1 Introduction↩︎

1.1 Background↩︎

Understanding the special values of \(L\)-functions is a fundamental problem in number theory. The Tamagawa number conjecture of Bloch-Kato [1], generalized by Kato [2], [3], Fontaine and Perrin-Riou [4], [5] and Burns-Flach [6], is the most general and sophisticated conjecture on the special values of \(L\)-functions. This conjecture includes the Birch and Swinnerton-Dyer conjecture as a special case, which still remains largely open, and is considered extremely difficult.

The first significant contribution to the (equivariant) Tamagawa number conjecture was given by Burns-Greither [7] and Huber-Kings [8]. They proved the conjecture for Tate motives associated with abelian fields. An important aspect of their result is that they showed that Iwasawa theory can be effectively used to tackle the conjecture. Their method belongs to what is known as “descent theory". It would be worth noting that the basic idea behind such a method can already be seen in [1].

More recently, their method was generalized by Burns, Kurihara and the present author in [9] to general base number fields. They developed descent theory and gave a strategy for proving the (equivariant) Tamagawa number conjecture for Tate motives over general number fields. An analogue of this work for elliptic curves was later given in [10]. Moreover, Kataoka and the present author [11] studied a generalization of [10], and gave a general strategy for proving the Tamagawa number conjecture for a general motive. The key to this work is to consider (higher rank) Euler systems for a general motive, whose existence was predicted by a conjecture formulated by Burns, Sakamoto and the present author [12].

The underlying philosophy behind these works is that one should consider Iwasawa theory of Euler systems in order to approach the Tamagawa number conjecture. On the other hand, in Iwasawa theory, \(p\)-adic \(L\)-functions are intensively studied, and a variety of new constructions have been found. However, it seems that the relation between \(p\)-adic \(L\)-functions and the Tamagawa number conjecture has not been thoroughly investigated. In particular, an application of the Bertolini-Darmon-Prasanna (BDP) \(p\)-adic \(L\)-function (constructed in [13], [14], [15]) to the Tamagawa number conjecture has hardly been studied so far. (However, we remark that it has been applied to the Birch and Swinnerton-Dyer conjecture, most notably in the work of Jetchev-Skinner-Wan [16].)

In this article, we make a first attempt to apply the BDP \(p\)-adic \(L\)-function to the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters.

1.2 Main results↩︎

We set some notation. Let \(f = \sum_{n=1}^\infty a_n q^n \in S_{2r}(\Gamma_0(N))\) be a normalized newform of weight \(2r\geq 2\) and level \(N\). Let \(p\) be an odd prime number such that \(p\nmid N\). Let \(K\) be an imaginary quadratic field of class number one with odd discriminant \(-D_K <-3\). We assume that every prime divisor of \(N\) splits in \(K\) (Heegner hypothesis). We also assume that \(p\) splits in \(K\): we write \((p)=\mathfrak{p}\overline{\mathfrak{p}}\). Let \(\overline{\mathbb{Q}}\) be the algebraic closure of \(\mathbb{Q}\) in \(\mathbb{C}\) and fix an embedding \(\iota_p: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p\) which induces \(K_\mathfrak{p}\xrightarrow{\sim} \mathbb{Q}_p\). Let \(F\) be a finite extension of \(\mathbb{Q}_p\) which contains \(\iota_p(a_n)\) for all \(n\geq 1\). Let \(V_{f,F}\) be the \(p\)-adic Galois representation of \(G_\mathbb{Q}:=\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q})\) attached to \(f\) with coefficients in \(F\). Let \(K_\infty/K\) be the anticyclotomic \(\mathbb{Z}_p\)-extension and set \(\Gamma:=\mathop{\mathrm{Gal}}(K_\infty/K)\). Let \(\chi : \Gamma \to \mathcal{O}_F^\times\) be a character which is crystalline at both \(\mathfrak{p}\) and \(\overline{\mathfrak{p}}\) and corresponds to an anticyclotomic Hecke character \(\chi_\mathbb{A}: \mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times\) of infinity type \((j,-j)\). By a slight abuse of notation, we denote \(\chi_\mathbb{A}\) also by \(\chi\).

Let \(M_f\) be the motive attached to \(f\). We consider the critical motive \(M:=M_f(r)\otimes \chi\) defined over \(K\), whose \(p\)-adic realization is the \(G_K\)-representation \(V_{f,F}(r) \otimes \chi\). The \(L\)-function attached to \(M\) is \(L(M,s)=L(f,\chi^{-1},s+r)\). We consider the Tamagawa number conjecture of Bloch-Kato [1] for the pair \((M,\mathcal{O}_F)\) (in the sense of Burns-Flach [6]), which determines the leading term of \(L(M,s)\) at \(s=0\) (i.e., the leading term of \(L(f,\chi^{-1},s)\) at \(s=r\)) up to a unit in \(\mathcal{O}_F^\times\). (See Conjecture 5 for the precise statement.)

The main result of this article is the following.

In the case of elliptic curves (i.e., when \(r=1\) and \(f\) corresponds to an elliptic curve \(E\) over \(\mathbb{Q}\)), the Iwasawa main conjecture is proved in [19] and [20] under mild hypotheses. Thus we obtain the following unconditional result on the Tamagawa number conjecture.

1.3 Idea of the proof↩︎

We shall sketch an idea of the proof of Theorem 1.

Fix a finite set \(S\) of places of \(K\) containing the infinite place and the primes dividing \(pN\). Let \(G_{K,S}\) be the Galois group of the maximal extension of \(K\) unramified outside \(S\). Let \(T_f\) be a stable lattice of the \(p\)-adic Galois representation attached to \(f\). Let \(\Lambda\) be the anticyclotomic Iwasawa algebra and set \(\Lambda^{\rm ur}:=\widehat \mathbb{Z}_p^{\rm ur}\widehat \otimes \Lambda\), where \(\widehat \mathbb{Z}_p^{\rm ur}\) denotes the completion of the ring of integers of the maximal unramified extension of \(\mathbb{Q}_p\). We consider the deformation \(\mathbb{T}:=\Lambda \otimes T_f(r)\).

The key idea is to construct an element \[\label{zs} \mathfrak{z}_S \in {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\tag{1}\] which “interpolates" special values \(L(f,\chi^{-1},r)\). In the construction, we use the Bertolini-Darmon-Prasanna \(p\)-adic \(L\)-function \[L_\mathfrak{p}^{\rm BDP} \in \Lambda^{\rm ur}\] and the”local epsilon element" \[\varepsilon_\mathfrak{p}\in \Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda(\mathbf{R}\Gamma(K_\mathfrak{p},\mathbb{T})),\] whose existence is predicted by Kato’s local epsilon conjecture [24]. (In our case, this conjecture is proved by Loeffler-Venjakob-Zerbes [25], Nakamura [26] and Rodrigues Jacinto [27].) By the general philosophy of Fukaya-Kato in [28], local epsilon elements should play the role of “Coleman maps". Roughly speaking, we define \(\mathfrak{z}_S\) to be the preimage of \(L_\mathfrak{p}^{\rm BDP}\) under the”Coleman map" \(\varepsilon_\mathfrak{p}\). A delicate point is to check that \(\mathfrak{z}_S\) has coefficients in \(\Lambda\), and to do this we need to study the behavior of \(L_\mathfrak{p}^{\rm BDP}\) under the Frobenius action (see Lemma 3). The Iwasawa main conjecture ensures that \(\mathfrak{z}_S\) is a \(\Lambda\)-basis. See §3.2 for the precise construction of \(\mathfrak{z}_S\). It would be worth noting that \(\mathfrak{z}_S\) is essentially an Euler system of rank two: see §1.4.2 below.

