) such that the Frobenius action is given by \(\mathop{\mathrm{Frob}}= u \circ (\varphi \otimes\sigma)\). Let us fix these identifications.

Using the above description, together with Ogus’ results on supersingular K3 crystals ([32]), we can express F-crystal \(\mathbb{L}_\mathrm{cris}\) over \(\mathscr{S}^{/P}_W\) in terms of local coordinates. This has been done in [7] so we briefly recap the main points for readers’ convenience and set up some notations. First, the Tate module \(\mathcal{L}:= \mathbb{L}^{\varphi = 1} =\) orthogonally splits into two parts \(\mathcal{L}_0 \oplus \mathcal{L}_1\) such that , if we denote by \(\langle-, -\rangle_i\) the symmetric pairing on \(\mathcal{L}_i\), then \(\langle-, -\rangle_1\) is self-dual over \(\mathbb{Z}_p\) whereas \(\langle-, -\rangle_0\) is divisible by \(p\) and \(p^{-1} \langle-, -\rangle_0\) is self-dual. Define \(\mathbb{L}_1 = \mathcal{L}_1 \otimes W\). Then \(\mathbb{L}_1\) is a direct summand of \(\mathbb{L}\), which splits into \(\mathbb{L}_0 \oplus \mathbb{L}_1\). The \(W\)-lattice \(\mathbb{L}_0\) is self-dual such that \(\mathbb{L}_0^{\varphi = 1} = \mathcal{L}_0\) and \(\mathcal{L}_0 \otimes W \subseteq \mathbb{L}_0 \subseteq p^{-1}(\mathcal{L}_0 \otimes W)\). Set \(t_P := \mathrm{rk}_{\mathbb{Z}_p} \mathcal{L}_0\). Then \(2 \mid t_P\) and \(a_P := t_P / 2\) is called the Artin-invariant of \(\mathbb{L}\). The point \(P\) is defined to be superspecial if \(a_P = 1\). In this section, we address the case when \(P\) is a superspecial point, and will deal with more general supersingular points in a later section.

As in [7], we have the following trichotomy.

1. \(\textrm{rank } \mathcal{L}_1\) is odd. We call this the odd case, and it corresponds to \(b\) odd.

2. \(\textrm{rank } \mathcal{L}_1 = 2m\) and \(\mathcal{L}_1\) is a split quadratic space. We call this the split case, and it corresponds to \(b\) even and \(L\otimes \mathbb{Z}_p\) not being split.

3. \(\textrm{rank } \mathcal{L}_1 = 2m\) and \(\mathcal{L}_1\) is not a split quadratic space. We call this the inert case, and it corresponds to \(b\) even and \(L\otimes \mathbb{Z}_p\) being split.

We will briefly need the odd case (despite the statement of Theorem 3 being trivial) to analyze singularities of Newton strata. For the rest of this section, and in Sections 5, 6, and 7, we will work in the split case unless explicitly stated otherwise.

4.2 Equations for Newton strata↩︎

4.2.1 ↩︎

In order to compute with coordinates, we need to introduce bases to \(\mathbb{L}\). For \(\mathbb{L}_1\), we just take any basis \(\{ e_i, f_i \}\) for \(\mathcal{L}_1\) such that the intersection matrix is given by \(\left[ \begin{array}{c|c} 0 & I \\ \hline I & 0 \end{array} \right]\). Using work of Ogus as in [17] and [7], \(\mathcal{L}_0\) has a basis \(e',f'\) such that \(\langle e', e' \rangle_0 = 2p, \langle f', f' \rangle_0 = -2 \lambda^2 p\) and \(\langle e', f' \rangle_0 = 0\), where \(\lambda \in W(\mathbb{F}_{p^2})^{\times}\) satisfies \(\sigma(\lambda) = -\lambda\) (cf. [7]). Then, a basis of \(\mathbb{L}_0\) is given by \(w' = \frac{1}{2\lambda p}(\lambda e' + f'), v' = \frac{1}{2\lambda}(\lambda e' - f')\). From [7], we can extract that for some choice of coordinates \[\label{eqn:32coords32of32R} \mathscr{S}^{/P}_W = R \simeq W[\![x_1, \cdots, x_m, y_1, \cdots, y_m]\!]\tag{10}\] the operator \(\mathop{\mathrm{Frob}}\) on \(\mathbb{L}\otimes R\) (identified with \(\mathbb{L}_\mathrm{cris}(R)\)) is given by the matrix \((u B) \circ \sigma\) where \[\label{eqn:32u32and32B} u = \left[ \begin{array}{cc|cc} 1 & Q & \mathbf{x} & \mathbf{y} \\ & 1 & & \\ \hline & - \mathbf{y}^t & I_m & \\ & - \mathbf{x}^t & & I_m \end{array} \right] \text{ and } B = \left[ \begin{array}{cc|cc} & p & & \\ p^{-1} & & & \\ \hline & & I_m & \\ & & & I_m \end{array} \right] \Rightarrow uB = \left[ \begin{array}{cc|cc} p^{-1} Q & p & \mathbf{x} & \mathbf{y} \\ p^{-1} & & & \\ \hline - p^{-1} \mathbf{y}^t & & I_m & \\ - p^{-1} \mathbf{x}^t & & & I_m \end{array} \right]\tag{11}\] with respect to the basis \(\{ v', w', e_1, \cdots, e_m, f_1, \cdots, f_m \}\). Note that compared to loc. cit., we switched the role played by the letters \(e_i, f_i, x_i, y_i\)’s with \(e'_i, f'_i, x'_i, y'_i\), so that now basis vectors or coordinates related to \(\mathbb{L}_0\) are labelled with a (\('\)). In the above, \(\mathbf{x}\) and \(\mathbf{y}\) are the row vectors \([x_1, \cdots, x_m]\) and \([y_1, \cdots, y_m]\) respectively. Here, \(Q=-\sum_{k=1}^m x_ky_k = - \mathbf{x}\mathbf{y}^t\).

For a matrix \(M\) we denote \(M^{(i)}\) the matrix whose entries are twisted by \(\sigma^{i}\), set \[\label{eqn:32define32F32infty32and32Q} Q_{j}=-\sum_{k=1}^m\left( x_ky^{(j)}_k+x^{(j)}_ky_k \right) = - \left( \mathbf{x}^{(j)} \mathbf{y}^t + \mathbf{x}(\mathbf{y}^{(j)})^t \right) \in W[\![x_i,y_i]\!].\tag{12}\]

In particular, we have that \(Q =\frac{1}{2} Q_0\).

5 Decay of special endomorphisms: superspecial points.↩︎

Let us keep using the notations in the preceeding section. In particular, \(P\) is a superspecial point on \(\mathscr{S}_\mathbb{F}\) and assume that we are in the split case. Suppose that \(v \in \mathcal{L}= \mathcal{L}(\mathcal{A}_P)\) is a formal special endomorphism and \(C\) is a formal curve at \(P\) on \(\mathscr{S}_\mathbb{F}\) which is not generically ordinary. In this section, we study the decay rate of \(v\), which measures how far \(v\) extends along \(C\).

To formulate this notion, let us make a preliminary definition.

For the fact that the locus to which \(v\) extends is indeed cut out by a single equation, the reader may refer to [31]. Technically, \(x_v\) is only defined up to \(\mathbb{F}^\times\). For convenience we fix one by asking its leading term to be \(1\) with respect to the lexicographic order of \(x_1, \cdots, x_m, y_1, \cdots, y_m\). Note that then the coordinates we have chosen for \(\mathscr{S}^{/P}_W\) can be re-interpreted by \(x_i = x_{e_i}, y_i = x_{f_i}\).

