Introduction↩︎

Given a prime number \(\ell\) and a number field \(k\), Manin showed in [1] that there exists an integer \(r=r(\ell,k)\) such that for any non-CM elliptic curve \(E\) over \(k\), \(E[\ell^r]\simeq (\mathbf{Z}/\ell^r\mathbf{Z})^2\) does not contain a \(k\)-rational line, or equivalently that the image of the reduction modulo \(\ell^r\) of its \(\ell\)-adic Galois representation \[\mathop{\mathrm{Gal}}_k=\mathop{\mathrm{Gal}}(\bar{k}/k)\longrightarrow \mathop{\mathrm{Aut}}_{\mathbf{Z}/\ell^r\mathbf{Z}}\left(E[\ell^r]\right)\simeq \mathop{\mathrm{GL}}(2,\mathbf{Z}/\ell^r\mathbf{Z})\] is not contained in a Borel subgroup. Manin’s original proof can be greatly simplified using Faltings’ proof of Mordell’s conjecture, which came later. In a series of papers Cadoret and Tamagawa established a definitive result regarding the uniform boundedness of the \(\ell\)-primary torsion for \(1\)-dimensional families of abelian varieties. In this paper we prove an analogous statement for certain \(2\)-dimensional families of abelian \(3\)-folds which we believe to be the first result over a genuine two-dimensional base.

Henceforth we fix an imaginary quadratic field \(M\) of odd fundamental discriminant \(-D\ne -3\) and denote by \(\mathcal{O}_M\) its ring of integers. An abelian \(3\)-fold of Picard type over a field \(k\) containing \(M\) will always stand for a principally polarized abelian variety over \(k\) of dimension \(3\) having multiplication by \(\mathcal{O}_M\) defined over \(k\). Its \(\ell\)-adic Tate module \(T_\ell A\) is free of rank \(3\) over \(\mathcal{O}:=\mathbf{Z}_\ell\otimes\mathcal{O}_M\) endowed with a continuous \(\mathcal{O}\)-linear action of \(\mathop{\mathrm{Gal}}_k\). By a line (resp. plane) in \(T_\ell A\), we would mean a \(\mathcal{O}\)-submodule of rank \(1\) (resp. \(2\)) which is a direct factor. More generally, given a positive integer \(r\), a line (resp. plane) in \(A[\ell^r]\) will always be assumed be the image, under the natural reduction map, of a line (resp. plane) in \(T_\ell A\). Finally, by a full flag we would mean a tuple of a line sitting as direct factor in a plane. Lines (resp. planes) will be called \(k\)-rational if they are stable by \(\mathop{\mathrm{Gal}}_k\) (but not necessarily point-wise fixed).

Our first main result addresses the semi-stable case.

As in the case of elliptic curves, the conclusion of Theorem 1 asks the image of the attached Galois representation \[\mathop{\mathrm{Gal}}_k\longrightarrow \mathop{\mathrm{Aut}}_{\mathcal{O}/\ell^r\mathcal{O}}\left(A[\ell^r]\right)\simeq \mathrm{GU}(3,\mathbf{Z}/\ell^r\mathbf{Z})\] not to be contained in a Borel subgroup. Also, as in the case of elliptic curves, it is necessary to cast aside the CM abelian varieties, as their \(\ell\)-adic representations are potentially reducible.

We next show how one can relax the semi-stability assumption by adding a tiny bit of level structure at \(D\). Given a prime \(v\) of \(M\) above some \(p\mid D\), the projective \(\mathop{\mathrm{Gal}}_k\)-action on the \(\mathbf{F}_p\)-vector space \(A[v]\) yields a homomorphism \(\widetilde{\rho}_{A,p}: \mathop{\mathrm{Gal}}_k \to \mathop{\mathrm{PGL}}(2,\mathbf{F}_p)\) (see 15 ). Taking quotient by the unique index two subgroup \(\mathop{\mathrm{PSL}}(2,\mathbf{F}_p)\) of \(\mathop{\mathrm{PGL}}(2,\mathbf{F}_p)\) yields a canonical homomorphism \(\varepsilon_{A,p}: \mathop{\mathrm{Gal}}_k \to\{ \pm 1 \}\) and we let \(\varepsilon_{A,D}=\prod_{p \mid D} \varepsilon_{A,p}: \mathop{\mathrm{Gal}}_k \to\{ \pm 1 \}\).

Theorem 2 is the main result of this paper and implies Theorem 1 as follows. Let \(k'\) be the compositum of the (finitely many) quadratic extensions of \(k\) which are unramified outside \(S\) and the primes dividing \(D\). Given any abelian \(3\)-fold \(A\) as in Theorem 1, we claim that \(\varepsilon_{A,D}(\mathop{\mathrm{Gal}}_{k'})\) is trivial. Indeed, by a theorem of Grothendieck [2] the semi-stability of \(A\) at \(v\notin S\), \(v\nmid D\) implies that the inertia subgroup of \(\mathop{\mathrm{Gal}}_{k}\) at \(v\) acts unipotently on the \(D\)-adic Tate module of \(A\), in particular its image by \(\varepsilon_{A,D}\) is pro-\(D\) hence trivial (as \(D\) is odd). Therefore the base change of \(A\) to \(k'\) satisfies the additional assumption in Theorem 2, implying that Theorem 1 holds with \(r(\ell,k')\) from Theorem 2.

For an individual abelian variety \(A\), the conclusion of Theorem 2 is a consequence of the Mumford–Tate conjecture which is known for abelian \(3\)-folds (see §2.3), so the important feature of the result is its uniformity. As abelian \(3\)-folds of Picard type are parametrized by Shimura surfaces of Picard type, a natural way to proceed would be to show that the \(k\)-rational points are not Zariski dense in any of their connected components \(Y_\Gamma\). Let us for the moment consider the simpler situation from our earlier paper [3] where the congruence subgroups \(\Gamma\) were neat. Our method there had two principal steps. The first step involved showing the existence of three linearly independent global holomorphic \(1\)-forms on the toroidal compactification \(X_\Gamma\). By a theorem of Faltings concerning the associated Albanese variety this implies that the \(k\)-rational points on \(X_\Gamma\) are contained in a divisor \(Z\), as predicted by a conjecture of Bombieri and Lang as \(X_\Gamma\) turns out to be of general type. The second step consisted in applying a result of Nadel requiring \(\Gamma\) to be neat and the canonical divisor to be big (in his sense) to deduce that any curve \(C\) of genus \(\leqslant 1\) contained in \(X_\Gamma\) is in fact contained in the complement of \(Y_\Gamma\). Consequently, every curve in \(Z\) meeting the open surface \(Y_\Gamma\) must be of genus \(\geqslant 2\) thus, by Faltings’ proof of Mordell’s conjecture for curves, \(Y_\Gamma(k)\) is finite for any number field \(k\).

Let us now say a few words about the techniques involved in the proof of Theorem 2. As we are led to consider congruence subgroups of Iwahori type \(\Gamma_0(\ell^r)\), which are never neat as they have torsion, both steps mentioned above encounter difficulty and we have to resort to new methods. We produce irregularity by constructing explicit endoscopic automorphic forms in certain non-generic representations \(\pi\) on the unitary group in \(3\) variables. It is here that the index \(2\) projective Galois image condition at \(D\), suggested to us by Gross, is essential, as otherwise all of the Picard modular surfaces involved would have trivial Albanese and our approach would not apply for global reasons. Making this strategy actually work yet requires to address some delicate representation theoretic questions to which a significant part of the paper is devoted and on which we will elaborate now.

By Rogawski’s theory \(\pi\) is an element of an endoscopic Arthur packet parametrized by an anti-cyclotomic (more precisely, conjugate-symplectic) Hecke character \(\lambda\) of \(M\). Theorem 2 imposes conditions implying that \(\lambda\) must differ from Gross’ minimally ramified ‘canonical’ characters by a finite order character of \(M^1\) only ramified at \(\ell\). The local Arthur packet at \(\ell\) contains two representations, a supercuspidal \(\pi_{c,\ell}\) and a non-tempered \(\pi_{n,\ell}\), both non-generic. The difficulty of finding \(\Gamma_0(\ell^r)\)-invariants in \(\pi_{c,\ell}\) forces us to work with the global \(\pi_{n}\), which is automorphic if, and only if, the global root number \(W(\lambda^3)\) equals \(+1\). For \(D\equiv 3\pmod{8}\) Gross’ canonical characters work and a computation of matrix coefficients performed in §1.4 shows that the resulting \(\pi_{n,\ell}\) has invariants even by the hyperspecial maximal compact subgroup.

When \(D\equiv 7 \pmod{8}\) the canonical characters yield the wrong sign, leading us to consider \(\lambda\)’s which are tamely ramified at \(\ell\) to switch the sign. It remains however to show that the non-tempered representations \(\pi_{n,\ell}\) attached to such \(\lambda\)’s admit \(\Gamma_0(\ell^r)\)-invariants for some \(r\), which is significantly harder than \(\Gamma_1(\ell^r)\)-invariants. To that end, in §1.5 we devise a more involved argument relying on Jacquet modules and intertwining operators, and which requires precise computations of exponential sums beyond the reach of known estimates. It would be worthwhile exploring questions regarding levels of non-tempered representations in greater generality.

Once the irregularity of \(X_\Gamma\) has been shown to be at least \(3\) and the Bombieri–Lang conjecture established, one has to deal with the possible curves \(C\) of genus \(\leqslant 1\) contained in \(Y_\Gamma\). Using the key Lemma 6 on the lack of complex reflections, our Picard modular surfaces only admit a finite number of isolated singularities hence, after removing a finite number of points, \(C\) is endowed with an abelian family (see §2.2). This allows us apply the results of Cadoret and Tamagawa regarding the uniform boundedness of the Galois action on the Tate module of such \(1\)-dimensional families. Finally, each of the finitely many non-CM \(k\)-rational points are dealt with using the Mumford–Tate conjecture for abelian \(3\)-folds of Picard type recalled in §2.3, and which is not available in higher dimension.