The proof of Theorem 1 is then reduced to checking that \(\mathfrak{z}_S\) has the desired interpolation property predicted by the Tamagawa number conjecture. (This is the content of Theorem 15.) To do this we use the interpolation property of \(L_\mathfrak{p}^{\rm BDP}\): \[\chi(L_\mathfrak{p}^{\rm BDP}) =\Omega_p^{4j}\cdot \Gamma(j-r+1) \Gamma(j+r)\cdot (1-a_p \chi^{-1}(\overline{\mathfrak{p}}) p^{-r} + \chi^{-2}(\overline{\mathfrak{p}}) p^{-1})^2 \frac{L(f,\chi^{-1},r)}{ \Omega_\infty^{4j} (2\pi)^{1-2j}\sqrt{D_K}^{2j-1} }.\] Here \(\Omega_\infty \in \mathbb{C}^\times\) and \(\Omega_p \in (\widehat \mathbb{Z}_p^{\rm ur})^\times\) denote complex and \(p\)-adic CM periods respectively. We check that the \(p\)-adic period \(\Omega_p\) and the \(\Gamma\)-factor \(\Gamma(j-r+1) \Gamma(j+r)\) arise in the interpolation property of \(\varepsilon_\mathfrak{p}\). We also check that \(\Omega_\infty^{4j} (2\pi)^{1-2j}\sqrt{D_K}^{2j-1}\) is the period of the motive \(M_f(r)\otimes \chi\) in the sense of Deligne [29]. Such comparison of periods and calculations will be made in §3.5. (Here we use the simplifying assumption that \(K\) has class number one.)

We remark that, in a forthcoming article [30], we generalize the method used in the present work. More precisely, we consider a \(p\)-adic \(L\)-function for a general motive and give a general strategy for proving the Tamagawa number conjecture using the \(p\)-adic \(L\)-function.

1.4 Related topics↩︎

We shall discuss some related topics.

1.4.1 The Tamagawa number conjecture in analytic rank one↩︎

It is natural to also consider Hecke characters \(\chi\) such that \(\mathrm{ord}_{s=r}L(f,\chi^{-1},s)=1\). (This happens when \(-r < j <r\).) In fact, Bertolini-Darmon-Prasanna proved a “\(p\)-adic Gross-Zagier formula" which relates \(\chi(L_\mathfrak{p}^{\rm BDP})\) for such \(\chi\) with generalized Heegner cycles (see [13]). Combining this result with a generalization of the Gross-Zagier formula (as in [31], [32], [33], [34]), it would be possible to prove that the element (1 ) interpolates \(L'(f,\chi^{-1},r)\), which would lead to a proof of the Tamagawa number conjecture in analytic rank one (under the Iwasawa main conjecture). We do this in Appendix 5 in the special case when \(\chi\) is the trivial character and \(f\) corresponds to an elliptic curve. In particular, we give another proof of a result essentially obtained by Jetchev-Skinner-Wan [16] on the Birch and Swinnerton-Dyer conjecture in analytic rank one (see Theorem 39 and Remark 41). We also give a similar result when \(p\) is inert in \(K\) (see Theorem 42).

It would also be interesting to compare our method with the recent work by Castella [35], where the Bertolini-Darmon-Prasanna \(p\)-adic \(L\)-function is applied to the Tamagawa number conjecture for CM elliptic curves in analytic rank one.

1.4.2 Euler systems of rank two↩︎

The element \(\mathfrak{z}_S\) we construct in (1 ) is essentially an Euler system of rank two. In fact, by [36] (see also [11]), there is a canonical map \[\label{Theta} \Theta: {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \to {\bigcap}_\Lambda^2 H^1(G_{K,S},\mathbb{T}),\tag{2}\] where for \(r\geq 0\) we write \(\bigcap_\Lambda^r\) for the \(r\)-th “exterior power bidual" introduced in [36]. Using the element (eq. ¿eq:zs? ), we define \[\label{zinfty} z_{K_\infty} := \Theta(\mathfrak{z}_S) \in {\bigcap}_\Lambda^2 H^1(G_{K,S},\mathbb{T}).\qquad{(1)}\] We conjecture that this is (at least up to a certain normalization) the”\(K_\infty\)-component" of the Euler system whose existence is predicted by a general conjecture [11] (see also [12]). Main results of the present work can be regarded as a partial solution to this conjecture.

We remark that we can unconditionally construct \(z_{K_\infty}\) as an element of \(Q(\Lambda)\otimes_\Lambda {\bigwedge}_\Lambda^2 H^1(G_{K,S},\mathbb{T})\) (\(Q(\Lambda)\) denotes the quotient field of \(\Lambda\)), and the Iwasawa main conjecture for \(L_\mathfrak{p}^{\rm BDP}\) is equivalent to the equality \[\label{IMC32without32L} {\rm char}_\Lambda\left({\bigcap}_\Lambda^2 H^1(G_{K,S},\mathbb{T}) / \Lambda\cdot z_{K_\infty} \right) = {\rm char}_\Lambda(H^2(G_{K,S},\mathbb{T})).\tag{3}\] (See [11].)

We also remark that our construction of \(z_{K_\infty}\) is different from the construction of the “\(\Lambda\)-adic Heegner element" \(z_{\infty}^{\rm Hg}\) in [11], since \(z_{K_\infty}\) is constructed by using the Bertolini-Darmon-Prasanna \(p\)-adic \(L\)-function, while \(z_\infty^{\rm Hg}\) by Heegner points (in the elliptic curve case). (In the former case we assume \(p\) splits, while in the latter case \(p\) does not necessarily split but it must be ordinary.) Note also that the construction of \(z_\infty^{\rm Hg}\) in [11] is non-canonical. In §5.4, we improve the construction of \(z_\infty^{\rm Hg}\) by using a local epsilon element (see Remark 51).

1.4.3 \(p\)-adic Birch and Swinnerton-Dyer conjectures↩︎

In [37], Agboola-Castella formulated a \(p\)-adic analogue of the Birch and Swinnerton-Dyer conjecture for the Bertolini-Darmon-Prasanna \(p\)-adic \(L\)-function. On the other hand, the present author formulated a conjecture on derivatives of (higher rank) Euler systems for motives in [38]. In particular, if we specialize it to the rank two Euler system \(z_{K_\infty}\) in (?? ), we can formulate an analogue of the \(p\)-adic Birch and Swinnerton-Dyer conjecture concerning derivatives of \(z_{K_\infty}\) (as in [38]). In a forthcoming work [30], we prove that this conjecture is equivalent to the conjecture of Agboola-Castella. We also prove that, in the ordinary case, it is equivalent to the \(p\)-adic Birch and Swinnerton-Dyer conjecture for Heegner points formulated by Bertolini-Darmon [39]. By these results, we can view both the Agboola-Castella and Bertolini-Darmon conjectures as special cases of the general conjecture for motives.