For the purpose of this section, we shall express the Frobenius operator \(\mathop{\mathrm{Frob}}\) on \(\mathbb{L}[1/p] \otimes R\) with respect to the Frobenius-invariant basis \(\{e', f', e_i, f_i \}_{i = 1, \cdots, m}\) of \(\mathbb{L}[1/p]\), as opposed to \(\{ v', w', e_i, f_i \}_{i = 1, \cdots, m}\) used in the preceeding section.⁴ Now, with respect to \(\{e', f', e_i, f_i \}_{i = 1, \cdots, m}\), \(\mathop{\mathrm{Frob}}=(1+F)\comp \sigma\), where \[\label{eq:Fmatrix} \renewcommand{\arraystretch}{1.5} F = \left[ \begin{array}{cc|cc} \frac{Q}{2p} & \frac{-\lambda Q}{2p} & \frac{\mathbf{x}}{2p} & \frac{\mathbf{y}}{2p} \\ \frac{Q}{2p\lambda} & \frac{-Q}{2p} & \frac{\mathbf{x}}{2p \lambda} & \frac{\mathbf{y}}{2p \lambda} \\ \hline -\mathbf{y}^t & \lambda \mathbf{y}^t & & \\ -\mathbf{x}^t & \lambda \mathbf{x}^t & & \end{array}\right]\tag{14}\] Set \(F_\infty=\prod_{i=0}^\infty (I+F^{(i)})\). Recall that \(\mathbf{x}=[x_1,...,x_m]\), \(\mathbf{y}=[y_1,...,y_m]\) and we defined \(Q_i\)’s in (12 ).

5.1 Explicit computation of \(F_\infty\)↩︎

In this subsection, we explicitly compute \(F_\infty\) over the non-ordinary Newton strata. The assumption implies that \(Q=0\). So the upper left block of \(F\) in (14 ) is 0. Then the upper right, and lower left blocks of \(F\) can be written as \[\label{eq:blocks} \frac{1}{2p}\begin{bmatrix} 1\\ \lambda^{-1} \end{bmatrix}\begin{bmatrix} \mathbf{x} & \mathbf{y} \end{bmatrix},\text{ and }\begin{bmatrix} -\mathbf{y}^t\\ -\mathbf{x}^t \end{bmatrix}\begin{bmatrix} 1 & -\lambda \end{bmatrix},\tag{15}\] respectively. The following are some basic observations:

We now give a thorough description of \(F_\infty\):

We shall now use \(F_\infty\) to study the deformation theory of a special endomorphism. Recall that \(\{e', f', e_i, f_i\}\) is a \(\mathbb{Z}_p\)-basis for \(\mathcal{L}\).

Next we will show that \(D_n(v)\) controls the deformation of \(v\).

We use the standard terminology “MIC” for module with integrable connection. Recall that \(\widehat{\mathbb{L}}:=\mathbb{L}_{\mathscr{S}^{/P}_{\mathbb{F}}}\) can be identified with a filtered Frobenius MIC over \(W[\![x_i,y_i]\!]\), i.e., a quadruple \((\mathbb{L}(W)\otimes W[\![x_i,y_i]\!], \nabla,\mathop{\mathrm{Fil}}^\bullet\mathbb{L}(W)\otimes W[\![x_i,y_i]\!], F)\). Let \(\widehat{\mathbb{L}}_I\) be the base change of \(\mathbb{L}_{\mathscr{S}^{/P}_{\mathbb{F}}}\) to \(\hat{D}_I\) along the map \(W[\![x_i,y_i]\!]\rightarrow W[\![t]\!]\hookrightarrow \hat{D}_I\), which is again a quadruple \((\mathbb{L}(W)\otimes \hat{D}_I, \nabla\otimes \hat{D}_I, \mathop{\mathrm{Fil}}^\bullet\mathbb{L}(W)\otimes \hat{D}_I, F)\).

By [42], there is a natural functor from the category of \(p\)-divisible groups over \(\mathop{\mathrm{Spf}}(\mathbb{F}[\![t]\!]/I)\) to Dieudonné modules over \(\hat{D}_I\). Since \(\mathbb{F}[\![t]\!]/I\) is a complete intersection scheme, [43] further implies that the aforementioned natural functor is fully faithful. As a result, there exists a formal special endomorphism of \(\mathcal{A}_{\mathop{\mathrm{Spf}}(\mathbb{F}[\![t]\!]/I)}[p^\infty]\) deforming \(v\), if and only if there exists an endomorphism of the corresponding Dieudonné modules over \(\hat{D}_I\) deforming \(v\), if and only if there is a horizontal section of \(\widehat{\mathbb{L}}_I\) extending \(v\).

Recall that \(\tilde{v}\) is the unique horizontal section of the MIC \((\mathbb{L}(W)\otimes K[\![x_i,y_i]\!],\nabla\otimes K[\![x_i,y_i]\!])\) extending \(v\), and \({\iota}^*\tilde{v}\) is the unique horizontal section of the MIC \((\mathbb{L}(W)\otimes K[\![t]\!],\nabla\otimes K[\![t]\!])\) extending \(v\). If \(v\) deforms to \(\mathop{\mathrm{Spf}}(\mathbb{F}[\![t]\!]/I)\), then there is a horizontal section \(v_I\) of \(\widehat{\mathbb{L}}_I\) extending \(v\). Then the base change of \(v_I\) along the natural embedding \(\hat{D}_I\hookrightarrow K[\![t]\!]\) identifies with \({\iota}^*\tilde{v}\). In other words, \({\iota}^*\tilde{v}=v_I\in \mathbb{L}(W)\otimes \hat{D}_I\). Conversely, if \({\iota}^*\tilde{v}\in \mathbb{L}(W)\otimes \hat{D}_I\), the same argument shows that it must be the horizontal section of \(\widehat{\mathbb{L}}_I\) extending \(v\). \(\hfill\square\)

5.2 Setting up the local curve↩︎

Throughout, our curve \(C\) will be contained in the non-ordinary Newton strata. Therefore, the running assumption will be that \(Q_0 = 0\). For the decay calculation, we will choose a uniformizer so that \(C=\mathop{\mathrm{Spf}}\mathbb{F}[\![t]\!]\). The equation of the local curve is give by \[\label{eq:localequation1} C:t\rightarrow (x_1(t),...,x_m(t);y_1(t),...,y_m(t)),\;\;x_i(t),y_i(t)\in \mathbb{F}[\![t]\!].\tag{25}\]

As in [17], we will also use \(x_i(t), y_i(t)\) to denote the power series in \(W[\![t]\!]\) where all the coefficients are Teichmuller lifts of the coefficients of \(x_i(t), y_i(t)\) in (25 ). This abuse of notation won’t cause confusion if one keeps track of which base ring they work in. To ease notation, we often just write \(x_i,y_i\) for the corresponding power series in \(t\).

As the \(a_i\)’s play a pivotal rule in our analysis, we draw a picture for readers’ convenience. If we draw a bar chart for \(h_{a}, h_{a + 1}, \cdots\), then the \(a_i\)’s look like the following. The dashed lines will be “thrown away” in our analysis.

5.2.1 Setup with special endomorphisms↩︎

In this section, we always assume that \(v\in\mathcal{L}\) is a primitive formal special endomorphism, i.e., \(p^{-1} v\notin \mathcal{L}\). Write \(\tilde{v}=F_\infty v\).