As our Picard modular surfaces \(X_\Gamma\) have irregularity \(\geqslant 3\), the Enriques–Kodaira classification implies that they are either ruled of genus \(q\), or elliptic, or else they are of general type. In the last case, which according to Holzapfel [4] occurs for all odd \(D\notin\{3, 7,11,19, 23, 31, 39, 47, 71\}\), we show that the Bombieri–Lang Conjecture holds, i.e., that the \(k\)-rational points are not Zariski dense. Investigating small values of \(D\), as suggested by Mazur, seems even more interesting. It is established in loc. cit. that for all \(D\ne 71\) in the above list the level \(1\) Picard modular surfaces are rational and it would be natural to investigate the nature of their degree \(2\) Gross covers that we consider. A way to shed light on this question would be to find an explicit \(2\)-parameter family of abelian \(3\)-folds of Picard type to which our theorem applies.  

It might be worthwhile remarking that we could have also considered the simpler case of the moduli of principally polarized abelian surfaces \(A\) over \(k\) with multiplication by \(\mathcal{O}_M\), which will involve \({\rm U}(1,1)\). However, as \({\rm SU}(1,1)\simeq \mathop{\mathrm{SL}}(2)\) this essentially reduces to the modular curve case. On the other hand, if we consider principally polarized abelian surfaces \(A\) with real multiplication, then the family is parametrized by a Hilbert modular surface which has trivial \({\rm H}^1\), thus our methods, which rely on the Albanese variety, do not lead to an establishment the Bombieri–Lang Conjecture.

1 Levels for endoscopic non-tempered representations of \(\mathop{\mathrm{U}}(3)\)↩︎

It goes back to the work of Casselman that admissible irreducible representations having non-zero Iwahori invariants are exactly those occurring as sub-quotients in parabolic inductions of unramified characters. Whereas the dimension of the invariants by the depth \(r\) Iwahori subgroup in the full induced representation grows as \(r\) goes to infinity, this might not always be the case for all its sub-quotients, as shown by the example of the trivial representation of \(\mathop{\mathrm{GL}}(2)\), realized as a quotient of a unramified principal series representation.

Another challenging question is to determine which sub-quotient of a parabolically induced unramified character picks up the invariants by a given maximal open compact subgroup. Whereas MacDonald’s formula for zonal spherical functions yields an answer in the case of a maximal hyperspecial subgroup, the general case appears to be an open question.

In this section we fully answer those two natural questions in the case of certain non-tempered endoscopic representations of \(\mathop{\mathrm{U}}(3)\) attached to a quadratic extension \(E/\mathbf{Q}_p\). It will be later applied in a global setting to \(E=M_p\), where \(M\) is an imaginary quadratic field in which the prime \(p\) does not split.

In this section of our paper we will adopt local notations.

Let \(E\) be a quadratic field extension of \(\mathbf{Q}_p\), \(\mathcal{O}\) be its ring of integers, \(\mathcal{P}\) its maximal ideal and \(\varpi\) a uniformizer. We assume that \(E\) is not a ramified extension of \(\mathbf{Q}_2\) and we fix a generator \(\xi\) of its different ideal such that \(\bar\xi=-\xi\). Denote by \(x\mapsto \bar x\) the non-trivial element of \(\mathop{\mathrm{Gal}}(E/\mathbf{Q}_p)\). We fix an additive character \(\psi: \mathbf{Q}_p\to \mathbf{C}^\times\) of conductor \(0\), i.e. \(\ker(\psi)=\mathbf{Z}_p\), and we consider the additive character \(\psi^{}_E\) of \(E\) defined as \(\psi^{}_E(z)=\psi(\mathop{\mathrm{Tr}}_{E/\mathbf{Q}_p}(z))\).

Let \(G\) be the unique quasi-split unitary group in \(3\) variables relative to the extension \(E/\mathbf{Q}_p\). It can be realized as the automorphisms of \(E^3\) preserving the hermitian pairing \[\langle x,y\rangle = \bar x_1 y_3 + \bar x_2 y_2 +\bar x_3 y_1.\]

The standard Borel \(B\) of \(G\) is a product of its torus \[T=\left\{ \left(\begin{smallmatrix}\bar{\alpha} & & \\ &\beta & \\ & & \alpha^{-1}\end{smallmatrix} \right)\Big{|} \alpha\in E^\times, \beta\in E^1 \right\}\] with its unipotent subgroup \[N=\left\{ [z,x]= \left(\begin{smallmatrix}1 & -\bar{z}& \xi x -z\bar z /2 \\ &1 & z\\ & & 1\end{smallmatrix} \right)\Big{|} z\in E, x\in \mathbf{Q}_p \right\}.\]

1.1 The Bruhat-Tits tree of \(\mathop{\mathrm{U}}(3)\)↩︎

As \(G\) has rank \(1\), its Bruhat-Tits building is a tree. We will first describe its standard apartment. The relative roots of \(G\) are obtained by decomposing the adjoint action on the Lie algebra of the maximal \(\mathbf{Q}_p\)-split torus \(T_0=\left\{\mathrm{diag}(a,1,a^{-1}) | a\in \mathbf{Q}_p^\times \right\}\) of \(G\). The positive elements of the associated root system \(\Phi\) are \(\{\zeta, 2\zeta\}\). Let \(h: \mathbf{G}_m \to T_0 \subset G\) be the generator of the co-character lattice \(X_\ast(T_0) \simeq \mathbf{Z}\) such that \(\langle \zeta, h\rangle=1\). Then the co-root sub-lattice is generated by \(\zeta^\vee = 2h\), so that we have the standard normalization \(\langle\zeta, \zeta^\vee\rangle=2\). According to [5] the affine roots are \(\{ \pm \zeta+\mathbf{Z}\}\cup \{ \pm 2\zeta+\mathbf{Z}\}\) if \(E\) is unramified, and \(\{ \pm \zeta+\tfrac{1}{2}\mathbf{Z}\}\cup \{ \pm 2\zeta+\mathbf{Z}+\tfrac{1}{2}\}\) if \(E\) is ramified; note that \(\delta=0\) in loc. cit. as \(E\) is not a ramified extension of \(\mathbf{Q}_2\). The apartment associated to \(T_0\) is \(\mathbf{R}h\) and its walls are the vanishing sets of these (affine) roots, hence they are given by \(\tfrac{1}{2}\mathbf{Z}h= \mathbf{Z}h \cup \tfrac{1}{2}\mathbf{Z}h\), resp. \(\tfrac{1}{4}\mathbf{Z}h= \left(\tfrac{1}{2}\mathbf{Z}+\tfrac{1}{4}\right)h\cup \tfrac{1}{2}\mathbf{Z}h\), if \(E\) is unramified, resp. ramified. A conjugacy class of maximal compact subgroups can be represented by a wall in the standard apartment. By definition, a wall is hyperspecial if for every \(\zeta'\in \Phi\) there exists an affine root with gradient \(\zeta'\) vanishing on that wall. Since \(\left(\tfrac{1}{2}\mathbf{Z}+\tfrac{1}{4}\right)h\cap \tfrac{1}{2}\mathbf{Z}h=\varnothing\) this only can happen when \(E\) is unramified, in which case the hyperspecial walls are \(\mathbf{Z}h \cap \tfrac{1}{2}\mathbf{Z}h= \mathbf{Z}h\). All walls are special, as elements of \(\Phi\) are rational multiple of one another.

Let us describe the conjugacy classes of maximal compact subgroups in \(G\) in terms equivalent classes of \(\mathcal{O}\)-lattices \(\mathcal{L}\) in \(E^3\) modulo homothety. Recall the definition of the dual lattice \[\mathcal{L}^\perp=\mathop{\mathrm{Hom}}_{\mathcal{O}}(\mathcal{L}, \mathcal{O})= \{x\in E^3 | \langle x, \mathcal{L}\rangle \subset \mathcal{O}\}.\] There are two conjugacy classes of maximal compact subgroups in \(G\), those which are stabilizers of self-dual lattices, and those which are stabilizers of almost-self-dual lattices (i.e. \(\mathcal{L}\) such that \(\mathcal{L}\subsetneq \mathcal{L}^\perp \subsetneq\varpi^{-1}\mathcal{L}\)). We next give an explicit description of the maximal compact subgroups corresponding to the walls of a chamber in the standard apartment.

The standard maximal compact subgroup \(K^\circ=G(\mathcal{O})\) of \(G\) is defined as the stabilizer of the self-dual lattice \(\mathcal{L}^\circ=\mathcal{O}^3\). It is hyperspecial if and only if \(E\) is unramified. The reductive quotient \(\overline{G}{}^\circ\) is given by \(\mathop{\mathrm{U}}(3,\mathbf{F}_p)\) if \(E\) is unramified, and by \(\mathrm{O}(3,\mathbf{F}_p)\) if \(E\) is ramified.

The other standard maximal compact subgroup \(K'\) of \(G\), defined as the stabilizer of the almost self-dual lattice \(\mathcal{L}'=\mathcal{O}\oplus \mathcal{O}\oplus \mathcal{P}\), is given by \[K'=\left(\begin{smallmatrix} \mathcal{O}& \mathcal{O}& \mathcal{P}^{-1}\\ \mathcal{P}& \mathcal{O}^\times & \mathcal{O}\\ \mathcal{P}& \mathcal{P}& \mathcal{O}\end{smallmatrix} \right)\cap G.\] One has \(\mathcal{L}'^\perp=\mathcal{P}^{-1} \oplus \mathcal{O}\oplus \mathcal{O}\) and \(K'\) acts on \(\mathcal{L}'/\varpi\mathcal{L}'^\perp\simeq \mathcal{O}/\mathcal{P}\) via its middle coefficient. The reductive quotient \(\overline{G}{}'\) is isomorphic to \((\mathop{\mathrm{U}}(1,1)\times \mathop{\mathrm{U}}(1))(\mathbf{F}_p)\) if \(E\) is unramified, and to \(\pm \mathbf{1}_2 \cdot \mathop{\mathrm{SL}}(2,\mathbf{F}_p)\times\{\pm 1\}\), if \(E\) is ramified.