1.5 General notation↩︎

For a commutative ring \(R\) and an \(R\)-module \(X\), we set \[X^\ast:=\mathop{\mathrm{Hom}}_R(X,R).\] If \(X\) is a free \(R\)-module of rank one and \(x \in X\) is a basis, then the dual basis \[x^\ast \in X^\ast\] is the \(R\)-linear map satisfying \(x^\ast( x) =1\). If \(X\) is a \(\mathbb{Z}_p\)-module, then its Pontryagin dual is denoted by \[X^\vee:=\mathop{\mathrm{Hom}}_{\mathbb{Z}_p}(X,\mathbb{Q}_p/\mathbb{Z}_p).\]

Let \({\rm det}_R\) denote the determinant functor of Knudsen-Mumford [40]. This functor associates to a perfect complex \(C\) of \(R\)-modules a graded invertible \(R\)-module \({\rm det}_R(C)\). We often regard \({\rm det}_R(C)\) as an invertible \(R\)-module by forgetting the grading part. In particular, taking an element of \({\rm det}_R(C)\) makes sense. Note that, due to the sign issue, we need to regard \({\det}_R(-)\) as graded invertible modules when we use an isomorphism between \({\det}_R(C)\otimes_R {\det}_R(D)\) and \({\det}_R(D)\otimes_R {\det}_R(C)\) for two perfect complexes \(C\) and \(D\).

In this article, we use \({\det}_R\) when \(R\) is a regular local ring (e.g., any field, any discrete valuation ring, \(\mathbb{Z}_p[[T]]\), etc.). In this case, every bounded complex of finitely generated \(R\)-modules is perfect. In particular, we can define \({\rm det}_R(X)\) for any finitely generated \(R\)-module \(X\) by identifying \(X\) with its projective resolution. For basic properties of \({\det}_R\), see [38] for example.

For any field \(L\), its absolute Galois group is denoted by \(G_L\).

Let \(\overline{\mathbb{Q}}\) be the algebraic closure of \(\mathbb{Q}\) in \(\mathbb{C}\). Any number field \(K\) (i.e., a finite extension of \(\mathbb{Q}\)) is regarded as a subfield of \(\mathbb{C}\). The ring of integers of \(K\) is denoted by \(\mathcal{O}_K\).

For a number field \(K\) and a finite place \(v\) of \(K\), we fix a place of \(\overline{\mathbb{Q}}\) lying above \(v\). We regard \(G_{K_v}\) as the decomposition subgroup of \(v\) in \(G_K\). The maximal unramified extension of \(K_v\) is denoted by \(K_v^{\rm ur}\). The inertia subgroup of \(v\) in \(G_K\) is defined by \(I_v:=G_{K_v^{\rm ur}}\). The arithmetic Frobenius of \(v\) is denoted by \({\rm Fr}_v \in \mathop{\mathrm{Gal}}(K_v^{\rm ur}/K_v)=G_{K_v}/I_v\).

We use the standard notation of (continuous) Galois cohomology. For the definitions of \(H^1_f\) and \(H^1_{\rm ur}\), see [41] for example. We set \(H^1_{/f}:=H^1/H^1_f\) and \(H^1_{/{\rm ur}}:=H^1/H^1_{\rm ur}\). \(H^0_f\) is understood to be \(H^0\). For a finite place \(v\) and a \(G_{K_v}\)-module \(X\), the unramified cohomology complex is defined by \[\mathbf{R}\Gamma_{\rm ur}(K_v, X):=\mathbf{R}\Gamma(K_v^{\rm ur}/K_v, X^{I_v}) = [ X^{I_v}\xrightarrow{1-{\rm Fr}_v^{-1}} X^{I_v}].\] Let \({\rm Inf}_{K_v^{\rm ur}/K_v}: \mathbf{R}\Gamma_{\rm ur}(K_v,X) \to \mathbf{R}\Gamma(K_v,X)\) be the inflation morphism and set \[\mathbf{R}\Gamma_{/{\rm ur}}(K_v,X):= {\rm Cone}\left(\mathbf{R}\Gamma_{\rm ur}(K_v,X)\xrightarrow{-{\rm Inf}_{K_v^{\rm ur}/K_v}} \mathbf{R}\Gamma(K_v,X)\right).\]

Let \(S\) be a finite set of places of \(K\) containing all infinite and \(p\)-adic places of \(K\). Let \(S_f\subset S\) be the subset of finite places. Let \(K_S/K\) be the maximal extension unramified outside \(S\) and set \(G_{K,S}:=\mathop{\mathrm{Gal}}(K_S/K)\). Let \(V\) be a finite dimensional \(\mathbb{Q}_p\)-vector space endowed with a continuous linear action of \(G_{K,S}\). We frequently use the Poitou-Tate exact sequence \[\begin{gather} 0\to H^1_f(K,V)\to H^1(G_{K,S},V)\to \bigoplus_{v\in S_f}H^1_{/f}(K_v,V)\\ \to H^1_f(K,V^\ast(1))^\ast \to H^2(G_{K,S},V)\to \bigoplus_{v\in S_f}H^0(K_v,V^\ast(1))^\ast. \end{gather}\] (See [5] for example.)

2 Statement of the main results↩︎

The aim of this section is to state the main results precisely (Theorem 9 and Corollary 3). In §2.1, we review the formulation of the Tamagawa number conjecture in the setting as in Introduction. In §2.2, after reviewing the Bertolini-Darmon-Prasanna \(p\)-adic \(L\)-function and the Iwasawa main conjecture, we state the main results.

Throughout this article, let \(p\) be an odd prime number. Let \(f = \sum_{n=1}^\infty a_n q^n \in S_{2r}(\Gamma_0(N))\) be a normalized newform of weight \(2r\geq 2\) and level \(N\). We assume \(p\nmid N\). Let \(\mathcal{F}\) be a number field which contains all \(a_n\). Let \(\lambda\) be the prime of \(\mathcal{F}\) lying above \(p\) corresponding to the fixed embedding \(\iota_p :\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p\). We set \(F:=\mathcal{F}_\lambda\). Let \(V_{f,F}\) be the \(\lambda\)-adic Galois representation of \(G_\mathbb{Q}\) attached to \(f\) with coefficients in \(F\).

Let \(K\) be an imaginary quadratic field of class number one with odd discriminant \(-D_K <-3\)¹. We assume that every prime divisor of \(pN\) splits in \(K\). We write \((p)=\mathfrak{p}\overline{\mathfrak{p}}\) in \(K\) so that \(\mathfrak{p}\) corresponds to \(\iota_p\).

Let \(\chi : \mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times\) be an anticyclotomic Hecke character of infinity type \((j,-j)\) which takes values in \(\mathcal{F}\). (See §4.5.6 for the definition of Hecke characters.) By Remark 2, we are interested in the case \(j\geq r\) or \(j\leq -r\). We may assume \(j\geq r\), since the other case is treated in the same way by considering \(\overline{\chi}\) instead of \(\chi\). The \(p\)-adic avatar of \(\chi\) is a character \(G_K\to \mathcal{O}_F^\times\), which is also denoted by \(\chi\).