The concrete meaning of \(d_n(v)\) is given in the following lemma. The quantity turns out to be equal to \(v_t(x_{p^{n-1}v})\) and \(v_t(\overline{D}_n(v))\). However, as compared to \(v_t(x_{p^{n-1}v})\), its description is more concrete; as compared to \(v_t(\overline{D}_n(v))\), its definition is more conceptual. This makes it easier to use in many situations.

Suppose that \(n\geq 2\). It is helpful to get a feeling of what \(\overline{D}_n(v)\) should be, and how large \(d_n(v)\) can get. By recursion formula Lemma 7(b) and Remark 1, \(\overline{D}_n(v)\) is a huge linear combination of terms of the following form \[\label{eq:term2} w = Q_{i_1<i_2<...<i_{2r}}\,x_v^{(1+i_{2r})}, \,\,\,\,\,\,r\geq n-1 \text{ and }2|i_1\tag{26}\] We can compute the \(v_t\)-valuation of a term in (26 ) as \[\label{eq:termdeg3} v_t(w(t)) = \sum_{k=1}^{r} p^{i_{2k-1}}h_{i_{2k}-i_{2k-1}}+p^{1+i_{2r}}v_t(x_v).\tag{27}\] Let \(H\) be the minimal \(v_t\)-valuation among all the terms in (26 ). To estimate how large \(d_n(v)\) is, the first step is to look at all possible terms in (26 ) with \(v_t\)-valuation \(H\).

5.3 Precise definition of rapid decay↩︎

In this section, we study the terms in (26 ) that achieve the minimal \(v_t\)-valuation \(H\); cf. Lemma 10. In order to be able to work with the combinatorics in a flexible way, we introduce the following notions:

The argument above also explains why in the above definition we treated the cases \(v \bmod p\) lie in \(\mathcal{L}_0 / p\) or \(\mathcal{L}_1/ p\) separately.

The lemma below is a basic reality check for the definition above.

Recall that we have used the symbol “ \(v'\) ” for an element in the basis \(\{v',w';e_i,f_i\}\) of \(\mathbb{L}(W)\). As we won’t use this basis again, in the rest of this section, we will abuse the symbol “ \(v'\) ” to mean a special endomorphism in \(\mathcal{L}\).

We have the following result.

Part [prop:32prelim32decay32results322] of the above has the following consequence.

Note that for any \(r \ge s\), \[I_s(v) = \Big[\min_{\mathcal{W}\in \mathbb{I}_r^{\min}(v)} a(\mathcal{W}_{r - s + 1}) , \max_{\mathcal{W}\in \mathbb{I}_r^{\min}(v)} a(\mathcal{W}_{r - s + 1}) \Big].\]

We have the following results about intervals.

5.4 Isotropic subspaces↩︎

We have the following result.

Lemma 14 yields the following corollary.

5.5 Proof of the decay Lemma↩︎

We are now ready to state and prove the decay Lemma.

6 First order decay and Eisentein series: superspecial points↩︎

In this section we prove the following main result on the Eisenstein series, which is a major middle step towards Theorem 7:

The theorem will be proved at the end of §6.7. There are two main inputs: the first input is the decay lemma established in §5. The second is a thorough understanding of the first order decay, which is the main contributor to the local Eisenstein series. The study of the first order decay is carried out in §6.2\(\sim\)§6.4, culminating in Proposition 1 and 1, which state that a decaying sequence of lattices must follow a strict pattern.

6.1 An example↩︎

Let’s assume \(m=2\). We will use the setup and notation in §5.2. Then the only case we care is that \(C\) lies generically on the almost supersingular stratum. Suppose that the local coordinates for \(C^{/P}\) is given by series \(x_1,y_1,x_2,y_2\in \mathbb{F}[\![t]\!]\). Then \[\begin{align} \tag{30} & Q_0 = x_1y_1+x_2y_2= 0, \\ \tag{31} &Q_1= x_1^py_1+ x_1y_1^p+x_2^py_2+ x_2y_2^p\neq 0. \end{align}\] In particular, we must have \(a=a_k=1\) in this case. Let \(c_i=v_t(x_i)\) and \(d_i=v_t(y_i)\). Note that (30 ) implies that \(c_1+d_1=c_2+d_2\). We also suppose that \(\alpha_i,\beta_i\) are the leading coefficients of the \(x_i,y_i\) respectively. Note that replacing the vectors \(e_i,f_i \in \mathcal{L}_1\) by an \(\textrm{SO}(\mathbb{Z}_p)\) coordinates where each matrix entry is the teichmüller-lift of its mod \(p\) reduction induces changing the variables \((x_i,y_i)\) by an element of \(\textrm{SO}(\mathbb{F}_p)\). Note that doing this leaves the equations of the height strata unchanged. We shall refer to transformations of this form as orthogonal change of coordinates. Clearly, there are only finitely orthogonal change of coordinates. Note that swapping \(x_i\) with \(y_i\) is such change of coordinates.

The proof consists of a fairly straightforward process of iteratively applying orthogonal change of coordinates. We prove (with full detail) a far more general statement in a far more general setting later in this section (Lemmas 18 and 19).

We will explain how to deduce Theorem 9 for \(m=2\). This amounts to upper-bounding the series \(S'(M)\). Following Remark 1, we can explicate each term of \(S'(M)\) as \[\frac{q_{L_{P,r}'}(M)}{g_P(M)}=\frac{p^{2}-1}{h_P}\frac{1}{p(1+p^{-3})}\frac{\delta(p,L'_{P,r},M)}{[L'_{P,1}\otimes\mathbb{Z}_p:L'_{P,r}\otimes\mathbb{Z}_p]}.\] Note that in the setting of Lemma 15, we have that \[\label{eq:estimateh95Pm612} h_P = v_t(Q_1) \geq pc_1 + d_1,\tag{32}\] so the term \(\frac{p^2-1}{h_P}\) can be bonded from above accordingly. Therefore, in theory we can bound \(S'(M)\) by bounding the local density \(\delta(p,L'_{P,r},M)\) and covolume \({[L'_{P,1}\otimes\mathbb{Z}_p:L'_{P,r}\otimes\mathbb{Z}_p]}\). However, in practice, the shape of \(L'_{P,r}\) is somewhat evasive. Instead of directly bounding the local density and covolume for \(L'_{P,r}\), we will construct a sequence of auxiliary lattices \(L''_{P,r}\) which we can describe exactly. For each lattice \(L''_{P,r}\) and each integer \(M\) not a multiple of \(p\), we can also precisely compute \(\delta(p,L''_{P,r},M)\) and \([L''_{P,1}\otimes\mathbb{Z}_p:L''_{P,r}\otimes\mathbb{Z}_p]\). This allows us to explicitly upper-bound the auxiliary series \[S''(M):=\sum_{r=1}^{\infty}\frac{q_{L_{P,r}''}(M)}{g_P(M)}=\frac{p^{2}-1}{h_P}\frac{1}{p(1+p^{-3})}\sum_{r=1}^\infty\frac{\delta(p,L''_{P,r},M)}{[L''_{P,1}\otimes\mathbb{Z}_p:L''_{P,r}\otimes\mathbb{Z}_p]}.\] We will then show that \(S'(M)\leq S''(M)\). Upon computing \(S''(M)\) explicitly, we obtain that \(S'(M)<1\) for all \(M\) coprime to \(p\) unless \(\mathcal{A}_{C^{/P}}\) admits a rank-2 saturated lattice of formal special endomorphisms. Since \(\delta(p,L'_{P,r},M)\) depends only on \(\left(\frac{M}{p}\right)\) and \({[L'_{P,1}\otimes\mathbb{Z}_p:L'_{P,r}\otimes\mathbb{Z}_p]}\) is independent of \(M\), there exists a constant \(\alpha<1\) independent of \(M\), such that \(S'(M)<\alpha\). This yields the theorem.