The standard Iwahori subgroup of \(G\), defined as \(I=K^\circ\cap K'=\left(\begin{smallmatrix} \mathcal{O}^\times & \mathcal{O}& \mathcal{O}\\ \mathcal{P}& \mathcal{O}^\times &\mathcal{O}\\ \mathcal{P}&\mathcal{P}& \mathcal{O}^\times \end{smallmatrix} \right)\cap G\), is the stabilizer of a chamber in the standard apartment in the Bruhat-Tits tree of \(G\): \[\xymatrix@C=40pt{ \ar@{--}[r] & \gamma^{-1} K' \gamma \ar@{-}[r] & K^\circ \ar@{-}_{I}[r] \ar@/^1pc/@{.}[rr]^{I_{2,1}}& K' \ar@{-}[r] & \gamma K^\circ \gamma^{-1} \ar@{--}[r] &}\] where \(\gamma=\left(\begin{smallmatrix}\varpi^{-1} & & \\ & 1 & \\ & & \varpi \end{smallmatrix}\right)\) and \(I_{2,1}=K^\circ \cap \gamma K^\circ \gamma^{-1}=\left(\begin{smallmatrix} \mathcal{O}^\times & \mathcal{O}& \mathcal{O}\\ \mathcal{P}& \mathcal{O}^\times &\mathcal{O}\\ \mathcal{P}^2 & \mathcal{P}& \mathcal{O}^\times \end{smallmatrix} \right)\cap G\).

One has \(G\supset K'\supset I \supset I_{2,1}\supset I_2\), where \(I_r=\left(\begin{smallmatrix}\mathcal{O}^\times &\mathcal{O}&\mathcal{O}\\\mathcal{P}^r &\mathcal{O}^\times &\mathcal{O}\\ \mathcal{P}^r & \mathcal{P}^r &\mathcal{O}^\times\end{smallmatrix} \right)\cap G\). Finally, we let \(K_1\subset K^\circ\) be the principal congruence subgroup of matrices \(\equiv \mathbf{1}_3\pmod{\mathcal{P}}\) and \(K_T=\left(\begin{smallmatrix}\mathcal{O}^\times &\mathcal{P}&\mathcal{P}\\\mathcal{P}&\mathcal{O}^\times &\mathcal{P}\\ \mathcal{P}& \mathcal{P}&\mathcal{O}^\times \end{smallmatrix} \right)\cap G\).

1.2 Review of \(L\)-parameters and \(A\)-packets↩︎

For any integer \(n\geqslant 1\) there are exactly two (up to isomorphism) \(n\)-dimensional hermitian spaces over \(E\), depending on the image of the discriminant in \(\mathbf{Q}_p^\times/\mathrm{N}_{E/\mathbf{Q}_p}(E^\times)\), and the corresponding unitary groups \(\mathop{\mathrm{U}}(n)\) are isomorphic if and only if \(n\) is odd. When \(n=2\), by analogy with the Archimedean case, we will denote by \(\mathop{\mathrm{U}}(1,1)\) the quasi-split form and by \(\mathop{\mathrm{U}}(2)\) the compact one.

The \(L\)-group of \(\mathop{\mathrm{U}}(n)\) is given by \(\mathop{\mathrm{GL}}(n,\mathbf{C})\rtimes W_{\mathbf{Q}_p}\) with the Weil group \(W_{\mathbf{Q}_p}\) acting on \(\mathop{\mathrm{GL}}(n,\mathbf{C})\) through its quotient \(\mathop{\mathrm{Gal}}(E/\mathbf{Q}_p)\) whose non-trivial element sends \(g\) to \(w_n{}^{t}g^{-1} w_n^{-1}\), where \(w_n\) denotes the anti-diagonal matrix \((1,-1,1,\dots,(-1)^{n-1})\). By definition, an \(L\)-parameter for the quasi-split \(\mathop{\mathrm{U}}(n)\) is a homomorphism \(W_{\mathbf{Q}_p}\times \mathop{\mathrm{SL}}(2,\mathbf{C}) \longrightarrow \mathop{\mathrm{GL}}(n,\mathbf{C})\rtimes W_{\mathbf{Q}_p}\), but as one knows (see [6]) it is equivalent to ask for its restriction \[\phi: W_E\times \mathop{\mathrm{SL}}(2,\mathbf{C}) \longrightarrow \mathop{\mathrm{GL}}(n,\mathbf{C}),\] to be conjugate-orthogonal if \(n\) is odd and conjugate-symplectic if \(n\) is even. Recall that \(\phi\) is conjugate-self-dual if \(\overline{\phi}\simeq \phi^\vee\), or equivalently, if the induced representation \(\mathop{\mathrm{Ind}}_{W_E}^{W_{\mathbf{Q}_p}}(\phi)\) is self-dual. Furthermore, \(\phi\) is conjugate-orthogonal, resp. conjugate-symplectic, if it preserves a non-degenerate symmetric, resp. skew-symmetric, bilinear form. Note that while Schur’s Lemma implies that any irreducible self-dual (or conjugate-self-dual) parameter has a well defined sign, this need not be always the case for reducible parameters.

For \(n=1\), a character of \(E^\times\) is conjugate-orthogonal (resp. conjugate-symplectic) if its restriction to \(\mathbf{Q}_p^\times\) is trivial (resp. is the quadratic character attached to \(E/\mathbf{Q}_p\)). For \(n\in \mathbf{Z}_{\geqslant 0}\), the \(n\)-th symmetric power of the standard \(2\)-dimensional representation \(\mathop{\mathrm{St}}\) of SL\((2,\mathbf{C})\), with \(W_E\) acting trivially, is conjugate-symplectic if \(n\) is odd and conjugate-orthogonal if \(n\) is even.

The base change \(\nu^{}_E(z)=\nu(z/\overline{z})\) from \(\mathop{\mathrm{U}}(1)\) to \(\mathop{\mathrm{GL}}(1)/E\) of a character \(\nu\) of \(E^1\) is conjugate-orthogonal and conversely any conjugate-orthogonal character of \(E^\times\) is obtained in that way. For \(\lambda\) a conjugate-symplectic character of \(E^\times\), the conjugate-orthogonal representation \[(\lambda \otimes \mathop{\mathrm{St}}) \oplus \nu^{}_E : W_E\times \mathop{\mathrm{SL}}(2,\mathbf{C}) \longrightarrow \mathop{\mathrm{GL}}(3,\mathbf{C})\] would be of key relevance to us. It yields an \(L\)-parameter \(\phi_{\lambda, \nu}\) of \(G\), coming from an \(L\)-parameter of the (unique) cuspidal endoscopic subgroup \(H= \mathop{\mathrm{U}}(1,1) \times \mathop{\mathrm{U}}(1)\) of \(G\). The cardinality of the corresponding \(L\)-packet \(\Pi_L(\phi_{\lambda, \nu})\) is given by the order of the centralizer of \(\phi_{\lambda, \nu}\) (modulo center) which turns out to be \(2\). More precisely, \(\Pi_L(\phi_{\lambda, \nu})\) contains two discrete series representations \(\pi_2\) and \(\pi_c\) of \(\mathop{\mathrm{U}}(3)\), exactly one of them, namely \(\pi_c\), being supercuspidal (see [7] where this \(L\)-packet is denoted \(\Pi_L(\mathop{\mathrm{St}}_H(\xi))\)). There is another endoscopic \(L\)-packet for \(G\) consisting of a single non-tempered representation \(\pi_n\) whose the \(L\)-parameter is given by \[\lambda |\cdot |_E^{1/2}\oplus \lambda |\cdot|_E^{-1/2}\oplus \nu^{}_E: W_E\times \mathop{\mathrm{SL}}(2,\mathbf{C}) \longrightarrow \mathop{\mathrm{GL}}(3,\mathbf{C}).\]

Rogawski’s theory [7], [8] describes the automorphic representations contributing to the \(\mathop{\mathrm{H}}^1\) of Shimura surfaces of Picard type in terms global Arthur packets (see [3] for a summary). The corresponding local Arthur packet at \(p\) has \(2\) elements \(\Pi(\lambda,\nu)=\{\pi_n,\pi_c\}\) (see [7], where \(\pi_c\) is denoted \(\pi^s\)), and the restriction to \(W_E\) of its \(A\)-parameter is given by \[(\lambda\otimes \mathbf{1} \otimes \mathop{\mathrm{St}}) \oplus \nu^{}_E : W_E\times \mathop{\mathrm{SL}}(2,\mathbf{C})\times \mathop{\mathrm{SL}}(2,\mathbf{C}) \longrightarrow \mathop{\mathrm{GL}}(3,\mathbf{C}),\] while the \(A\)-parameter of \(\pi_2\) is given by \((\lambda \otimes \mathop{\mathrm{St}}\otimes \mathbf{1}) \oplus \nu^{}_E\).

Crucial for us would be the description \(\pi_n\) and \(\pi_2\) as the Jordan–Hölder constituents of a principal series representation \(\pi\). Indeed, by [8], \(\pi_n\) is the Langlands quotient of the (unitarily normalized) parabolic induction of the character \[\label{eq:mu} \mu(\bar\alpha, \beta, \alpha^{-1})= \lambda(\bar\alpha)\nu(\beta)|\alpha|_E^{1/2},\tag{1}\] with \(\pi_2\) the unique non-zero irreducible sub-representation. The sub and quotient are switched when \(\mu\) is replaced by \(\mu^{\mathsf{w}}(\bar\alpha, \beta, \alpha^{-1})= \lambda(\bar\alpha)\nu(\beta)|\alpha|_E^{-1/2}\), where \(\mathsf{w}=\left(\begin{smallmatrix} & & 1 \\ &1 & \\ 1 & & \end{smallmatrix} \right)\) is the non-trivial element of the Weyl group of \(G\). The Jacquet functor \(\pi\to \pi_N\) is exact and it sends \(\pi_2\) (resp. \(\pi_n\)) to \(\mu\delta^{1/2}\) (resp. \(\mu^{\mathsf{w}}\delta^{1/2}\)), where \(\delta(\bar\alpha, \beta, \alpha^{-1})= |\alpha|_E^2\) is the modulus character. The extension \[\label{induction} 0\to \pi_2 \to \pi = \mathop{\mathrm{Ind}}_B^G(\mu)\xrightarrow{\mathrm{pr}} \pi_n \to 0\tag{2}\] does not split since, by Frobenuis reciprocity, one has \[\mathop{\mathrm{End}}_G\left(\mathop{\mathrm{Ind}}_B^G(\mu)\right)=\mathop{\mathrm{Hom}}_B\left(\mathop{\mathrm{Ind}}_B^G(\mu), \mu\delta^{1/2}\right)= \mathop{\mathrm{Hom}}_T\left(\mathop{\mathrm{Ind}}_B^G(\mu)_N, \mu\delta^{1/2}\right)\simeq \mathbf{C}.\] One knows by Rodier [9] that the image of \(\mathop{\mathrm{Ind}}_B^G(\mu)\) by a twisted (by a non-degenerate character of \(N\)) Jacquet functor, singling out generic representations, is a line. Since \(\pi_n\) is non-generic (see [7]), the exactness of this functor implies that \(\pi_2\) is generic.