To simplify the notation, we set \[V:=V_{f,F}(r)\otimes \chi^{-1}.\] Let \(S\) be a finite set of places of \(K\) which contains the infinite place, the \(p\)-adic primes, and the primes at which \(V\) ramify. Let \(G_{K,S}\) be the Galois group of the maximal Galois extension of \(K\) unramified outside \(S\). Let \(V^\ast(1):=\mathop{\mathrm{Hom}}_F(V,F(1)) \simeq V_{f,F}(r)\otimes \chi\) be the Kummer dual of \(V\).

In this article, we regard a motive as a collection of realizations and comparison isomorphisms (see Appendix 4 for details). Let \(M_f\) and \(M(\chi)\) be the motives attached to \(f\) and \(\chi\) respectively (see §§4.5.5 and 4.5.6). We regard \(M_f\) as a motive defined over \(K\) (see Remark 19). We consider the motive \[M:=M_f(r)\otimes M(\chi),\] which is defined over \(K\) of rank two with coefficients in \(\mathcal{F}\). The weight of \(M\) is \(-1\) and so \(M\) is critical. Note that its \(\lambda\)-adic realization is \(V^\ast(1)\), and the \(L\)-function is \(L(M,s)=L(f,\chi^{-1},s+r)\).

2.1 The Tamagawa number conjecture↩︎

In this subsection, we review the statement of the Tamagawa number conjecture [6] for the pair \((M,\mathcal{O}_F)\) in analytic rank zero. (This is a special case of Conjecture 33.)

Note that the Betti realization \(H_B(M)\) and the de Rham realization \(H_{\rm dR}(M)\) of \(M\) are \(\mathcal{F}\)-vector spaces of dimension four. We have the comparison isomorphisms: \[\mathbb{C}\otimes_\mathbb{Q}H_B(M)\xrightarrow{\sim} \mathbb{C}\otimes_\mathbb{Q}H_{\rm dR}(M),\] \[F\otimes_\mathcal{F}H_B(M)^+ \simeq V^\ast(1),\] \[F\otimes_\mathcal{F}H_{\rm dR}(M) \simeq D_{{\rm dR},\mathfrak{p}}(V^\ast(1))\oplus D_{{\rm dR},\overline{\mathfrak{p}}}(V^\ast(1)).\] Here we set \(D_{{\rm dR},v}(-):=H^0(K_v, B_{\rm dR}\otimes_{\mathbb{Q}_p}-)\) for \(v\in \{\mathfrak{p}, \overline{\mathfrak{p}}\}\). The tangent space of \(M\) is defined by \[t(M):=H_{\rm dR}(M)/{\rm Fil}^0 H_{\rm dR}(M).\] Since Hodge-Tate weights of \(V^\ast(1)\) at \(\mathfrak{p}\) (resp. \(\overline{\mathfrak{p}}\)) are positive (resp. non-positive), we have \[F\otimes_\mathcal{F}t(M) \simeq D_{{\rm dR},\mathfrak{p}}(V^\ast(1)).\] Recall that \(M\) is critical, i.e., the period map is an isomorphism: \[\alpha_M: \mathbb{R}\otimes_\mathbb{Q}H_{B}(M)^+ \xrightarrow{\sim} \mathbb{R}\otimes_\mathbb{Q}t(M).\] Take \(\mathcal{F}\)-bases \(\gamma \in \bigwedge_\mathcal{F}^2 H_B(M)^+\) and \(\delta \in \bigwedge_\mathcal{F}^2 t(M)\). Let \[\label{period32C} \alpha_{M,\mathbb{C}}: \mathbb{C}\otimes_\mathcal{F}{\bigwedge}_\mathcal{F}^2 H_B(M)^+ \xrightarrow{\sim} \mathbb{C}\otimes_\mathcal{F}{\bigwedge}_\mathcal{F}^2 t(M)\tag{4}\] be the isomorphism induced by \(\alpha_M\). We define the period \[\label{complex32period} \Omega_{\gamma,\delta} \in \mathbb{C}^\times\tag{5}\] with respect to \(\gamma\) and \(\delta\) by \[\alpha_{M,\mathbb{C}}(\gamma)=\Omega_{\gamma,\delta}\cdot \delta.\] Deligne’s conjecture [29] for the motive \(M\) (see Conjecture 26) states that \[\frac{L(f,\chi^{-1},r)}{\Omega_{\gamma,\delta}} \in \mathcal{F},\] which is known to be true (essentially due to Shimura [42], see [13] and Lemma 6 below).

We now state the Tamagawa number conjecture for the pair \((M,\mathcal{O}_F)\). We fix a stable \(\mathcal{O}_F\)-lattice \(T\subset V\) and set \(T^\ast(1):=\mathop{\mathrm{Hom}}_{\mathcal{O}_F}(T,\mathcal{O}_F(1))\). Take an \(\mathcal{F}\)-basis \(\gamma \in \bigwedge_\mathcal{F}^2 H_B(M)^+\) so that its image under the comparison isomorphism \[F\otimes_\mathcal{F}{\bigwedge}_\mathcal{F}^2 H_B(M)^+\simeq {\bigwedge}_F^2 V^\ast(1)\] is an \(\mathcal{O}_F\)-basis of \({\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1)\). If we assume \(H^1_f(K,V)=H^1_f(K,V^\ast(1))=0\), then by Corollary 2 we have a canonical identification \[{\det}_F^{-1}(\mathbf{R}\Gamma(G_{K,S},V)) = {\bigwedge}_F^2 H^1(G_{K,S},V)\] and the localization isomorphism \[{\rm loc}_\mathfrak{p}: {\bigwedge}_F^2 H^1(G_{K,S},V) \xrightarrow{\sim} {\bigwedge}_F^2 H^1(K_\mathfrak{p},V).\] Also, the Bloch-Kato dual exponential map induces an isomorphism \[\exp_\mathfrak{p}^\ast: {\bigwedge}_F^2 H^1(K_\mathfrak{p},V) \xrightarrow{\sim} {\bigwedge}_F^2 D_{{\rm dR},\mathfrak{p}}(V) .\] Via the isomorphism \(F\otimes_\mathcal{F}t(M) \simeq D_{{\rm dR},\mathfrak{p}}(V^\ast(1))\), we regard the \(\mathcal{F}\)-basis \(\delta \in {\bigwedge}_\mathcal{F}^2 t(M)\) as an \(F\)-basis of \({\bigwedge}_F^2 D_{{\rm dR},\mathfrak{p}}(V^\ast(1))\). Let \(\delta^\ast \in {\bigwedge}_F^2 D_{{\rm dR},\mathfrak{p}}(V^\ast(1))^\ast \simeq {\bigwedge}_F^2 D_{{\rm dR},\mathfrak{p}}(V)\) denote the dual basis.

2.2 The Iwasawa main conjecture↩︎

We review the formulation of the Iwasawa main conjecture for the Bertolini-Darmon-Prasanna (BDP) \(p\)-adic \(L\)-function and state the main results of this article.

Let \(F_f\) be the minimal extension of \(\mathbb{Q}_p\) which contains the Fourier coefficients of \(f\) via \(\iota_p: \overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p\). Let \(\mathcal{O}_f:=\mathcal{O}_{F_f}\) be the ring of integers of \(F_f\). Let \(V_f\) be the \(p\)-adic Galois representation attached to \(f\) with coefficients in \(F_f\). Fix a stable \(\mathcal{O}_f\)-lattice \(T_f \subset V_f\).