We will now carry out this procedure in case 3 of Lemma 15 (the hardest case). In other words, we will describe \(L''_{P,r}\) in case 3 of Lemma 15, and will explicitly compute \(S''(M)\). Note that in case 3 we must have \(c_1=c_2=d_2=d_1\), and we will use \(d\) to denote this number. In the following, we construct \(L''_{P,r}\). First, let \(\mathcal{L}_{\mathrm{brd}}:=\mathrm{Span}_{\mathbb{Z}_p}\{e_1-\gamma_1f_1, e_2-\gamma_2f_2\}\), where by abuse of notation \(\gamma_i\in \mathbb{Z}_p^*\) denotes the Teichmüller lift of \(\gamma_i\) in Lemma 15(3). Second, for each \(j\geq 1\), define \(H_{j}:=h_P(1+p^2+...+p^{2(j-1)})\) and \(H_{0}:=0\). Finally, define \(L''_{P,r}\) to be the \(\mathbb{Z}\)-lattice such that \(L''_{P,r}\otimes \mathbb{Z}_l=L'_{P,r}\otimes \mathbb{Z}_l\) for \(l\neq p\), and \[L''_{P,r}\otimes\mathbb{Z}_p= \left\{\begin{align} & \mathcal{L}_{\mathrm{brd}}+p^j\mathcal{L},\,\,\,\, &r\in (H_{j},H_{j}+p^{2j}d],\\ & \mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}_0 +p^{j+1}\mathcal{L}_1,\,\,\,\, &r\in (H_{j}+p^{2j}d,H_{j+1}]. \end{align}\right.\] The lattice \(L''_{P,r}\) can be thought of as \(L'_{P,r}\) in the special case where all vectors in \(\mathcal{L}_{\mathrm{brd}}\) never decay. For \(M\) coprime to \(p\), one computes that \[\delta(p,L''_{P,r},M)=\left\{\begin{align} & 1-p^{-2},\,\,\,\, &r\in (0,a],\\ & 1+p^{-1},\,\,\,\, &r\in (a,\infty). \end{align}\right.\] \[[L''_{P,1}\otimes\mathbb{Z}_p:L''_{P,r}\otimes\mathbb{Z}_p]=\left\{\begin{align} & p^{4j},\,\,\,\, &r\in (H_{j},H_{j}+p^{2j}d],\\ & p^{4j+2},\,\,\,\, &r\in (H_{j}+p^{2j}d,H_{j+1}]. \end{align}\right.\] It follows that \[\begin{align} S''(M)&\leq \frac{p^2-1}{d(p+1)}\frac{1}{p(1+ p^{-3})}\left[d(1-p^{-2})+ (1+p^{-1})\sum_{j=1}^\infty \frac{dp^{2j}}{p^{4j}} +(1+p^{-1})\sum_{j=0}^\infty\frac{(h_P-d)p^{2j}}{p^{4j+2}} \right]\\ & \leq \frac{p^2-1}{d(p+1)}\frac{d}{p(1+ p^{-3})}\left[(1-p^{-2})+\frac{(1+p)^2}{p(p^2-1)}\right] =1. \end{align}\] Each inequality in the above formula comes from (32 ).

Since \(\mathcal{A}_{C^{/P}}\) does not admit a rank-2 saturated lattice of formal special endomorphisms, follows from the lemma that \(S'(M) < S'(M)\leq 1\). We have seen that this implies Theorem 9.

6.2 \(Q\)-degeneracy and line configurations↩︎

We introduce the notions of \(Q\)-degeneracy and line configuration that effectively captures the combinatorics of the Frobenius-twisted quadratic forms \(\{Q_i(t)\}_{i\geq 0}\) associated to a collection of power series \(\{\underline{x}(t),\underline{y}(t)\}\) that arise in the context of §5.2. This can be thought of as an analogue of the Newton polygon. For maximal flexibility, we will build our tools separately from the setup in §5.2.

The following notion will be extremely important through out the section:

6.2.1 Basic definitions↩︎

6.2.2 \(Q\)-isometry and maximal elements↩︎

The \(Q\)-isometric class can be best understood in the following context: Fix \(n\). We can form a frame space \[V_{\mathrm{frame}}=\mathrm{Span}_{\mathbb{F}_p}\{X_1,...,X_n,Y_1,...,Y_n\},\] where \(X_i,Y_i\) are merely symbols. This is a quadratic \(\mathbb{F}_p\)-space with pairing \[Q\left(\sum_{i=1}^na_iX_i+b_iY_i\right)= \sum_{i=1}^n a_ib_i.\] We can then say that a collection \(\{\underline{x},\underline{y}\}\) of power series in \(\mathbb{F}[\![t]\!]\) is an \(\mathbb{F}[\![t]\!]\)-realization of \(\{\underline{X},\underline{Y}\}\), written as \(\{\underline{X},\underline{Y}\}\rightarrowtail \{\underline{x},\underline{y}\}\). Then any element in the \(Q\)-isometry class of \(\{\underline{x},\underline{y}\}\) is induced from an isometry of \(V_{\mathrm{frame}}\): \[\{\underline{X},\underline{Y}\} \xrightarrow{R} \{\underline{X},\underline{Y}\}\rightarrowtail \{\underline{x},\underline{y}\}.\]

Maximal collections have nice properties. The following lemmas will be extremely useful:

6.3 Critical points and shape of the \(Q\)-polygon↩︎

In the following, we understand the behavior of critical points (Definition 16([def:critpoints])) of a line configuration. The main result is Corollary 7.

6.3.1 Dependence results↩︎

If \(s_1,...,s_n\in \mathbb{F}\), the Moore matrix \((s_j^{(i-1)})_{1\leq i,j\leq n}\) will be denoted by \(M(s_1,...,s_n)\).

6.3.2 Application to critical points↩︎

6.4 Pattern of the first order decay↩︎

Let recall the setup in §5.2: we have a collection \[\label{eq:collection} \{x_1,...,x_m, y_1,...,y_m\}\tag{37}\] of nonzero series in \(\mathbb{F}[\![t]\!]\) given by the equation of the formal curve \(C\). We also have \(Q_0(t)=...=Q_{a-1}(t) = 0\) and \(Q_a(t)\neq 0\). We have defined \(h_i=v_t(Q_i(t))\). Furthermore, \(a_1 = a < a_2 \ldots < a_k\) is the maximal sequence of integers such that \(a_k \leq m-1\), and \(h_{a_1} \geq h_{a_2} \ldots \geq h_{a_k}\). Note that any collection in the \(Q\)-isometry class of (37 ) is \(Q_{a_k-1}\)-degenerate.

We have the following two key propositions.

6.5 Estimating \(S'(M)\): preparation↩︎

We continue to use the setup in §6.4, and will assume that our maximal collection satisfies Assumption 1[ass:red2].

In this section, we make various preparations for computing \(S'(M)\). First, we split out cases from three perspectives. Second, we define auxiliary lattices \(L_{P,r}''\) generalizing the construction in the toy example §6.1, based on a cases by case study (however, the explicit computation of \(L_{P,r}''\), and the relation between \(S'(M)\) and \(S''(M)\) will be carried out in the next two sections §6.6\(\sim\)§6.7). Third, we compute local density for \(L_{P,r}''\). Fourth, we estimate a quantity \(D_0\) which will appear in various computation later on.