As \(\pi_n\) is non-tempered, the subspace \(\pi_2\) consists of \(f\in\pi\) such that for all \(f^\vee\in \pi ^\vee\) the matrix coefficient \(g\mapsto \langle g\cdot f, f^\vee \rangle\) belongs to \(\mathop{\mathrm{L}}^2(G)\). Conversely the following lemma holds.

We will be mostly interested in the following \(A\)-packets having trivial central characters: \[\label{A-packet-lambda} \Pi(\lambda)=\Pi(\lambda,\lambda_{| E^1}^{-1}).\tag{3}\]

1.3 The Gross subgroup \(K''\)↩︎

In this subsection, \(E\) is assumed ramified (hence \(p\) is odd), \(\mathcal{O}/\mathcal{P}=\mathbf{F}_p\) and \(\mathcal{P}=(\xi)\). As \(|\mathop{\mathrm{PGL}}(2,\mathbf{F}_p)|=|\mathop{\mathrm{SL}}(2,\mathbf{F}_p)|\) all vertices in the tree of \(G\) have valence \(p^3+1\). The reductive quotient \(\overline{G}{}^\circ\) of \(K^\circ\) is isomorphic to the orthogonal group \(\mathrm{O}(3,\mathbf{F}_p)\) with respect to the quadratic form represented by \(\left(\begin{smallmatrix} & & 1 \\ & 1 & \\ 1 & & \end{smallmatrix}\right)\). The adjoint action on matrices \(\left(\begin{smallmatrix} y & x \\ z & -y \end{smallmatrix}\right)\) preserving the determinant \(-(y^2+xz)\) allows us to identify \(\mathop{\mathrm{PGL}}(2,\mathbf{F}_p)\) and \(\mathrm{SO}(3,\mathbf{F}_p)\) as follows: \[\label{eq:adjoint} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\mapsto \frac{1}{ad-bc} \begin{pmatrix} a^2 & -ab & -b^2/2\\ -2ac & ad+bc &bd \\ -2c^2 &2cd & d^2 \end{pmatrix}.\tag{4}\] One has \(\mathrm{O}(3,\mathbf{F}_p)=\pm \mathbf{1}_3 \cdot \mathrm{SO}(3,\mathbf{F}_p)\). By the above description, \(\mathrm{SO}(3,\mathbf{F}_p)\) is generated by the set \[\left\{\left(\begin{smallmatrix} & & -1/2\\ & -1 &\\ -2 & & \end{smallmatrix}\right), \left(\begin{smallmatrix} a & &\\ & 1 & \\ & & a^{-1} \end{smallmatrix}\right), \left(\begin{smallmatrix} 1 & -b & -b^2/2\\ & 1 &b \\ & & 1 \end{smallmatrix}\right)\Big{|} a\in \mathbf{F}_p^\times, b\in\mathbf{F}_p\right\}.\]

As \(E/\mathbf{Q}_p\) is a ramified quadratic extension with \(p\) odd, a conjugate-symplectic character \(\lambda\) of \(E^\times\) is necessarily ramified and its restriction to \(\mathbf{Z}_p^\times\) is given by its unique quadratic character. If \(\lambda\) is tamely ramified, then its restriction to \(\mathcal{O}^\times\) is also given by its unique quadratic character, and the equation \(\lambda (\xi)^2= \lambda(-\xi\bar\xi)=\lambda(-1)=(-1)^{(p-1)/2}\) shows that there are precisely two such characters.

Interested in determining a level for an element of the \(A\)-packet \(\Pi(\lambda)\) considered in 3 , we are indebted to B. Gross for generously sharing a suggestion that led to the following proposition.

1.4 Higher Iwahori invariants via matrix coefficients↩︎

In this subsection, we assume that \(E/\mathbf{Q}_p\) is unramified and we let \(\lambda_0\) denote the unique quadratic unramified character of \(E^\times\). By Iwasawa decomposition, the corresponding \(\pi=\mathop{\mathrm{Ind}}_B^G(\mu^{}_0)\) has one dimensional invariants by any given maximal open compact subgroup \(K\) of \(G\). The following proposition states, depending on \(K\), whether the \(K\)-invariant line belongs its sub-representation \(\pi_2\) or maps non-trivially to the quotient \(\pi_n\) (see §1.2 for notation). We recall that \(K^\circ\) and \(K'\) are the two standard maximal compact subgroups, \(K^\circ\) being the hyperspecial one, and that the standard Iwahori subgroup \(I\) equals \(K^\circ \cap K'\).

As a consequence we obtain the following lower bound, in the unramified case.

While the unramified \(A\)-packet \(\Pi(\lambda_0)\) would suffice for our global applications when \(D\equiv 3\pmod{8}\), the case of discriminants \(D\equiv 7\pmod{8}\) would require the use of certain tamely ramified \(A\)-packets \(\Pi(\lambda)\) and providing explicit levels for them is the object of the next subsection.

1.5 Intertwinings and exponential sums↩︎

Our arithmetic applications will require to show existence of non-zero \(I_r\)-invariants, for some \(r\in \mathbf{Z}_{\geqslant 1}\), in certain ramified \(A\)-packets. This is delicate because of the lack of a new-vector theory for non-generic representations \(\pi\) (see Remark 7). Also Casselman’s result asserting that \(\pi^{I_r}\) surjects onto \(\pi_N^{T\cap I_r}\) is inconclusive here as the latter vanishes, in contrast with the situation in [3] where the open compact is a pro-\(p\)-Iwahori subgroup of a sufficiently deep level. We will instead employ explicit methods and prove in Theorem 6 that \(\pi_n\) has non-zero invariants by \(K_T\), which contains a conjugate of \(I_3\). It is relatively straightforward to determine all such vectors in the full induced representation, but it becomes a thorny comptutation to find a non-square integrable matrix coefficient as in §1.4. By another result of Casselman, matrix coefficients can be expressed in terms of the corresponding ones in the Jacquet module, given here by an explicit scalar product on \(\pi_N\simeq\mathbf{C}^2\). Making this actually work requires non-vanishing of the second coordinate of the image of a \(K_T\)-invariant vector under the Jacquet functor which, once verified, leads directly to the result we seek.

We reduce the computation of the Jacquet functor to a precise statement about the intertwining operator at the level of finite reductive groups. The latter involves showing non-vanishing of some explicit exponential sums, bringing out the arithmetic nature of the problem. Although these sums seem extremely hard to compute individually, we manage to deduce it by precisely computing a suitable average of such sums corresponding to the trace of the finite intertwining operator.

Recall that \(\pi_n\) is the Langlands quotient of \(\mathop{\mathrm{Ind}}_B^G(\mu_\lambda)\) whose other Jordan-Hölder constituent is the discrete series \(\pi_2\). As taking invariants by an open compact subgroup is an exact functor in the category of admissible representations, Lemma 4 implies that \[\label{eq:dim-sum} \dim \pi_n^{K_T}+ \dim \pi_2^{K_T}=\begin{cases} p & \text{ if } \chi^2\ne \mathbf{1},\\p+1 & \text{if } \chi \text{ quadratic}.\end{cases}None\tag{6}\]

In order to show that \(\pi_n\) admits non-zero \(K_T\)-invariants, it would be enough to consider the image of a \(K_T\)-invariant vector from Lemma 4 under the Jacquet functor and show that its second coordinate does not vanish. As well known, this functor is given by \[\mathop{\mathrm{Ind}}_B^G(\mu_\lambda) \longrightarrow \mathbf{C}\cdot \mu_\lambda \delta^{1/2} \oplus \mathbf{C}\cdot\mu_\lambda^{\mathsf{w}}\delta^{1/2}, \quad f\mapsto (f(1),(Mf)(1))\] where the standard intertwining operator \(M:\mathop{\mathrm{Ind}}_B^G(\mu_\lambda) \to \mathop{\mathrm{Ind}}_B^G(\mu_\lambda^{\mathsf{w}})\) is defined via analytic continuation, as follows. For \(s\in \mathbf{C}\), letting \(\mu_{\lambda,s}=\mu_\lambda\cdot \delta^{s/2}\), the intertwining operator \[M_s:\mathop{\mathrm{Ind}}_B^G(\mu_{\lambda,s}) \to \mathop{\mathrm{Ind}}_B^G(\mu_{\lambda,s}^{\mathsf{w}}), \quad f_s\mapsto \int\limits_N f_s(\mathsf{w}n\cdot)dn\] is absolutely convergent for \(\Re(s)\gg 0\) and \(G\)-equivariant. Moreover, for any section \(f_s\in \mathop{\mathrm{Ind}}_B^G(\mu_{\lambda,s})\) such that for all \(g\in G\) the function \(f_s(g)\) is analytic in \(s\in \mathbf{C}\), the function \((M_s(f_s))(g)\), a priori only defined for \(\Re(s)\gg 0\), is a rational function in \(p^{-s}\), hence extends to a meromorphic function on all of \(\mathbf{C}\) with only possibly a finite number of poles independent of \(f\) and of \(g\). In fact, it continues as an intertwining operator, i.e. \[(M_s(f_s))(gg')= (M_s(f_s(\cdot g'))(g), \text{ for all } g,g'\in G.\] We refer to [13] for more detail and proofs. We will computing explicitly the right hand side of \[M(f_0)=\lim_{s\to 0} M_s(f_{s})\] as a rational function in \(p^{-s}\), simultaneously justifying the absence of pole at \(s=0\).