Let \(K_\infty/K\) be the anticyclotomic \(\mathbb{Z}_p\)-extension and set \(\Gamma:=\mathop{\mathrm{Gal}}(K_\infty/K)\). Let \(\widehat \mathbb{Z}_p^{\rm ur}\) be the completion of the ring of integers of the maximal unramified extension \(\mathbb{Q}_p^{\rm ur}\) of \(\mathbb{Q}_p\). We set \[\Lambda:=\mathcal{O}_{f}[[\Gamma]] \text{ and }\Lambda^{\rm ur}:= \widehat \mathbb{Z}_p^{\rm ur}\widehat \otimes \Lambda.\]

The BDP \(p\)-adic \(L\)-function for \(f\) \[L_\mathfrak{p}^{\rm BDP} \in \Lambda^{\rm ur}\] satisfies the following interpolation property: for a Hecke character of infinity type \((j,-j)\) with \(j\geq r\) such that its \(p\)-adic avatar \(\chi\) factors through \(\Gamma\) and is crystalline at both \(\mathfrak{p}\) and \(\overline{\mathfrak{p}}\), we have \[\label{BDPinterpolation} \chi(L_\mathfrak{p}^{\rm BDP}) = \Omega_p^{4j}\cdot \Gamma(j-r+1) \Gamma(j+r)\cdot (1-a_p \chi^{-1}(\overline{\mathfrak{p}}) p^{-r} + \chi^{-2}(\overline{\mathfrak{p}}) p^{-1})^2 \frac{L(f,\chi^{-1},r)}{ \Omega_\infty^{4j} (2\pi)^{1-2j}\sqrt{D_K}^{2j-1} }\tag{6}\] Here \(\Omega_\infty \in \mathbb{C}^\times\) and \(\Omega_p \in (\widehat \mathbb{Z}_p^{\rm ur})^\times\) denote CM periods, whose definitions are given in §3.3 below. (Note that \(\Omega_K\) in [15] satisfies \(\Omega_\infty= 2\pi i \cdot \Omega_K\). Our \(L_\mathfrak{p}^{\rm BDP}\) is the involution of \(\mathscr{L}_\mathfrak{p}(f,{\boldsymbol{1}})^2\), where \(\mathscr{L}_\mathfrak{p}(f,{\boldsymbol{1}})\) is as in [35].) See [13], [14], [15] for the construction of \(L_\mathfrak{p}^{\rm BDP}\).

Let \[\mathbb{T}:= \Lambda \otimes_{\mathcal{O}_f}T_f(r)\] be the deformation of \(T_f(r)\), on which \(G_K\) acts by \[\sigma \cdot (x\otimes y) := \overline{\sigma}^{-1} x\otimes \sigma y \quad (\sigma \in G_K, \;x\in \Lambda, \;y\in T_f(r)),\] where \(\overline{\sigma} \in \Gamma\) denotes the image of \(\sigma\). We consider the following Selmer complex: \[\label{def32selmer} \widetilde{\mathbf{R}}\Gamma_\mathfrak{p}(K,\mathbb{T}):= {\rm Cone}\left(\mathbf{R}\Gamma(G_{K,S}, \mathbb{T}) \to \mathbf{R}\Gamma(K_\mathfrak{p}, \mathbb{T}) \oplus \bigoplus_{v\in S, v\nmid p}\mathbf{R}\Gamma_{/{\rm ur}}(K_v,\mathbb{T})\right)[-1].\tag{7}\] Here we set \[\mathbf{R}\Gamma_{/{\rm ur}}(K_v,\mathbb{T}):= {\rm Cone}\left(\mathbf{R}\Gamma_{\rm ur}(K_v,\mathbb{T})\xrightarrow{-{\rm Inf}_{K_v^{\rm ur}/K_v}} \mathbf{R}\Gamma(K_v,\mathbb{T})\right).\] Note that \(\widetilde{\mathbf{R}}\Gamma_\mathfrak{p}(K,\mathbb{T})\) coincides with the Selmer complex defined in [43] for the local condition \[U_v^+(\mathbb{T}):= \begin{cases} 0 &\text{if v=\mathfrak{p},}\\ \mathbf{R}\Gamma(K_{\overline{\mathfrak{p}}},\mathbb{T}) &\text{if v=\overline{\mathfrak{p}},}\\ \mathbf{R}\Gamma_{\rm ur}(K_v,\mathbb{T}) &\text{if v\nmid p.} \end{cases}\]

For a commutative ring \(R\), let \(Q(R)\) be the total quotient ring of \(R\). The Iwasawa main conjecture for the BDP \(p\)-adic \(L\)-function is stated as follows.

The following is the main result of this article.

3 Proof↩︎

In this section, we give a proof of Theorem 9.

As explained in §1.3, the key idea is to construct a \(\Lambda\)-basis \[\mathfrak{z}_S \in {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\] which is related with the value \(L(f,\chi^{-1},r)\). We first give a review on the local epsilon element in §3.1, which is necessary in the construction of \(\mathfrak{z}_S\). In §3.2, we construct \(\mathfrak{z}_S\) by using \(\mathfrak{z}_\mathfrak{p}\) in the Iwasawa main conjecture (Conjecture 7) and the local epsilon element. (Note that in this construction we do not need to assume \(K\) has class number one.) Theorem 9 is then reduced to the interpolation property of \(\mathfrak{z}_S\), which is Theorem 15. We prove Theorem 15 by comparing various periods. We give preliminaries on CM periods and choice of bases of Betti/de Rham cohomology in §§3.3 and 3.4. We complete the proof of Theorem 15 (and hence Theorem 9) in §3.5.

3.1 The local epsilon element↩︎

We first take a natural basis of \({\bigwedge}_\Lambda^2 \mathbb{T}= \Lambda \otimes_{\mathcal{O}_f} {\bigwedge}_{\mathcal{O}_f}^2 T_f(r)\) in the following way. Since we regard \(\overline{\mathbb{Q}}\subset \mathbb{C}\), we have a canonical \(p^n\)-th root of unity \(\zeta_{p^n}:=e^{2\pi i/p^n} \in \overline{\mathbb{Q}}\). The collection of these gives a \(\mathbb{Z}_p\)-basis \(\xi := (\iota_p(\zeta_{p^n}))_n \in H^0(\overline{\mathbb{Q}}_p ,\mathbb{Z}_p(1))\). Since we have a canonical isomorphism \({\bigwedge}_{\mathcal{O}_f}^2 T_f(r)\simeq \mathcal{O}_f (1)\), we can regard \(\xi\) as a \(\Lambda\)-basis of \(\bigwedge_{\Lambda}^2 \mathbb{T}\), which we denote by \(\gamma_\mathbb{T}\).