6.5.1 Split out cases and set up notations↩︎

We have three different rules to split out cases: depending on whether \(a_k>a\), we have two cases; depending on the dimension of \(\overline{\mathcal{L}}_{\mathrm{bd}}\), we have three cases; depending on whether \(n>a_k+1\), we have two cases. Let’s spell them out:

1. If \(a_k=a\) (resp. \(a_k>a\)), we say that we are in tight (resp. loose) case.

2. If \(\dim\overline{\mathcal{L}}_{\mathrm{bd}}=i\in \{0,1,2\}\), we say that we are in borderline case \(i\).

3. If \(n> a_k+1\) (resp. \(n= a_k+1\)), we say that we are in thick (resp. thin) case. This is already defined in Definition 21, and the fact that \(n\leq a_k+1\) implies \(n=a_k+1\) is proved in Proposition 1.

The three different rules do not depend on each other, so we have \(2\times 3\times 2=12\) subcases in total. For example, when \(a_k=a\), \(\dim\overline{\mathcal{L}}_{\mathrm{bd}}=2\) and \(n>a_k+1\), we are in tight thick borderline case 2.

6.5.2 Auxiliary lattices: the tight case↩︎

Assume that \(a_k=a\). In this section, we define a sequence of auxiliary lattices \(L_{P,r}''\), generalizing the construction in the toy example §6.1.

Recall we have the borderline subspace \(\overline{\mathcal{L}}_{\mathrm{bd}}\) and redundant subspace \(\overline{\mathcal{L}}_{\mathrm{rd}}\) as per Definition 21. We now define \(\mathbb{Z}_p\)-lattices \(\mathcal{L}_{\mathrm{bd}}\) and \(\mathcal{L}_{\mathrm{rd}}\) as follows. We define \(\mathcal{L}_{\mathrm{rd}}:=\mathrm{Span}_{\mathbb{F}_p}\{e_j,f_j\}_{n<j\leq m}\). If \(\dim\overline{\mathcal{L}}_{\mathrm{bd}}=0\), we define \({\mathcal{L}}_{\mathrm{bd}}=\mathbf{0}\). If \(\dim\overline{\mathcal{L}}_{\mathrm{bd}}=1\) (resp. 2), then up to a \(Q\)-isometry that keeps Assumption 1[ass:red2], we can assume \(\overline{\mathcal{L}}_{\mathrm{bd}}=\{\overline{e_n}-\gamma_n\overline{f_n}\}\) (resp. \(\{\overline{e_{n-1}}-\gamma_{n-1}\overline{f_{n-1}}, \overline{e_n}-\gamma_n\overline{f_n}\}\)), and we define \({\mathcal{L}}_{\mathrm{bd}}=\mathrm{Span}_{\mathbb{Z}_p}\{e_{n}-\gamma_{n}f_{n}\}\) (resp. \(\mathrm{Span}_{\mathbb{Z}_p}\{e_{n-1}-\gamma_{n-1}f_{n-1}, e_{n}-\gamma_{n}f_{n}\}\)), where by abuse of notation \(\gamma_{n-1},\gamma_{n}\in \mathbb{Z}_p\) are Teichmuller lifts of \(\gamma_{n-1},\gamma_{n}\in \mathbb{F}_p\). Let \(\mathcal{L}_{\mathrm{brd}}:=\mathcal{L}_{\mathrm{bd}}\oplus \mathcal{L}_{\mathrm{rd}}\). For \(0\leq i\leq n\), we define \[\begin{align} &\mathcal{L}_{> i}=\mathrm{Span}_{\mathbb{Z}_p}\{e',f',{e}_{j},{f}_{j'}\}_{i<j\leq n, 1\leq j' \leq n }.\\&\mathcal{L}'_{\leq i}=\mathrm{Span}_{\mathbb{Z}_p}\{e',f',{f}_{j'}\}_{1\leq j' \leq i}. \end{align}\] Note that by definition, we have \(\mathcal{L}_{> n}=\mathcal{L}'_{\leq n}\). These lattices are closely related to the description in Proposition 1.

Moreover, for each \(j\geq 1\), define \(H_{j}:=h_P(1+p^{a_k+1}+...+p^{(a_k+1)(j-1)})\) and \(H_{0}:=0\).

Finally, define \(L''_{P,r}\) to be the \(\mathbb{Z}\)-lattice such that \(L''_{P,r}\otimes \mathbb{Z}_l=L'_{P,r}\otimes \mathbb{Z}_l\) for \(l\neq p\), and \[L''_{P,r}\otimes\mathbb{Z}_p= \left\{\begin{align} & \mathcal{L}_{\mathrm{brd}}+p^j\mathcal{L}_{>0}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j},H_{j}+p^{(a_k+1)j}c_1],\\ & \mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}_{>1}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j}+p^{(a_k+1)j}c_1,H_{j}+p^{(a_k+1)j}c_2],\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\;\;\;\;\;\;\;\;\;\;\;&\vdots\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\\ &\mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}_{>n-1}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j}+p^{(a_k+1)j}c_{n-1},H_{j}+p^{(a_k+1)j}c_n],\\ &\mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}'_{\leq n}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j}+p^{(a_k+1)j}c_{n},H_{j}+p^{(a_k+1)j}{d_n}],\\ &\mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}'_{\leq n-1}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j}+p^{(a_k+1)j}d_{n},H_{j}+p^{(a_k+1)j}{d_{n-1}}],\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots\;\;\;\;\;\;\;\;\;\;\;\;\;\;,&\vdots\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\\ &\mathcal{L}_{\mathrm{brd}}+ p^{j}\mathcal{L}'_{\leq 0}+ p^{j+1}\mathcal{L},\,\,\,\, &r\in (H_{j}+p^{(a_k+1)j}d_{1},H_{j+1}].\\ \end{align}\right.\]

6.5.3 Auxiliary lattices: the loose case↩︎

Assume that \(a_k>a\). We will again construct a sequence of auxiliary lattices \(L_{P,r}''\). First, define \(\mathcal{L}_{\mathrm{brd}}\), \(\mathcal{L}_{> i},\mathcal{L}'_{\leq i}\) and \(H_j\) in the same way as §6.5.2. For \(r\leq H_1=h_P\), we define \(L''_{P,r}\) the same way as §6.5.2. We won’t need auxiliary lattices when \(r>h_{a_k}\).

6.5.4 Local density computation↩︎

Below we state a local density formula, using the following notation. Let \(\mathsf{Q}\) be the set of isomorphism classes of self-dual quadratic forms over \(\mathbb{F}_p\). Define \(\chi : \mathsf{Q} \to \{-1,0,1\}\) by setting \(\chi = 0\) for odd-dimensional forms, \(1\) (resp. \(-1\)) for even-dimensional split (resp. nonsplit) forms. For \(\alpha \in \mathbb{F}_p^\times\), let \(\mathbb{F}_p(\alpha)\) denote \(\mathbb{F}_p\) equipped with the quadratic form \(q(x) = \alpha x^2\). For \(W \in \mathsf{Q}\), write \(W(\alpha) = W \otimes \mathbb{F}_p(\alpha)\).

Now we are ready to compute the local density of \(L''_{P,r}\otimes\mathbb{Z}_p\) (in both tight and loose cases).

6.5.5 The mysterious quantity \(D_0\)↩︎

Define the following quantity \[D_0=d_n+ p^{n-1}c_1+\sum_{i=1}^{n-1} p^{n-i-1}(c_{i+1}-c_{i}).\] Later on, this quantity will appear mysteriously in various computations, but we don’t have a very good geometric explanation of it.