For any \(y\in \mathbf{F}_p\), Lemma 4 yields a unique function \(f_y\in\left(\mathop{\mathrm{Ind}}_B^G(\mu_\lambda)\right)^{K_T}\) supported on \(B\sigma_y K_T\) and normalized by \(f_y(\sigma_y)=1\). In addition, when \(\chi\) is quadratic, we let \(f_\infty\) be the unique function supported on \(B\sigma_\infty K_T\) and normalized by \(f_\infty(\sigma_\infty)=1\). Consider a \(K^\circ\)-flat section \(f_{y,s}\) passing through \(f_{y,0}=f_{y}\).

 

The coefficients of the intertwining automorphism of \(\mathop{\mathrm{Ind}}_{\overline{B}}^{\overline{G}}(\overline{\mu}_\lambda)^{\overline{T}}\) in the basis \((f_{y})_y\) are given by \[M_{y,y'}=\sum_{[z,x]\in \mathbf{F}_{p^2}\times\mathbf{F}_p }f_{y',0}\left(\mathsf{w}[z,x]\sigma_y\right).\]

2 Galois representations for \(3\)-folds of Picard type↩︎

From this point onwards, we will use global notations from the introduction. The local results of the previous section can be applied to the completion \(E\) of \(M\) at any prime number which does not split in that field. We denote by \(\mathbf{A}_f\) the ring of finite adeles of \(\mathbf{Q}\), so that \(\mathbf{A}=\mathbf{A}_f\times \mathbf{R}\).

2.1 Abelian \(3\)-folds and Shimura surfaces of Picard type↩︎

Let \(k\) be any field containing \(M\). Consider an abelian \(3\)-fold \(A/k\) together with an injection \(\iota^0: M \hookrightarrow \mathrm{End}^0(A/k)=\mathrm{End}(A/k)\otimes\mathbf{Q}\), or equivalently with an injection \(\iota\) of an order of \(M\) into \(\mathrm{End}(A/k)\), the most important for us case being when \(\iota^0\) comes from \(\iota: \mathcal{O}_M\hookrightarrow \mathrm{End}(A/k)\).

A polarization on \(A\) is an isogeny \(\theta: A\to A^\vee\), where \(A^\vee\) denotes the dual abelian variety. By positivity, since \(k\) is a field, the Rosati involution induced by \(\theta\) on \(\iota^0(M)\) is given by the complex conjugation (see [14]). A polarization is called principal, if it is an isomorphism, and it can always be acquired over a finite extension of \(k\).

By an abelian \(3\)-fold of Picard type over \(k\) we will mean a principally polarized abelian variety over \(k\) of dimension \(3\) having multiplication by \(\mathcal{O}_M\) defined over \(k\).

The action of \(M\) splits the \(3\)-dimensional \(k\)-vector space \(\mathop{\mathrm{Lie}}(A/k)\) in a direct sum of two sub-spaces: one on which the actions of \(M\) and \(k\) agree, and one on which they differ by the complex conjugation, the pair of their dimension being called the signature.

To define a level structure on \(A\) we need to consider its Tate module. Given a finite place \(v\) of \(M\), the \(v\)-adic Tate module \(T_v A=\varprojlim\limits_r A[v^r]\) of \(A\) is free of rank \(3\) over \(\mathcal{O}_v\). Denote \(V_v A=M_v \otimes_{\mathcal{O}_v}T_v A\). One also considers the adelic Tate module \[V_f A=\mathbf{Q}\otimes_{\mathbf{Z}} \varprojlim\limits_{n}A[n],\] which is free of rank \(3\) over \(\mathbf{A}_{M,f}\). Given a polarization \(\theta: A\to A^\vee\), the Weil pairing endows \(V_f A\) with a non-degenerate skew-hermitian form, i.e., a non-degenerate alternating pairing \[\langle \cdot, \cdot \rangle_A:V_f A\times V_f A\to \mathbf{A}_f\] such that \(\langle a\cdot v, v' \rangle_A=\langle v, \bar a\cdot v' \rangle_A\) for all \(a\in M\). If \(\theta\) is principlal, then \(\langle \cdot, \cdot \rangle_A\) is a perfect pairing.

Let \((V,\langle \cdot, \cdot \rangle)\) be a \(3\)-dimensional (non-degenerate) hermitian space over \(M\). The unitary similitude group \(\widetilde{G}=\mathrm{GU}(V)\) is the reductive group over \(\mathbf{Q}\) characterized by the property that \[\widetilde{G}(R)= \{ g\in \mathop{\mathrm{GL}}(V\otimes_\mathbf{Q}R) \mid \forall v,v'\in V\otimes_\mathbf{Q}R, \langle g(v), g(v') \rangle=\nu(g) \langle v, v' \rangle\},\] for any \(\mathbf{Q}\)-algebra \(R\), where \(\nu: \mathrm{GU}(V)\to \mathbf{G}_{m,\mathbf{Q}}\) is a homomorphism whose kernel is the unitary group \(G= \mathop{\mathrm{U}}(V)\). Since any hermitian form in \(3\) variables over a non-archimedean local field is isotropic, the group \(\widetilde{G}(\mathbf{Q}_p)\) is unique up to isomorphism, while at infinity \(\langle \cdot, \cdot \rangle\) is uniquely determined by its signature, hence there are two possibilities for \(\widetilde{G}(\mathbf{R})\) (as opposite signatures define isomorphic groups). Hasse’s Principle applied to the semi-simple simply connected derived group \(G^1=\mathrm{SU}(V)\) implies then that, up to an isomorphism, there exists a unique quasi-split unitary group, denoted \(\widetilde{G}=\mathrm{GU}(2,1)\), and a unique definite unitary group, denoted \(\mathrm{GU}(3)\).

We will now introduce the Shimura variety for \(\widetilde{G}\) represented by the matrix \(\left(\begin{smallmatrix} & & 1 \\ & \sqrt{-D} & \\ -1 & & \end{smallmatrix} \right)\). The homomorphism \[\tilde{h}: \mathrm{Res}^\mathbf{C}_\mathbf{R}\mathbf{G}_{m,\mathbf{R}} \to \widetilde{G}_\mathbf{R}\text{ , } z\mapsto \begin{pmatrix} \Re(z) & 0 & \Im(z) \\ 0 & z & 0 \\ -\Im(z) & 0 & \Re(z) \end{pmatrix}\] of \(\mathbf{R}\)-algebraic groups satisfies Deligne’s axioms [15] for a Shimura variety, hence for any open compact subgroups \(\widetilde{K}\) of \(\widetilde{G}(\mathbf{A}_f)\) one can consider the Shimura surface \[Y_{\widetilde{K}}(\mathbf{C}) =\widetilde{G}(\mathbf{Q})\backslash \left(\mathcal{H} \times \widetilde{G}(\mathbf{A}_f)/\widetilde{K}\right),\] where \(\mathcal{H}\) is identified with the \(\widetilde{G}(\mathbf{R})\)-conjugacy classes of \(\tilde{h}\). By a fundamental result of Shimura, \(Y_{\widetilde{K}}\) admits a canonical model over the reflex field \(M\). The connected components of \(Y_{\widetilde{K}}\) are Picard modular surfaces.

For the anisotropic form \(\mathrm{GU}(3)\), one can analogously define Shimura sets which are finite and therefore will not alter the uniformity of our results in §3.

The Shimura surfaces of Picard type are coarse moduli spaces of abelian \(3\)-folds. Namely, \(Y_{\widetilde{K}}(\mathbf{C})\) is in bijection with isogeny classes of \((A,\iota^0,\theta, \eta\cdot \widetilde{K})\), where \((A,\iota^0, \theta)\) is an abelian \(3\)-fold over \(\mathbf{C}\) as above, and \(\eta: \mathbf{A}_f\otimes_{\mathbf{Q}}V \xrightarrow{\sim} V_f A\) is an isomorphism sending \(\langle \cdot, \cdot \rangle_A\) to a \(\mathbf{A}_f^\times\)-multiple of \(\langle \cdot, \cdot \rangle_V\). Note that the usual \(\mathbf{Q}^\times\)-multiple condition is automatically satisfied as we are in the type C case. When \(\widetilde{K}^\circ\) is the standard maximal open compact subgroup of \(\widetilde{G}(\mathbf{A}_f)\), the points of \(Y_{\widetilde{K}^\circ}(k)\) correspond to isomorphism classes of abelian \(3\)-folds of Picard type.

Let \((A,\iota, \theta)\) be an abelian \(3\)-fold of Picard type over a number field \(k\). The action of \(\mathop{\mathrm{Gal}}_k\) on the adelic Tate module \(V_f A\), together with the choice of \(\eta\) as above, yields a continuous homomorphism: \[\begin{align} \label{eq:galois-action} \rho_{A,f}: \mathop{\mathrm{Gal}}_k\longrightarrow \widetilde{G}(\mathbf{A}_f). \end{align}\tag{12}\] Moreover, the point \((A,\iota,\theta, \eta \cdot \widetilde{K})\) on \(Y_{\widetilde{K}}(\mathbf{C})\) is defined over \(k\) if, and only if, \(\rho_{A,f}(\mathop{\mathrm{Gal}}_k)\subset \widetilde{K}\).