Kato’s local epsilon conjecture (see [24] or [28]) predicts the existence of a \(\Lambda^{\rm ur}\)-basis \[\varepsilon_{\Lambda,\xi}(\mathbb{T}) \in \Lambda^{\rm ur} \otimes_\Lambda {\det}_\Lambda(\mathbf{R}\Gamma(K_\mathfrak{p},\mathbb{T})) \otimes_\Lambda {\bigwedge}_\Lambda^2 \mathbb{T}\] (“local epsilon element", see Remark 11 below) satisfying certain interpolation properties. In our setting, this conjecture is known, thanks to work of Loeffler-Venjakob-Zerbes [25], Nakamura [26] and Rodrigues Jacinto [27]. In fact, since \(f\) is good at \(p\), \(\mathbb{T}=\Lambda\otimes_{\mathcal{O}_f}T_f(r)\) is a deformation of a crystalline representation of \(G_{\mathbb{Q}_p}\), which is treated in [25]. (Note that \(K_\mathfrak{p}\simeq \mathbb{Q}_p\).) Alternatively, since \(T_f(r)\) is a representation of \(G_{\mathbb{Q}_p}\) of rank two, the validity of the local epsilon conjecture is covered by [26], [27].

Define \[\label{def32local32epsilon} \varepsilon_\mathfrak{p}\in \Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda(\mathbf{R}\Gamma(K_\mathfrak{p},\mathbb{T}))\tag{8}\] to be the \(\Lambda^{\rm ur}\)-basis such that \[\varepsilon_\mathfrak{p}\otimes\gamma_\mathbb{T}= \varepsilon_{\Lambda,\xi}(\mathbb{T}).\]

Recall \(V:=V_f(r)\otimes_{F_f}F(\chi^{-1})\). Let \(\widehat \mathbb{Q}_p^{\rm ur}\) be the completion of the maximal unramified extension of \(\mathbb{Q}_p\) and set \[\widetilde{F}:=\widehat \mathbb{Q}_p^{\rm ur}\otimes_{\mathbb{Q}_p}F.\] We set \(D_{\rm dR}(V)=D_{{\rm dR},\mathfrak{p}}(V):=H^0(K_\mathfrak{p}, B_{\rm dR}\otimes_{\mathbb{Q}_p} V)\). We recall the definition of the isomorphism \[\label{dr32isom} \varepsilon_{\rm dR}=\varepsilon_{F,\xi,{\rm dR}}(V): \widetilde{F} \otimes_F {\bigwedge}_F^2 D_{{\rm dR}}(V)\xrightarrow{\sim} \widetilde{F} \otimes_F {\bigwedge}_F^2 V\tag{9}\] constructed in [28]. Note that Hodge-Tate weights of \(V\) (as a \(G_{K_\mathfrak{p}}\)-representation) are \(r-j\) and \(1-r-j\). Hence we have \[m:=\sum_{i\in \mathbb{Z}}i \dim_F({\rm gr}^i D_{{\rm dR}}(V))= (-r+j)+(-1+r+j) =2j-1.\] Let \[{\rm can}: (B_{\rm dR}\otimes_{\mathbb{Q}_p} F) \otimes_F {\bigwedge}_F^2 D_{{\rm dR}}(V) \xrightarrow{\sim} (B_{\rm dR}\otimes_{\mathbb{Q}_p}F) \otimes_F {\bigwedge}_F^2 V\] be the canonical isomorphism. We define \[\varepsilon_{\rm dR}:= t_\xi^{-m} \cdot {\rm can},\] where \(t_\xi \in B_{\rm dR}^+\) denotes the uniformizer corresponding to \(\xi\). (Since \(V\) is crystalline, the linearized action of the Weil group on \(D_{\rm pst}(V)\) is unramified (see [25]), and hence the epsilon constant \(\varepsilon_F(D_{\rm pst}(V),\psi)\) (as in [28]) is \(1\) by [28].) One can check that \(\varepsilon_{\rm dR}\) has coefficients in \(\widetilde{F}\) (see [28]).

3.2 Construction of a basis↩︎

In the following, we assume the Iwasawa main conjecture (Conjecture 7).

By the definition of \(\widetilde{\mathbf{R}}\Gamma_\mathfrak{p}(K,\mathbb{T})\) (see (7 )), we have a canonical isomorphism \[\label{can32det} {\det}_\Lambda^{-1}( \mathbf{R}\Gamma(G_{K,S}, \mathbb{T}) ) \otimes_\Lambda {\det}_\Lambda( \mathbf{R}\Gamma(K_\mathfrak{p}, \mathbb{T}) ) \otimes_\Lambda \bigotimes_{v\in S, v\nmid p}{\det}_\Lambda(\mathbf{R}\Gamma_{/{\rm ur}}(K_v,\mathbb{T})) \simeq {\det}_\Lambda^{-1}( \widetilde{\mathbf{R}}\Gamma_\mathfrak{p}(K,\mathbb{T})) .\tag{10}\]

The following is well-known.

By (10 ) and Lemma 2, we obtain a canonical isomorphism \[\left(\Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \right)\otimes_{\Lambda^{\rm ur}} \left(\Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda(\mathbf{R}\Gamma(K_\mathfrak{p}, \mathbb{T})) \right)\simeq \Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda^{-1}(\widetilde{\mathbf{R}}\Gamma_\mathfrak{p}(K,\mathbb{T})).\] Assuming the Iwasawa main conjecture (Conjecture 7), we define a \(\Lambda^{\rm ur}\)-basis \[\mathfrak{z}_S \in \Lambda^{\rm ur}\otimes_\Lambda {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\] to be the element such that \(\mathfrak{z}_S \otimes \varepsilon_\mathfrak{p}\) corresponds to \(\mathfrak{z}_\mathfrak{p}\) under the isomorphism above.

By Proposition 14, we can define an \(\mathcal{O}_F\)-basis \[\mathfrak{z}_S^\chi \in {\det}_{\mathcal{O}_F}^{-1}(\mathbf{R}\Gamma(G_{K,S},T))\] to be the image of \(\mathfrak{z}_S\) under the \(\chi\)-twisting map \[{\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \xrightarrow{a \mapsto a\otimes 1}{\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \otimes_{\Lambda,\chi} \mathcal{O}_F \simeq {\det}_{\mathcal{O}_F}^{-1}(\mathbf{R}\Gamma(G_{K,S},T)).\] (The last isomorphism is due to [28].)

Recall that in §2.1 we took an \(\mathcal{O}_F\)-basis \(\gamma \in {\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1)\) and an \(F\)-basis \(\delta^\ast \in {\bigwedge}_F^2 D_{{\rm dR}}(V^\ast(1))^\ast \simeq {\bigwedge}_F^2 D_{{\rm dR}}(V)\). Let \(\Omega_{\gamma,\delta} \in \mathbb{C}^\times\) be the period with respect to these bases (see (5 )). Theorem 9 is now reduced to the following.

The rest of this section is devoted to the proof of Theorem 15.

3.3 CM periods↩︎

We review the definitions of CM periods \(\Omega_\infty \in \mathbb{C}^\times\) and \(\Omega_p \in (\widehat \mathbb{Z}_p^{\rm ur})^\times\).