6.6 Estimating \(S'(M)\): the tight case↩︎

In this section we assume \(a_k=a\). We will compare \(S'(M)\) to \(S''(M)\) and explicitly compute \(S''(M)\), and use this to prove Theorem 9 in the tight case. Following Remark 1, we can write \(S'(M)\) as the following series (we will use \(h_a\) instead of \(h_P\), the two quantities are equal; cf. Notation 1): \[S'(M)=\frac{p^{a+1}-1}{h_a}\frac{1}{p(1+p^{-(m+1)})}\sum_{r=1}^\infty\frac{\delta(p,L'_{P,r},M)}{|L'_{P,1}\otimes\mathbb{Z}_p:L'_{P,r}\otimes\mathbb{Z}_p|}.\] We then define \[S''(M):=\frac{p^{a+1}-1}{h_a}\frac{1}{p(1+p^{-(m+1)})}\sum_{r=1}^\infty\frac{\delta(p,L''_{P,r},M)}{[L''_{P,1}\otimes\mathbb{Z}_p:L''_{P,r}\otimes\mathbb{Z}_p]},\] where the auxiliary lattices \(L''_{P,r}\) are as in §6.5.2. In the following we prove a partial generalization of Lemma 16 that compares \(S'(M)\) to \(S''(M)\):

In the rest of this section we conduct a case by case study. Even though \(a_k=a\), we will still keep the notation \(a_k\). This is because some computations hold even when \(a_k>a\), and will be reused in the loose situation. To ease the computation of \(S''(M)\), we decompose it to two parts: \[\label{eq:decompositionofS393940M41} S''(M)=S''_1(M)+ S''_{>1}(M),\tag{46}\] where \(S''_1(M)\) is the sub-sum from \(r=1\) to \(h_{a_k}\), while \(S''_{>1}(M)\) is the sub-sum from \(r=h_{a_k}+1\) to \(\infty\).

6.6.1 Borderline case 2↩︎

The case where \(\dim\overline{\mathcal{L}}_{1,\mathrm{bd}}=2\) is harder than others, since there are potentially more special endomorphisms that resist decay. We will treat this case first. Once this is done, the remaining cases are nothing but easier variants.

First we have \(c_{n-1}=c_n=d_{n}=d_{n-1}\). We then define \[\begin{align} &D_\delta:=c_1\delta_0+\sum_{i=1}^{n-1}(c_{i+1}-c_i){p}^{-i} \delta_{i}+\sum_{i=1}^{n-1} (d_i-d_{i+1})p^{2+i-2n}\delta_{i}'+ (h_{a_k}-d_1)p^{2-2n}\delta_{0}',\\ &D:=c_1+\sum_{i=1}^{n-1}(c_{i+1}-c_i){p}^{-i}+\sum_{i=1}^{n-1} (d_{i}-d_{i+1})p^{2+i-2n}+ (h_{a_k}-d_1)p^{2-2n}. \end{align}\] Here \(\delta_i,\delta_i'\) are from Definition 22. Then it is easy to see from Remark 1 that \[\label{eq:S13939S6213939border2} \begin{align} &S_1''(M)= \frac{p^{a+1}-1}{h_{a}}\frac{ D_\delta}{p(1+p^{-{(m+1)}})}.\\ &S_{>1}''(M)=\frac{p^{a+1}-1}{h_{a}}\frac{\delta_0'D}{p(1+p^{-{(m+1)}})} \sum_{j=1}^\infty\frac{p^{(a_k+1)j}}{p^{2nj}}. \end{align}\tag{47}\]

Now plug lemma 27 into (47 ), and equate \(a_k=a\). We obtain \[\begin{align} \tag{55}S''(M)=S_1''(M)+S_{>1}''(M)&\leq \frac{1-p^{-a-1}}{1+p^{-{(m+1)}}}\left(1+ \frac{1+p^{n-m-1}}{p^{2n-a-1}-1}\right)\\ &\tag{56}=1-\frac{ (p^n - p^{a+1}) (p^{a + m + 1} + p^{a + n} + p^a + p^{m + n})}{p^a(p^{m + 1} + 1) (p^{2 n} - p^{a + 1})}. \end{align}\] In the thick case, (56 ) is strictly smaller than \(1\). By Lemma 26, we get \(S'(M)\leq S''(M)<1\). In the thin case, (56 ) equals \(1\). Therefore we need the following extra input (which is a partial generalization of the second half of Lemma 16):

The following consequence is immediate:

6.6.2 Borderline case 1↩︎

Suppose that \(\dim\overline{\mathcal{L}}_{1,\mathrm{bd}}=1\). We then have \(c_n=d_{n}\). Define \[\begin{align} &D_\delta=c_1\delta_0+\sum_{i=1}^{n-1}(c_{i+1}-c_i){p}^{-i} \delta_{i}+\sum_{i=1}^{n-1} (d_i-d_{i+1})p^{1+i-2n}\delta_{i}'+ (h_{a_k}-d_1)p^{1-2n}\delta_{0}',\\ & D=c_1+\sum_{i=1}^{n-1}(c_{i+1}-c_i)p^{-i}+ \sum_{i=1}^{n-1} (d_i-d_{i+1})p^{1+i-2n}+ (h_{a_k}-d_1)p^{1-2n}. \end{align}\] Then \(S_1''(M)\) and \(S_{>1}''(M)\) can still be computed by the formula (47 ) (with the meaning of \(D_\delta\) and \(D\) replaced by the new definition given above).

Use Lemma 29 and equate \(a_k=a\). We have \[\begin{align} S''(M)&= S_1''(M)+S_{>1}''(M) \stackrel{(n\geq a+1)}{<} \frac{1-p^{-a-1}}{1+p^{-{(m+1)}}}\left(1+ \frac{2}{p^{a+2}-1}\right) <1. \end{align}\]

6.6.3 Borderline case 0↩︎

Define \[\begin{align} & D_\delta=c_1\delta_0+\sum_{i=1}^{n-1}(c_{i+1}-c_i){p}^{-i} \delta_{i}+(d_n-c_n)p^{-n}\delta_n+\sum_{i=1}^{n-1} (d_i-d_{i+1})p^{i-2n}\delta_{i}'+ (h_{a_k}-d_1)p^{-2n}\delta_{0}',\\ & D=c_1+\sum_{i=1}^{n-1}(c_{i+1}-c_i)p^{-i}+ (d_n-c_n)p^{-n}+\sum_{i=1}^{n-1} (d_i-d_{i+1})p^{i-2n}+ (h_{a_k}-d_1)p^{-2n}. \end{align}\] Then \(S_1''(M)\) and \(S_{>1}''(M)\) can still be computed by the formula (47 ) (with the meaning of \(D_\delta\) and \(D\) replaced by the new definition given above).

Use Lemma 30 and equate \(a_k=a\), we find that

\[\begin{align} S''(M)=S_1''(M)+S_{>1}''(M)& \stackrel{(n\geq a+1)}{<}\frac{1-p^{-a-1}}{1+p^{-(m+1)}}\left(1+\frac{2}{p^{a+3}-1}\right)<1. \end{align}\]

6.7 Estimating \(S'(M)\): the loose case↩︎

Suppose that \(a_k>a\). As compared to the tight situation, this is much easier, and we can treat all borderline cases all at once.

We will make the decomposition \(S'(M)=S'_1(M)+ S'_{>1}(M)\) similar to (46 ), where \(S'_1(M)\) is the sub-sum of \(S'(M)\) from \(r=1\) to \(h_{a_k}\), while \(S'_{>1}(M)\) is the sub-sum from \(r=h_{a_k}+1\) to \(\infty\). On the other hand, since \(L''_{P,r}\) is only defined for \(r\leq h_{a_k}\), we can only make sense of \(S''_1(M)\).