As for each \(\ell\) the principal polarization \(\theta\) induces a perfect pairing on \(T_\ell A\simeq \mathop{\mathrm{H}}_1(A(\mathbf{C}), \mathbf{Z}_\ell)\), it follows that one can choose \(\eta\) so that \(\rho_{A,f}(\mathop{\mathrm{Gal}}_k)\subset \widetilde{K}^\circ\), i.e., \((A,\iota, \theta, \eta\cdot \widetilde{K}^\circ)\) defines a \(k\)-rational point on \(Y_{\widetilde{K}^\circ}\). By the Brauer–Nesbitt Theorem the semi-simplification \(\overline{\rho}_{A,\ell}\) of the composition of \(\rho_{A,\ell}: \mathop{\mathrm{Gal}}_k\longrightarrow \widetilde{K}_\ell^\circ\) with the natural surjection of \(\widetilde{K}_\ell^\circ\) onto its reductive quotient \(\overline{G}_\ell^\circ\) is uniquely determined by its characteristic polynomial. Moreover, \(\overline{\rho}_{A,\ell}\) is absolutely irreducible if, and only if, the self-dual lattice fixed by \(\widetilde{K}_\ell^\circ\) is the unique, up to homothecy, \(\rho_\ell(\mathop{\mathrm{Gal}}_k)\)-stable lattice.

In analogy with the index \(2\) Gross subgroups \(\widetilde{K}''_p\) of maximal compacts \(K^\circ_p\) at ramified primes introduced in §1.3, we consider the open compact subgroup \[\label{K-double-prime} \widetilde{K}''= \widetilde{K}''_D \prod_{p\nmid D} \widetilde{K}_p^\circ \subset \widetilde{G}(\mathbf{A}_f),\tag{13}\] where \(\widetilde{K}''_D\) is defined as the kernel of the composed homomorphism \[\label{KD-double-prime} \prod_{p\mid D} \widetilde{K}_p^\circ \twoheadrightarrow \prod_{p\mid D} \widetilde{K}_p^\circ/\widetilde{K}''_p= \prod_{p\mid D} \{\pm 1\} \xrightarrow{\Pi} \{\pm 1\}.\tag{14}\]

Let \((A,\iota,\theta)\) be a principally polarized abelian \(3\)-fold of Picard type over \(k\). For \(v\) the prime of \(M\) above \(p\mid D\), the action of \(\mathop{\mathrm{Gal}}_k\) on \(A[v]\) factors through \(\mathrm{GO}(3,\mathbf{F}_p)\). Using the exceptional isomorphism \(\mathrm{PGO}(3,\mathbf{F}_p)\xrightarrow{\sim} \mathrm{SO}(3,\mathbf{F}_p)\xrightarrow{\sim} \mathop{\mathrm{PGL}}(2,\mathbf{F}_p)\), one defines its projectivization \[\label{eq:star} \widetilde{\rho}_{A,p}: \mathop{\mathrm{Gal}}_k \to \mathop{\mathrm{PGL}}(2,\mathbf{F}_p).\tag{15}\] Taking quotient by the unique index two subgroup \(\mathop{\mathrm{PSL}}(2,\mathbf{F}_p)\) of \(\mathop{\mathrm{PGL}}(2,\mathbf{F}_p)\) yields a canonical homomorphism \(\varepsilon_{A,p}: \mathop{\mathrm{Gal}}_k \to\{ \pm 1 \}\) and we let \(\varepsilon_{A,D}=\prod_{p \mid D} \varepsilon_{A,p}: \mathop{\mathrm{Gal}}_k \to\{ \pm 1 \}\).

Using the observation made after 12 , the points in \(Y_{\widetilde{K}''}(k)\) corresponds precisely to an abelian \(3\)-fold \(A\) over \(k\) of Picard type having trivial \(\varepsilon_{A,D}\).

2.2 Complex reflexions and families of abelian threefolds of Picard type↩︎

It is important to observe that even though for \(\widetilde{K}\subset \widetilde{K}^\circ\) each point of \(Y_{\widetilde{K}}(\mathbf{C})\) is associated to an abelian \(3\)-fold of Picard type, there does not exist such a family over the entire \(Y_{\widetilde{K}}(\mathbf{C})\) unless there are no points with extra automorphisms, in which case \(Y_{\widetilde{K}}(\mathbf{C})\) would be a fine moduli space. In our cases of interest \(\widetilde{K}\) is not neat, and therefore \(Y_{\widetilde{K}}\) is not a fine moduli space. In this subsection we prove that there exists a natural family of abelian \(3\)-folds of Picard type over \(Y_{\widetilde{K}''}\) minus a finite number of points (see 13 ). This will be crucially used in the proof of Theorem 2.

Recall that an element \(\gamma\) of the discrete subgroup \(\Gamma=\widetilde{G}(\mathbf{Q})\cap \widetilde{K}\widetilde{G}(\mathbf{R})\) has a fixed point in \(\mathcal{H}\) if, and only if, \(\gamma\) has finite order (this is because the stabilizers in \(\widetilde{G}(\mathbf{R})\) of points in \(\mathcal{H}\) are maximal compact subgroups). Such a \(\gamma\) is called elliptic if it only fixes an isolated point in \(\mathcal{H}\), otherwise it is called a complex reflexion. By [4], the set of singular points of \(Y_\Gamma\) is finite and consists of isolated fixed points of elliptic elements of \(\Gamma\), all defined over a finite extension of \(M\). Moreover, smooth points at which universal cover \(\mathcal{H}\to Y_\Gamma\) is not étale are fixed points of complex reflexions, i.e.  order \(2\) elements in \(\Gamma\) fixing hyperbolic planes (see [4]).

Given a geometrically connected component \(Y''\) of \(Y_{\widetilde{K}''}\times_{M} k\) , we have \(Y''(\mathbf{C})=\Gamma''\backslash\mathcal{H}\) with \(\Gamma''=\widetilde{G}(\mathbf{Q})\cap g_f\widetilde{K}'' g_f^{-1} \widetilde{G}(\mathbf{R})\) for some \(g_f\in \widetilde{G}(\mathbf{A}_f)\).

 

 

2.3 Étale fundamental groups and Mumford–Tate groups↩︎

Let \(k\) be a number field containing \(M\) over which the connected component of \(Y_{\widetilde{K}}\) are defined. Fix a connected component \(Y\) of \(Y_{\widetilde{K}}\times_M k\) and a smooth open \(U\) of \(Y\) endowed with an abelian scheme \(f:A\to U\) of Picard type. Denote by \(\eta\) the generic point of the smooth surface \(U\). Fixing a closed geometric point \(\bar x\) of \(U\) the étale fundamental group sits in the middle of a short exact sequence \[\begin{align} \label{fund-group-es} 1\to \Pi_1(U_{\bar k}, \bar x)\to \Pi_1(U, \bar x) \to \mathop{\mathrm{Gal}}_k \to 1. \end{align}\tag{16}\] The morphism \(f:A\to U\) being proper and smooth, one can consider the étale sheaf \(\mathrm{R}^1f_*\mathbf{Z}_\ell\) on \(U\). As \(U\) is geometrically connected, we have \(\Pi_1(U, \bar x)\simeq \Pi_1(U, \bar \eta)\) and the latter acts on \[(\mathrm{R}^1f_*\mathbf{Z}_\ell)_{\bar \eta}= \mathop{\mathrm{H}}^1(A_{\bar \eta}, \mathbf{Z}_\ell)=(T_\ell A_{\eta})^\vee,\] yielding a continuous representation \[\mathop{\mathrm{Gal}}(\bar \eta/\eta)\twoheadrightarrow \Pi_1(U, \bar x)\xrightarrow{\rho_{U,\ell}} \mathop{\mathrm{Aut}}_{\mathbf{Z}_\ell}(T_\ell A_{\eta}).\] Any closed point \(x\in U(k)\) yields a section \(s_x:\mathop{\mathrm{Gal}}_k\to \Pi_1(U, \bar x)\) of 16 allowing one to consider \[\rho_{x,\ell}=\rho_{U,\ell}\circ s_x: \mathop{\mathrm{Gal}}_k \to \mathop{\mathrm{Aut}}_{\mathbf{Z}_\ell}(T_\ell A_{\eta}).\] Finally for any closed curve \(C\subset U\) defined over \(k\), there is a natural map \(\Pi_1(C, \bar x)\to \Pi_1(U, \bar x)\) whose composition with \(\rho_{U,\ell}\) is denoted \(\rho_{C,\ell}\). As \(f:A\to U\) is of Picard type, for any \(x\in C(k)\), \[\Gamma_x=\mathrm{im}(\rho_{x,\ell})\subset\Gamma_C=\mathrm{im}(\rho_{C,\ell})\subset \Gamma_U=\mathrm{im}(\rho_{U,\ell})\subset K^\circ_\ell.\]

By a series of results of Cadoret–Tamagawa (see [18], [19]), the set \(C_\rho\) of all \(x\in C(k)\) for which \(\Gamma_x\) is not open in \(\Gamma_C\) is finite and the index \([\Gamma_C: \Gamma_x]\) is uniformly bounded for \(x\in C(k){\smallsetminus} C_\rho\).

Recall that the Mumford–Tate group \(\mathop{\mathrm{MT}}(A)\) of a polarized abelian variety \(A\) over \(\mathbf{C}\) is the smallest connected reductive subgroup of \(\mathop{\mathrm{GL}}(\mathop{\mathrm{H}}_1(A, \mathbf{Q}))\) over \(\mathbf{Q}\), whose \(\mathbf{R}\)-points contain the image of the associated \(\mathbf{R}\)-morphism \(h: \mathbf{C}^\times \to \mathop{\mathrm{GL}}(\mathop{\mathrm{H}}_1(A(\mathbf{C}), \mathbf{R}))\) coming from the Hodge decomposition. If \(A\) is defined over a field \(k\) finitely generated over \(\mathbf{Q}\), then the image \(\Gamma_\ell\) of \(\mathop{\mathrm{Gal}}_k\) acting on \(T_\ell A\) is an \(\ell\)-adic Lie group. Denoting \(\mathfrak{g}^{}_{\mathbf{Z}_\ell}\) its Lie algebra, it is a theorem of Deligne [20] that \(\mathfrak{g}^{}_{\mathbf{Q}_\ell}=\mathfrak{g}^{}_{\mathbf{Z}_\ell}\otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell \subset \mathop{\mathrm{Lie}}(\mathop{\mathrm{MT}}(A)\otimes_{\mathbf{Q}}\mathbf{Q}_\ell)\) and the Mumford–Tate conjecture, known for abelian varieties of dimension at most \(3\), asserts that they are equal (see e.g. [21]).