Let \(A\) be the canonical elliptic curve defined over \(K\) with complex multiplication by \(\mathcal{O}_K\) as in [48]. (Note that \(K\) has class number one and \(D_K\) is odd. Also, \(A\) descends to an elliptic curve over \(\mathbb{Q}\).) Fix a global minimal Weierstrass model of \(A\) over \(\mathcal{O}_K\) and let \(\omega_A \in \Gamma(A,\Omega_{A/K}^1)\) be the corresponding Néron differential. We also fix an \(\mathcal{O}_K\)-basis \(\gamma_A \in H_1(A(\mathbb{C}),\mathbb{Z})\). We define the complex CM period by \[\Omega_\infty:=\int_{\gamma_A}\omega_A.\]

Next, we define the \(p\)-adic CM period. Let \(\widehat A\) be the formal group of \(A\) over \(\mathcal{O}_{K_\mathfrak{p}}\simeq \mathbb{Z}_p\) with respect to the parameter \(-x/y\) (as in [49]). Let \(T_\mathfrak{p}(A)\) be the \(\mathfrak{p}\)-adic Tate module of \(A\). Then we have a canonical isomorphism \(H_1(A(\mathbb{C}),\mathbb{Z})\otimes_{\mathcal{O}_K}\mathcal{O}_{K_\mathfrak{p}} \simeq T_\mathfrak{p}(A)\). Also, note that the \(p\)-adic Tate module \(T_p(\widehat A)\) of \(\widehat A\) is identified with \(T_\mathfrak{p}(A)\) as \(G_{K_\mathfrak{p}}\)-representations. Thus we can regard \(\gamma_A \in H_1(A(\mathbb{C}),\mathbb{Z})\) as a \(\mathbb{Z}_p\)-basis of \(T_p(\widehat A)\), which we denote by \(\gamma_{A,p}\). Let \[\eta_A: \widehat \mathbb{G}_m \xrightarrow{\sim} \widehat A\] be the isomorphism of formal groups over \(\widehat \mathbb{Z}_p^{\rm ur}\) which corresponds (by [50]) to the isomorphism \[\mathbb{Z}_p(1)=T_p(\widehat \mathbb{G}_m)\xrightarrow{\sim} T_p(\widehat A); \;\xi \mapsto \gamma_{A,p}.\] Regarding \(\eta_A \in \widehat \mathbb{Z}_p^{\rm ur}[[X]]\), we define the \(p\)-adic CM period by \[\Omega_p:= \eta_A'(0).\] In other words, we have \[\eta_A^\ast (\omega_A) = \Omega_p\frac{dX}{1+X}.\]

For later purpose, we shall give another description of \(\Omega_p\).

3.4 Choice of bases↩︎

In order to compare various periods, we make specific choices of \(\gamma \in {\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1)\) and \(\delta^\ast \in {\bigwedge}_F^2 D_{{\rm dR}}(V)\).

Let \(\psi =\psi_A\) be the Hecke character associated with the CM elliptic curve \(A\) fixed in §3.3. We first need the following.

By Lemma 4, we can write \(\chi= \psi^j\overline{\psi}^{-j}\). Since \(\psi\overline{\psi}\) is the norm Hecke character \(\mathop{\mathrm{N}}\), we can also write \(\chi= \psi^{2j}\mathop{\mathrm{N}}^{-j}\). Note that the \(p\)-adic avatar of \(\mathop{\mathrm{N}}\) is the cyclotomic character \(\chi_{\rm cyc}\).

We choose \(\gamma \in {\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1)\) in the following way. Note that we have an isomorphism \[{\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1) \simeq \mathbb{Z}_p(1)\otimes_{\mathbb{Z}_p}\mathcal{O}_F( \chi^2) = T_\mathfrak{p}(A)^{\otimes 4j} \otimes_{\mathbb{Z}_p} \mathcal{O}_F(1-2j).\] Let \(\gamma_{A,p} \in T_\mathfrak{p}(A)=T_p(\widehat A)\) be the basis chosen in §3.3. We take \(\gamma \in {\bigwedge}_{\mathcal{O}_F}^2 T^\ast(1)\) to be the element corresponding to \(\gamma_{A,p}^{\otimes 4j} \otimes \xi^{\otimes (1-2j)}\) under the isomorphism above.

Next, we choose \(\delta^\ast \in {\bigwedge}_F^2 D_{\rm dR}(V)\). We similarly have \[\begin{align} \label{ddr32isom} {\bigwedge}_F^2 D_{\rm dR}(V) = D_{\rm dR}\left({\bigwedge}_F^2 V\right) &\simeq D_{\rm dR}\left(V_\mathfrak{p}(A)^{\otimes (-4j)} \otimes_{\mathbb{Q}_p} F(2j+1)\right) \\ &= D_{\rm dR}(V_\mathfrak{p}(A)^{\otimes (-4j)}) \otimes_{\mathbb{Q}_p} D_{\rm dR}(F(2j+1)) ,\nonumber \end{align}\tag{13}\] where we set \(V_\mathfrak{p}(A):=\mathbb{Q}_p\otimes_{\mathbb{Z}_p} T_\mathfrak{p}(A)\). Let \[\omega_{A,p}\in D_{\rm dR}(V_\mathfrak{p}(A)^\ast)\] be the image of \(1\otimes \omega_A\) under the canonical isomorphism \(K_\mathfrak{p}\otimes_K \Gamma(A,\Omega_{A/K}^1)\simeq D_{\rm dR}(V_\mathfrak{p}(A)^\ast)\). Explicitly, we have \[\omega_{A,p} = t_\xi \Omega_p \otimes \gamma_{A,p}^\ast\] by Proposition 16. We set \[e_k := t_\xi^{-k}\otimes \xi^k \in D_{\rm dR}(F(k))\] for any \(k\in \mathbb{Z}\). We define \(\delta^\ast \in {\bigwedge}_F^2 D_{\rm dR}(V)\) to be the element corresponding to \[\omega_{A,p}^{\otimes 4j} \otimes e_{2j+1} \in D_{\rm dR}(V_\mathfrak{p}(A)^{\otimes (-4j)}) \otimes_{\mathbb{Q}_p} D_{\rm dR}(F(2j+1))\] under the isomorphism (13 ). Explicitly, we have \[\delta^\ast = t_\xi^{2j-1} \Omega_p^{4j}\otimes \gamma_{A,p}^{\otimes (-4j)} \otimes \xi^{\otimes (2j+1)}\] as an element of \(B_{\rm dR}\otimes_{\mathbb{Q}_p} V_\mathfrak{p}(A)^{\otimes (-4j)} \otimes_{\mathbb{Q}_p} F(2j+1)\).

3.5 Comparison of periods↩︎

Let \(\gamma_T \in {\bigwedge}_{\mathcal{O}_F}^2 T\) be the \(\mathcal{O}_F\)-basis obtained from the fixed basis \(\gamma_\mathbb{T}\in {\bigwedge}_\Lambda^2 \mathbb{T}\) via \(\mathbb{T}\otimes_{\Lambda,\chi}\mathcal{O}_F\simeq T\). We define a \(p\)-adic period \[\Omega_{p, \gamma_T,\delta^\ast} \in \widetilde{F}^\times\] by \[\varepsilon_{\rm dR}(\delta^\ast)= \Omega_{p,\gamma_T,\delta^\ast}\cdot \gamma_T,\] where \(\varepsilon_{\rm dR}: \widetilde{F} \otimes_F {\bigwedge}_F^2 D_{{\rm dR}}(V)\xrightarrow{\sim} \widetilde{F} \otimes_F {\bigwedge}_F^2 V\) is the isomorphism defined in (9 ). We regard \(\Omega_{p,\gamma_T,\delta^\ast} \in \widehat F^{\rm ur}\) via the natural map \(\widetilde{F} = \widehat \mathbb{Q}_p^{\rm ur}\otimes_{\mathbb{Q}_p}F \to \widehat F^{\rm ur}\), where \(\widehat F^{\rm ur}\) denotes the completion of the maximal unramified extension of \(F\).