Define the quantity \(D_\delta\) exactly the same way as in the tight situation. As noted in Remark 1, we still have the bound \(D_\delta\leq p^{-a_{k}}h_{a_k}\) (in all thin and tight and all borderline cases). Note that \(h_a\geq h_{a_k}\). The same computation as in §6.6 shows that in all cases we have \[S'_1(M)\leq S_1''(M)\leq \frac{p^{a+1}-1}{h_a}\frac{D_\delta}{p(1+p^{-{(m+1)}})} \leq \frac{p^{a-a_k}-p^{-a_k-1}}{1+p^{-{(m+1)}}}\frac{h_{a_k}}{h_a}<\frac{p^{-1}}{1+p^{-{(m+1)}}}.\] On the other hand, it is easy to estimate \(S_{>1}'(M)\): one can just use the coarsest bounds. By Theorem 8, we have a rank \(a+a_k+2\) saturated sublattice of rapid decay. Each primitive vector in this sublattice can at most lift to \(h_{a_k}\). Upon using the coarsest upper bound for the local density (which is \(2\)), we find that \[S_{> 1}'(M)\leq \frac{p^{a+1}-1}{h_a}\frac{2}{p(1+p^{-(m+1)})} \sum_{j=1}^\infty\frac{p^{(a_k+1)j}h_{a_k}}{p^{(a+a_k+2)j}}\leq \frac{2p^{-1}}{1+p^{-(m+1)}}.\] As a result, we have \[S'(M)=S_1'(M)+S_{>1}'(M) < \frac{3p^{-1}}{1+p^{-(m+1)}} \stackrel{(p\geq 3)}{<} 1.\]

7 Decay of special endomorphisms: supersingular but not superspecial↩︎

We closely follow the exposition and notation in [7]. The computations are similar to the ones in loc. cit. and the ones carried out in the superspecial case above, and so we will be content with only providing an outline of the calculations.

We work at a supersingular point \(P\). Set \(n = \frac{t_P}{2}\) and \(m = (b-2n)/2\). To ease exposition, we set \(n' = n-1\). Note that the case \(n' =0\) is precisely the superspecial case, and so we have that \(n' >0\). As recalled in 4.1, we have a decomposition of \(W\)-lattices \(\mathbb{L}= \mathbb{L}_0 \oplus \mathbb{L}_1\) and hence a decomposition of their Tate modules \(\mathcal{L}= \mathcal{L}_0 \oplus \mathcal{L}_1\). Recall that \(\mathcal{L}_1\) is self-dual over \(\mathbb{Z}_p\), so \(\mathrm{rank}_W \mathbb{L}_0 = \mathrm{rank}_{\mathbb{Z}_p} \mathcal{L}_0 = 2n\) and \(\mathrm{rank}_W \mathbb{L}_1 = \mathrm{rank}_{\mathbb{Z}_p} \mathcal{L}_1 = 2m\). In order to keep notation consistent within our paper, we diverge from the notation in [7] and make the following changes.

Note that \(\{e_j, f_j : 1 \le j \le m \}\) is a basis for \(\mathbb{L}_1\) as well, and the basis \(\{v'_i, w'_i : 1 \le i \le n \}\) for \(\mathbb{L}_0\) is provided by [7]. Again let \(R := W[\![x'_i, y'_i, x_j, y_j]\!]_{1 \le i \le n', 1 \le j \le m}\) be the structure ring of \(\mathscr{S}^{/P}_W\). The Frobenius operator \(\mathop{\mathrm{Frob}}\) on \(\mathbb{L}_{\mathrm{cris}}(R) \simeq \mathbb{L}\otimes R\) is given by \((uB) \circ \sigma\) with respect to \(\{ v'_i, w'_i, e_j, f_j \}\), where \(u\) is a matrix with entries in \(R\) and \(B\) is a matrix with entries in \(\frac{1}{p} W \subset w[1/p]\). The general formulas for \(u\) and \(B\) for a supersingular \(P\) that extends 11 in §4.2.1 for the superspecial case is spelled out in [7].

Combining these formulae with the argument of 1, we find that the local equations of the newton strata through \(P\) are defined by vanishing of the \(Q_i\)’s.

We will recall notation for future use. Let \(A\) (resp. \(B\),\(D\)) denote the top left \(2n \times 2n\) (resp. top right \(2n \times 2m\), bottom left \(2m \times 2n\)) block of \(u' - I\) where \(u'\) is as in [7]. As in [7], we let \(S_0\) be the change-of-coordinates matrix from \(\{e'_i,f'_i \}\) to \(\{ v'_i,w'_i\}\), i.e. the first (resp. last) \(n\) columns of \(S_0\) are the coordinates of the \(v'_i\) (resp. \(w'_i\)) expressed in terms of the \(e'_i,f'_i\). As in loc. cit., there is a unique matrix \(S'_0 \in \textrm{GL}_{2n}(W)\) such that\[S_0 = S'_0 \left[ \begin{array}{c|c} p^{-1} I & 0 \\ \hline 0 & I \end{array} \right].\]

7.1 Setup with local curve↩︎

We will assume that \(Q_0(t), \ldots Q_{a-1}(t) = 0\) and that \(Q_a(t)\) is the first non-zero function of \(t\). We refer to [7] for the definition of the matrices \(K_i\). The matrices \(K_i\) are defined for \(i = 1, \ldots n+1\) in loc. cit.. We define \(K_i\) for every \(i\). Indeed, set \(A_{n+i} = CD^{(i)}\). This is the \(2n\times 2n\) matrix which is zero everywhere except for the \((n,2n)\)-entry, and this entry is \(p^{-1}Q_{n'+i}\). Define \(K_{n+i} = S'_0 A_{n+i} ({S'}_0^{-1})^{(i)}\). We note that [7] is true for \(K_i\) with values of \(i\) greater than \(n+1\) as well, and the same proof goes through. The discussion immediately after loc. cit. also holds for arbitrary \(i\). We set \(h_i := v_t(Q_i(t))\).

Note that we have \(h_i = \infty\) for \(i< a\). Let \(a_1 = a < a_2 \ldots < a_k\) be the maximal sequence of integers such that \(a_k \leq n+m-1\), and \(h_{a_1} \geq h_{a_2} \ldots \geq h_{a_k}\). There are three main cases.

1. \(a_1<n', a_k \leq n'\).

2. \(n' \leq a_1\).

3. \(a_1 < n' < a_k\).

Cases 2 and 3 are treated identically to establish rapid decay of a subspace having rank at least \(2a+2\), and Cases 1 and 3 are treated identically to prove Theorem 7. We will therefore only analyze Cases 1 and 2, and only briefly comment on Case 3.

7.2 Rapid decay↩︎

Case 1↩︎

In this case, we will show that a sufficiently large-rank sublattice of \(\mathcal{L}_0\) decays, and this will be sufficient for our purposes.

7.2.1 Proof of rapid decay in Case 1↩︎

The treatment of rapid decay for Case 1 is almost identical to the analysis carried out in [7]. Indeed, it suffices to solely work with \(K_i\) with \(a_{1}+1 \leq i \leq a_{k}+1\). The results in loc. cit. through Lemma 5.12 are just general and therefore hold. Lemma 5.13 of loc. cit. has the following strengthening.

The proof is the same.

Now, the proof of the decay lemma goes through exactly as in Theorem 5.2 and Section 5.14 of [7]. We note that 5.14 requires \(a_k \leq n'\) – indeed, without this assumption, one must a-priori work with \(K_i\) for all values of \(i\) bounded above by \(n'+m-1\) (and this is precisely what we do in Cases 2 and 3). Therefore, [7] is only true in the setting of this case. However, the proof of [7] together with Lemma 32 in place of [7] actually shows that a saturated sub-lattice of \(\mathcal{L}_0\) having rank at least \(2n - (a_k-a_1)\) decays rapidly. We have that \(2n - (a_k - a_1) \geq 2a_1+2\) (with equality holding only when \(a_k = a_1 = n'\)).