As the Mumford–Tate group of the (generic point of the) universal family \(f:A\to U\) is given by \(\widetilde{G}=\mathrm{GU}(2,1)\), it follows from Deligne [20] that the Mumford–Tate group of any abelian \(3\)-fold of Picard type is a reductive subgroup of \(\widetilde{G}\). We have the following trichotomy.

If \(A\) is of CM type (resp. admits a CM factor), then \(\mathop{\mathrm{Lie}}(\mathop{\mathrm{MT}}(A)\otimes_{\mathbf{Q}}\mathbf{Q}_\ell)\) is of the first (resp. second) type in Lemma 7. Also clearly \(\rho_{A,\ell}\) is potentially abelian (resp. is potentially reducible) if, and only if, \(\mathfrak{g}^{}_{\mathbf{Q}_\ell}\) is of the first (resp. second) type in Lemma 7. The following proposition bridges the two sides.

3 Uniform irreducibility of Galois images↩︎

3.1 A lemma on surfaces with isolated singularities↩︎

We denote by \(q(X)\) the irregularity of a projective algebraic surface \(X\), given by the dimension of \(\mathop{\mathrm{H}}^1(X, \mathcal{O}_{X})\). If \(X\) is smooth and projective over \(\mathbf{C}\), then \(q(X)=\dim \mathop{\mathrm{H}}^0(X, \Omega^1_{X})\).

Let \(X\) be a projective irreducible algebraic surface over \(\mathbf{C}\) with isolated singularities, i.e., such that there exists a smooth open \(j:U\hookrightarrow X\) whose complement \(Z=X{\smallsetminus} U\) consists of finitely many closed points. There exists a smooth resolution \[\phi: \widetilde{X} \to X\] such that \(\phi^{-1}(Z)\) is a divisor with normal crossings with \(\phi\) restricting to an isomorphism from \(\phi^{-1}(U)\) onto \(U\). Thus we get an injection \(\widetilde{j}: U \hookrightarrow \widetilde{X}\) such that \(j= \phi \circ\widetilde{j}\), and we denote by \[\widetilde{j}^\ast: \mathop{\mathrm{H}}^1(\widetilde{X}, \mathbf{Q}) \to \mathop{\mathrm{H}}^1(U, \mathbf{Q})\] the pullback homomorphism on Betti cohomology. By [23] we know that \(\widetilde{j}^\ast\) is a homomorphism of mixed Hodge structure, with \(\mathop{\mathrm{H}}^1(\widetilde{X}, \mathbf{Q})\) being pure of weight \(1\).

3.2 Irregularity of Picard modular surfaces↩︎

Let \(z\mapsto \bar z\) be the non-trivial automorphism of \(M/\mathbf{Q}\). Put \(M^1=\{z\in M^\times\mid z\bar z=1\}\) viewed as an algebraic torus over \(\mathbf{Q}\) and denote by \(\mathbf{A}_M^1\) its adelic points.

For \(\widetilde{K}\subset \widetilde{G}(\mathbf{A}_f)\) an open compact subgroup, we recall the Shimura variety of Picard type \(Y_{\widetilde{K}}\) from §2.1. Let \(G^1=\mathrm{SU}(V)\) be the derived group of \(\widetilde{G}\). As \(G^1\) is simply connected and \(G^1_\infty\) is not compact, the Strong Approximation Theorem (see [25]) implies that \(G^1(\mathbf{Q})\) is dense in \(G^1(\mathbf{A}_f)\). It follows that the determinant map defines an isomorphism between the group of connected components \(\pi_0(Y_{\widetilde{K}})\) and the idele class group \(\mathbf{A}_M^\times/M^\times \det(\widetilde{K})M_\infty^\times\). Shimura’s theory of canonical models implies that the connected components of \(Y_{\widetilde{K}}\) are all Galois conjugates, hence share the same irregularity, and the same is true for the Shimura variety for \(G\) \[Y_K(\mathbf{C})= G(\mathbf{Q})\backslash \left(\mathcal{H} \times G(\mathbf{A}_f)/K\right),\] where \(K=\widetilde{K}\cap G(\mathbf{A}_f)\). Letting \(\Gamma= \widetilde{G}(\mathbf{Q})\cap \widetilde{K}\widetilde{G}(\mathbf{R})\) it follows from \(\nu(\Gamma)\subset \mathbf{Q}^\times \cap \widehat\mathbf{Z}^\times \widehat\mathbf{R}_+^\times=\{ 1\}\) that both \(Y_{\widetilde{K}}\) and \(Y_K\) share the same connected component of identity given by \(Y_\Gamma=\Gamma\backslash \mathcal{H}\) (see [3]). One should be careful to observe that the natural dominant map \(Y_{K^1}\to Y_\Gamma\), where \(Y_{K^1}\) is the Shimura variety of level \(K^1=K\cap G^1(\mathbf{A}_f)\) for \(G^1\), is an isomorphism precisely when either \(\det(\Gamma)=\{1\}\) or \(-1\in \Gamma\).

3.3 Twists of canonical characters and root numbers↩︎

Hecke characters \((\lambda,\nu)\in \Xi\) whose local components at each finite place have ‘minimal’ ramification are intimately related to the canonical characters studied by Gross and Rohrlich. They play a pivotal role in our production of automorphic forms contributing to the irregularity of the Picard modular surfaces of low level. We will now briefly recall some of their properties under the running assumption that \(D> 3\) is odd. Consider the character \(\lambda_\infty(z)=\bar{z}\cdot |z|^{-1}\) of \(M_\infty^\times\simeq \mathbf{C}^\times\) and let \(\lambda_f:\widehat{\mathcal{O}}_{M}^\times\to \mathbf{C}^\times\) be a continuous character whose restriction to \(\mathcal{O}_{M,p}^\times\) is given by the unique quadratic character \[\mathcal{O}_{M,p}^\times\to \mathbf{F}_p^\times\xrightarrow{\left(\tfrac{\cdot}{p}\right)}\{\pm 1\},\] for all \(p\) dividing \(D\), and is trivial otherwise. As \(\left(\tfrac{-1}{D}\right)=-1\), it follows that \(\lambda_\infty\) and \(\lambda_f\) agree on \(\mathcal{O}_M^\times=\{\pm 1\}\). The finiteness of the ideal class group \(\mathscr{C}\!\ell_M=\mathbf{A}_M^\times/M^\times \widehat{\mathcal{O}}_{M}^\times M_\infty^\times\) guarantees that the resulting character of \(M^\times \widehat{\mathcal{O}}_{M}^\times M_\infty^\times\) extends to a character \(\lambda\) of \(\mathbf{A}_M^\times\) and clearly two such extensions must differ by an ideal class character. As by construction the restriction of \(\lambda_f\) to \(\widehat{\mathbf{Z}}^\times\) agrees with the quadratic Dirichlet character \(\left(\tfrac{\cdot}{D}\right)\) viewed as a character of \(\mathbf{A}^\times/\mathbf{Q}^\times \mathrm{Nm}(\mathbf{A}_M^\times)=\mathop{\mathrm{Gal}}(M/\mathbf{Q})\) and \(\mathbf{A}^\times = \mathbf{Q}^\times \widehat{\mathbf{Z}}^\times \mathbf{R}_+^\times\), it follows that the restriction of \(\lambda\) to \(\mathbf{A}^\times\) equals \(\left(\tfrac{\cdot}{D}\right)\), i.e., \(\lambda\) is conjugate-symplectic. Such a character is called canonical and we will denote it by \(\lambda_c\), remembering that it is only unique up to a multiplication by a character of \(\mathscr{C}\!\ell_M\). The root number \(W(\lambda_c^3)=-W(\lambda_c)=\left(\tfrac{-2}{D}\right)\) is \(1\) if, and only if, \(D\equiv 3\pmod{8}\) (see [26]).

Assume henceforth that \(\det(K)=\widehat{\mathcal{O}}_{M}^1\), so that \(\pi_0(Y_K)=\mathscr{C}\!\ell_M^1:=\mathbf{A}_M^1/M^1 \widehat{\mathcal{O}}_{M}^1 M_\infty^1\), and that \(\widehat{\mathcal{O}}_{M}^1\) embeds centrally into \(K\), the central character \(\omega=\nu\cdot \lambda_{|M^1}\) of any \(\pi\) contributing to ?? has to be everywhere unramified, i.e., \[\label{q-formula-can} q(Y_\Gamma^\ast)= \frac{1}{|\mathscr{C}\!\ell_M^1|} \sum_{\chi\in \Xi^1, \omega\in \widehat{\mathscr{C}\!\ell_M^1}} \sum_{ \pi_f \in \Pi_f\left(\lambda_{c}\chi^{}_{M},{\lambda^{-1}_{c}}_{\!\!\!\!\!|M^1} \chi^{-2}\omega\right)} \dim(\pi_f^K) \frac{1+W(\lambda_c^3\chi^{3}_{M})(-1)^{s(\pi_f)}}{2},\tag{17}\] where \(\Xi^1\) denotes the set of finite order characters of \(\mathbf{A}_M^1/M^1\) (see [26] for the fact that multiplication by an ideal class character does not change the root number).

If \(3\) does not divide the class number \(h:=|\mathscr{C}\!\ell_M|\), then the action of \(\mu \in \widehat{\mathscr{C}\!\ell_M^1}\) sending \((\chi,\omega)\) to \((\chi\mu,\omega\mu^3)\) allows one to twist out the central character \(\omega\) and obtain the simpler formula \[q(Y_\Gamma^\ast)= \sum_{\chi\in \Xi^1} \sum_{ \pi_f \in \Pi_f(\lambda_{c}\chi^{}_{M})} \dim(\pi_f^K) \frac{1+W(\lambda_c^3\chi^{3}_{M})(-1)^{s(\pi_f)}}{2}.\]

Successfully applying 17 requires one to understand how root numbers behave under twisting. As we are interested in creating irregularity at level \(\Gamma''_0(\ell^r)\), we focus on characters \(\chi\) which are only ramified at the fixed inert prime \(\ell\).

3.4 The Bombieri–Lang Conjecture for Picard modular surfaces↩︎

In this part we follow the general strategy of [3] by adapting it to the case where the level is not neat, the main point being to show that the irregularity of the Picard modular surfaces under consideration is at least \(3\). This requires some new techniques, also providing new proofs to the cases treated in loc. cit.

We recall the index \(2\) subgroup \(\widetilde{K}''\) of the maximal open compact subgroup \(\widetilde{K}^\circ\) of \(\widetilde{G}(\mathbf{A}_f)\) introduced in 13 . Given \(r\in\mathbf{Z}_{\geqslant 1}\) and a prime \(\ell\) inert in \(M\), we let \(\widetilde{K}''_0(\ell^r)\) be the subgroup of \(\widetilde{K}''\) whose component at \(\ell\) is the depth \(r\) Iwahori subgroup \(I_r\) and we let \[K''_0(\ell^r)=\widetilde{K}''_0(\ell^r)\cap G(\mathbf{A}_f) \text{ and } \Gamma''_0(\ell^r)= G(\mathbf{Q}) \cap K''_0(\ell^r)\cdot G(\mathbf{R}).\]

Consider an automorphic representation \(\pi\in\Pi\left(\lambda_{c}\chi^{}_{M},{\lambda^{-1}_{c}}_{\!\!\!\!\!|M^1} \chi^{-2}\omega\right)\) having non-zero \(K''_0(\ell^r)\)-invariants. By Propositions 2 and 3, for all finites place \(v\ne \ell\) we must have \(\pi_v=\pi_{n,v}\) with \(\chi^{}_v\) unramified, in which case it follows from 14 that the line \(\bigotimes\limits_{p\mid D} \pi_{n,p}^{K''_p}\) is fixed by \(K''_D\). By 17 , \(\pi\) contributes to \(q\left(Y_{\Gamma''_0(\ell^r)}^\ast\right)\) if either \(\pi_\ell=\pi_{n,\ell}\) and \(W(\lambda_c^3\chi^{3}_{M})=1\), or \(\pi_\ell=\pi_{c,\ell}\) and \(W(\lambda_c^3\chi^{3}_{M})=-1\). As the choice of a character \(\chi^{}_\ell:\mathbf{Q}_{\ell^2}^1 \to \mathbf{C}^\times\) determines uniquely, up to an ideal class character, a global character \(\chi^{}_{M}\) unramified outside \(\ell\), the irregularity formula 17 becomes (we have omitted \(\omega\in \widehat{\mathscr{C}\!\ell_M^1}\) as it is trivial on \(\mathbf{Q}_{\ell^2}^1\)) \[\label{q-formula-at-ell} \tfrac{1}{h}\cdot q\left(Y_{\Gamma''_0(\ell^r)}^\ast\right)= \sum_{\chi^{}_\ell, \text{ s.t.} W(\lambda_c^3\chi_M^3)=1} \dim(\pi_{n,\chi^{}_\ell}^{I_r}) + \sum_{\chi^{}_\ell, \text{ s.t.} W(\lambda_c^3\chi_M^3)=-1} \dim(\pi_{c,\chi^{}_\ell}^{I_r}),\tag{20}\] where \(\Pi(\lambda_0 \chi^{}_{M,\ell}, \chi^{-2}_\ell)=\{\pi_{n,\chi^{}_\ell},\pi_{c,\chi^{}_\ell}\}\) denotes the local Arthur packet attached to the quadratic unramified character \(\lambda_0:\mathbf{Q}_{\ell^2}^\times \to \mathbf{C}^\times\) and to \(\chi^{}_\ell:\mathbf{Q}_{\ell^2}^1 \to \mathbf{C}^\times\).

Henceforth, we will only use a lower bound for the irregularity corresponding to the contribution of everywhere non-tempered automorphic representations. Namely, combining 20 with Lemma 9, we have \[\label{q-inequality-at-ell} q\left(Y_{\Gamma''_0(\ell^r)}^\ast\right)\geqslant h \cdot \sum_{\chi^{}_\ell} \dim(\pi_{n,\chi^{}_\ell}^{I_r}),\tag{21}\] where the sums runs over characters \(\chi^{}_\ell:\mathbf{Q}_{\ell^2}^1 \to \mathbf{C}^\times\) such that \(\chi^{}_\ell(-1)=(-1)^{a(\chi^{}_\ell)+(D-3)/4}\).

The following results from Faltings [29] (see [3] for details).

As \(2\) splits in \(M\) for \(D\equiv 7\pmod{8}\), Proposition 14 has the following consequence.

3.5 Proof of Main Theorem↩︎

At different stages of the proof we will remove finite sets and deal with them in the last step. By §2.1 an abelian variety \(A\) as in the Theorem 2 defines a \(k\)-rational point on \(Y_{\widetilde{K}''}\), where \(\widetilde{K}''\) is defined in 13 . If \(A[\ell^r]\) admits a full \(k\)-rational flag (or equivalently, a \(k\)-rational isotropic line), we claim that \(A\) defines a \(k\)-rational point on \(Y_{\widetilde{K}''_0(\ell^r)}\), where \(\widetilde{K}''_0(\ell^r)\subset\widetilde{K}''\) is the subgroup whose component at \(\ell\) consists of elements whose reduction modulo \(\ell^r\) belong to the standard Borel subgroup \(\widetilde{B}(\mathbf{Z}/\ell^r\mathbf{Z})\) of \(\widetilde{G}(\mathbf{Z}/\ell^r\mathbf{Z})\). Indeed, by assumption one knows that there is some Borel subgroup containing the image of \(\mathop{\mathrm{Gal}}_k\) acting on \(A[\ell^r]\). However \(\widetilde{G}(\mathbf{Q}_\ell)\) acts transitively on isotropic lines (because isometries between hermitian subspaces always extend), hence all Borel subgroups are conjugated by \(\widetilde{G}(\mathbf{Q}_\ell)\), and in fact by \(\widetilde{K}_\ell^\circ=\widetilde{G}(\mathbf{Z}_\ell)\) (using Iwasawa decomposition). As \(\widetilde{K}''\) is a normal subgroup of \(\widetilde{K}^\circ\) we deduce that the Galois image is contained in \(\widetilde{K}''\cap \widetilde{K}_0(\ell^r)=\widetilde{K}''_0(\ell^r)\), proving the claim.

By Corollary 16, \(Y^{\ast}_{\widetilde{K}''_0(\ell^{7})}\) satisfies the Strong Bombieri–Lang Conjecture. In particular all its \(k\)-rational points lie in a subvariety \(Z\) defined over \(k\) which is a finite union of points and curves.

Let us now take one of the (finitely many) geometrically connected curve \(C\) in \(Z\), and after removing finitely many of its points (which would not affect the wanted result), we may assume that \(C\) is contained in the smooth open \(U\) from §2.2. In particular, there exists a family \(f:A\to C\) of abelian \(3\)-folds of Picard type.

As in §2.3 let \(\Gamma_C\subset \widetilde{K}_\ell^\circ=\widetilde{G}(\mathbf{Z}_\ell)\) be the image of the étale fundamental group acting on the \(\ell\)-adic Tate module of the generic fiber of the family. By Cartan’s theorem (see [30]), \(\Gamma_C\) is an \(\ell\)-adic Lie group hence admits a Lie algebra \(\mathfrak{g}^{}_{\mathbf{Z}_\ell}\). By Bogomolov [31] the Lie algebra \(\mathfrak{g}^{}_{\mathbf{Q}_\ell} =\mathfrak{g}^{}_{\mathbf{Z}_\ell}\otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell\) is algebraic, namely it is the Lie algebra of the Zariski closure of \(\Gamma_C\) in \(\widetilde{G}(\mathbf{Q}_\ell)\), the latter being furthermore reductive over \(\mathbf{Q}_\ell\) by Faltings [32]. By the Mumford–Tate Conjecture, which is known for is known for abelian \(3\)-folds (see e.g. [21]), we know that \(\mathfrak{g}^{}_{\mathbf{Q}_\ell}\) is the Lie algebra of the Mumford–Tate group \(\mathop{\mathrm{MT}}(A)\otimes_{\mathbf{Q}} \mathbf{Q}_\ell\). As \(C\) has positive dimension, it has to contain non-CM points, whose Mumford–Tate group is not abelian. By Lemma 7, the Lie subalgebra \(\mathfrak{g}_{\mathbf{Q}_\ell}\cap \mathfrak{su}(3,\mathbf{Q}_\ell)\) contains a form of \(\mathfrak{sl}(2,\mathbf{Q}_\ell)\). By [19] applied to the abelian family \(f:A\to C\) there exist \(B>0\) such that for all \(x\in C(k)\) outside a finite set \(C_\rho\) we have \[\label{eq:uniform} [\Gamma_C: \Gamma_x]\leqslant B,\tag{22}\] where \(\Gamma_x=\rho_{A_x,\ell}(\mathop{\mathrm{Gal}}_k)\) with \(A_x\) the abelian \(3\)-fold of Picard type corresponding to \(x\).

It follows that for all \(r\geqslant r(C)\) and for all \(A\) as above corresponding to a \(k\)-rational point on \(C{\smallsetminus} C_\rho\), \(A[\ell^r]\) does not admit a full \(k\)-rational flag. Finally, applying Proposition 11 to the finitely many remaining \(k\)-rational points, yields an integer \(r\) such that all \(k\)-rational points in \(Y_{\widetilde{K}''_0(\ell^r)}\) are of CM type, completing the proof of the Theorem.

We would like to thank our respective institutions, and also the TIFR where part of the work was done. We would like to thank A.  Cadoret, T. Graber, M.  Goreski, B. Gross, H.  Hida, A. Jorza, K.-W. Lan, B. Mazur, J.  Nekovář, D.  Prasad, A.  Raghuram, D. Rohrlich, J.-P. Serre and J.  Tilouine for helpful discussions.

References↩︎