Cases 2 and 3↩︎

We will work in the setting of Case 2. We again note that the argument for rapid decay that we outline holds verbatim for Case 3.

The setup with words, valuations, etc is the same as in the superspecial case. Specifically, the definitions, lemmas, propositions, and proofs from Section 5.3 all hold with no change.

The results regarding isotropic vectors in \(\mathcal{L}_1\) also hold with little change in the statements and no change in the proofs. Indeed, we have the following results.

The proof is identical to the proof of Lemma 14.

The proof is idential to the proof of Corollary 5

We are now ready to outline the proof of the decay lemma in this setting.

7.3 Local-global comparison↩︎

7.3.1 Cases 1 and 3↩︎

The analysis in Cases 1 and 3 to compare the local and global Eisenstein series is identical. Indeed, there is no first-order decay analysis required. We are now ready to prove Theorem 7, which we restate for the reader’s convenience.

7.3.2 Case 2↩︎

The first order analysis in the superspecial case goes through verbatim in this setting. Indeed, the case that does not follow immediately from the arguments used in Cases 1 and 3 is when \(a_k = a_1 \geq n'\). In this setting, the sublattice \(\mathcal{L}_0\) decays rapidly. The analysis of first-order decay for vectors in \(\mathcal{L}_1\) is identical to the superspecial case. The analysis in Section 6 yields the following result.

8 The algebraization theorem↩︎

The main result of this section is to finish the proof of Theorem 7. We recall the setting. We have a supersingular point \(P\in C\), with local parameter \(t\). Let \(k[\![t]\!]\) denote the complete local ring of \(C\) at \(P\). We have shown the existence of \(\alpha < 1\) such that the conclusion of Theorem 7 holds unless the module of formal special endomorphisms at \(k[\![t]\!]\) has rank \(b-2a\). Recall that the Decay Lemma (Theorem 8, Theorem 10) implies that the module of formal special endomorphisms at \(k[\![t]\!]\) has rank at most \(b-2a\). We will now prove that under the assumptions of Theorem 3, the module of formal special endomorphisms at \(k[\![t]\!]\) has rank strictly less than \(b-2a\).

We now introduce our setup before stating our result.

Since \(\mathbb{L}_{\mathrm{cris},U_0}\) has constant Newton polygon, \(\mathbb{L}_{\mathrm{cris},U_0}\) admits a slope filtration in the category of \(F\)-crystals over \(U_0\): \[\label{eq:slopefiltration} 0=\mathop{\mathrm{Fil}}_{-2}\subseteq \mathop{\mathrm{Fil}}_{-1}\subseteq \mathop{\mathrm{Fil}}_{0}\subseteq \mathop{\mathrm{Fil}}_{1}=\mathbb{L}_{\mathrm{cris},U_0},\tag{57}\] where \(\mathop{\mathrm{Fil}}_{-1}\) is the slope \(-1/(a+1)\) part, \(\mathop{\mathrm{Fil}}_{0}\) is the slope \(\leq 0\) part. The rank of \(\mathrm{gr}_{-1},\mathrm{gr}_0\) and \(\mathrm{gr}_{1}\) are \(a+1\), \(b-2a\) and \(a+1\), respectively.

Formal Brauer groups↩︎

Note that the \(F\)-crystal \(\mathbb{L}_{\textrm{EB}} : = \mathop{\mathrm{Fil}}_{0}/\mathop{\mathrm{Fil}}_{-1}\) is a Dieudonné crystal, the main result of [42] implies that \(\mathbb{L}_{\textrm{EB}}=\mathbb{D}_{}(\mathbf{\Psi})\) where \(\mathbb{D}\) is the contravariant crystalline Dieudonné functor and \(\mathbf{\Psi}\) is a \(p\)-divisible group over \(U_0\), called the enlarged formal Brauer group. For a K3 surface over finite field, this recovers the enlarged formal Brauer group introduced in [11] and [12], and the connected subgroup of this group is just the classical formal Brauer group associated to the K3 surface.

Given the results of Sections 5, 6 and 7, Theorem 7 (and therefore Theorem 3) follows directly from Theorem 13.

8.1 Splittings↩︎

Let \(X\) be a \(U_0\)-scheme, we say that \(\mathbb{L}_{\mathrm{cris},X}\) splits in the middle, if \[\label{eq:splittingK31} \mathbb{L}_{\mathrm{cris},X}\simeq \mathrm{gr}_{0,X}\oplus \mathscr{E}_X,\tag{58}\] where \(\mathscr{E}_X\) is an extension of \(\mathop{\mathrm{Fil}}_{-1,X}\) by \(\mathrm{gr}_{1,X}\). Rationally, we also say that \(\mathbb{L}_{\mathrm{cris},X}[p^{-1}]\) splits in the middle, if (58 ) holds after up to inverting \(p\).

Note that the \(p\)-divisible group \(\mathbf{\Psi}\) sits inside the classical connected-étale sequence \[\label{ecs} 1\rightarrow \mathbf{\Psi}_{\text{con}}\rightarrow \mathbf{\Psi}\rightarrow \mathbf{\Psi}_{\text{ét}} \rightarrow 1.\tag{59}\] Applying \(\mathbb{D}(-)\) to (59 ), we recover \[\label{ecs2} 0\rightarrow \mathrm{gr}_0 \rightarrow \mathbb{L}_{\textrm{EB}}\rightarrow \mathrm{gr}_{1} \rightarrow 0.\tag{60}\]

9 Index of notations↩︎

Ruofan Jiang     University of California, Berkeley, Department of Mathematics, Evans Hall, Berkeley, CA 94720-3840, USA     Email: ruofanjiang@berkeley.edu
Ananth Shankar     Northwestern University, Department of Mathematics, Lunt Hall, Evanston, Il 60208, USA     Email: ananth@northwestern.edu
Ziquan Yang     The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.     Email: zqyang@cuhk.edu.hk

References↩︎

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1. i.e., the group \(G(\mathbb{L}_\mathrm{cris}[p^{-1}]|_U, \bar{\eta})\) if we use the notation of [23]. Here \(U\) is the open Newton stratum of \(C\), and \(\mathbb{L}_\mathrm{cris}\) is the natural \(F\)-crystal on \(\mathscr{S}\) recalled in 2 below.↩︎

2. Technically, in [9] and [31], \(\mathcal{A}\) is only defined as a sheaf of abelian schemes up to prime-to-\(p\) quasi-isogeny. However, for \(\mathbb{K}^p\) sufficiently small, we can take \(\mathcal{A}\) to be an actual abelian scheme (cf. [9]).↩︎

3. Technically, the lemma below uses that each \(\mathcal{F}_h\) is a local complete intersection. One easily checks that this holds for all cases in [prop:32Shen-Zhang], though later we only make use of case (2).↩︎

4. We again remark that the letter \(m\) is used in relation to the rank of \(\mathbb{L}\), unlike in Section 3. We will alert the reader if we resume using \(m\) in the context of special divisors.↩︎

5. We note that any isometry applied to the set \(\{x_1\ldots x_r, y_1\ldots y_r\}\) must necessarily produce lines \(\succeq\) the original lines, as we have the conditions \(v_t(x_i) = v_t(y_i) = d\).↩︎

6. This proof was found with the help of Gemini 3.0.↩︎