In the following, we write \[a\sim b\] if the equality \(a=b\) holds up to a unit in \(\mathcal{O}_F^\times\).

We now give a proof of Theorem 15.

4 Motives↩︎

In this article, we regard a “motive" as a collection of realizations and comparison isomorphisms. This is sufficient in order to formulate the Tamagawa number conjecture. In this appendix, we clarify the definition of motives in our sense. (Our treatment of motives is similar to [4], [5], where the category of”motivic structures" is considered.) We also review the formulation of the Tamagawa number conjecture in the critical case. In the last section we give basic examples of motives.

Throughout this appendix, let \(\mathcal{F}\) and \(K\) be number fields. Let \(S_p(\mathcal{F})\) be the set of \(p\)-adic places of \(\mathcal{F}\).

4.1 Motivic structures↩︎

Let \(d,w\in \mathbb{Z}\) with \(d\geq 0\). A (pure) motive \(M\) defined over \(K\) (of rank \(d\) and weight \(w\)) with coefficients in \(\mathcal{F}\) (denoted by \((M,\mathcal{F})\)) is the following collection of realizations and comparison isomorphisms. The realizations are the following.

• For each embedding \(\sigma: K\hookrightarrow \mathbb{C}\), we have the \(\sigma\)-Betti realization \[H_\sigma(M),\] which is an \(\mathcal{F}\)-vector space of dimension \(d\) endowed with a (pure) \(\mathbb{Q}\)-Hodge structure of weight \(w=w(M)\). For \(c \in G_\mathbb{R}=\mathop{\mathrm{Gal}}(\mathbb{C}/\mathbb{R})\), there is an isomorphism \[H_\sigma(M)\xrightarrow{\sim}H_{c\circ \sigma}(M),\] which is also denoted by \(c\). The Betti realization of \(M\) is defined by \[H_B(M):=\bigoplus_{\sigma:K\hookrightarrow \mathbb{C}}H_\sigma(M).\] The action of \(c \in G_\mathbb{R}\) on \(H_B(M)\) is defined by \(c\cdot (a_\sigma)_\sigma:=(c\cdot a_\sigma)_{c\circ \sigma}\).

• For each \(\lambda \in S_p(\mathcal{F})\), we have the \(\lambda\)-adic realization \[V_\lambda(M),\] which is an \(\mathcal{F}_\lambda\)-vector space of dimension \(d\) endowed with a continuous \(\mathcal{F}_\lambda\)-linear \(G_K\)-action, unramified outside a finite set \(S\) of places of \(K\). The \(p\)-adic realization of \(M\) is defined by \[V_p(M):=\bigoplus_{\lambda \in S_p(\mathcal{F})} V_\lambda(M).\]

• We have the de Rham realization \[H_{\rm dR}(M),\] which is a free \(\mathcal{F}\otimes_\mathbb{Q}K\)-module of rank \(d\) endowed with a decreasing filtration \(\{ {\rm Fil}^iH_{\rm dR}(M)\}_{i\in \mathbb{Z}}\) such that \({\rm Fil}^i H_{\rm dR}(M)=H_{\rm dR}(M)\) for \(i \ll 0\) and \({\rm Fil}^i H_{\rm dR}(M)=0\) for \(i \gg 0\). The tangent space of \(M\) is defined by \[t(M):=H_{\rm dR}(M)/{\rm Fil}^0H_{\rm dR}(M).\]

These realizations have the following comparison isomorphisms.

• For each \(\sigma:K\hookrightarrow \mathbb{C}\), we have \[\mathbb{C}\otimes_\mathbb{Q}H_\sigma(M) \simeq \mathbb{C}\otimes_{K,\sigma}H_{\rm dR}(M)\] which is compatible with the Hodge structure. This induces an isomorphism \[\mathbb{C}\otimes_\mathbb{Q}H_B(M)\simeq \mathbb{C}\otimes_\mathbb{Q}H_{\rm dR}(M).\]

• For each \(\sigma: K\hookrightarrow \mathbb{C}\) and \(\lambda \in S_p(\mathcal{F})\), we have \[\mathcal{F}_\lambda \otimes_\mathcal{F}H_\sigma(M)\simeq V_\lambda(M).\] This induces an isomorphism \[\mathbb{Q}_p\otimes_\mathbb{Q}H_\sigma(M)\simeq V_p(M).\]

• For each \(\sigma: K\hookrightarrow \overline{\mathbb{Q}}_p\), we have \[B_{\rm dR}\otimes_{K,\sigma} H_{\rm dR}(M)\simeq B_{\rm dR}\otimes_{\mathbb{Q}_p} V_p(M)\] which is compatible with filtration. For each \(p\)-adic place \(\mathfrak{p}\) of \(K\), this induces an isomorphism \[K_\mathfrak{p}\otimes_K H_{\rm dR}(M)\simeq D_{{\rm dR},\mathfrak{p}}(V_p(M)):= H^0(K_\mathfrak{p}, B_{\rm dR}\otimes_{\mathbb{Q}_p}V_p(M)).\]

We assume that \(M\) satisfies the following hypothesis. Fix a prime number \(p\) and \(\lambda \in S_p(\mathcal{F})\). For a finite place \(v\) of \(K\), let \({\rm Fr}_v\) denote the arithmetic Frobenius of \(v\) and \(I_v \subset G_{K_v}\) the inertia subgroup of \(v\). If \(v\mid p\), we set \(D_{{\rm cris},v}(-):=H^0(K_v,B_{\rm cris}\otimes_{\mathbb{Q}_p}-)\) and we write \(\varphi\) for the Frobenius acting on it. We set \[P_v(M,x):= \begin{cases} \det(1-{\rm Fr}_v^{-1} x \mid V_\lambda(M)^{I_v}) &\text{if v\nmid p,}\\ \det(1-\varphi x \mid D_{{\rm cris},v}(V_\lambda(M))) &\text{if v\mid p.} \end{cases}\] Note that \(P_v(M,x) \in \mathcal{F}_\lambda[x]\). We say that a finite place \(v\) of \(K\) is good if \(V_\lambda(M)\) is unramified at \(v\) (resp. crystalline at \(v\)) when \(v\nmid p\) (resp. \(v\mid p\)).

Finally, we define a critical motive. We set \[H_B(M)^+:=H^0(\mathbb{R}, H_B(M)).\] The period map for \(M\) \[\alpha_M: \mathbb{R}\otimes_\mathbb{Q}H_B(M)^+\to \mathbb{R}\otimes_\mathbb{Q}t(M)\] is defined to be the map induced by the comparison isomorphism \(\mathbb{C}\otimes_\mathbb{Q}H_B(M)\simeq \mathbb{C}\otimes_\mathbb{Q}H_{\rm dR}(M)\).

4.2 Motivic cohomology↩︎

Let \((M,\mathcal{F})\) be a motive defined over \(K\) with coefficients in \(\mathcal{F}\). We have the following standard conjectures.

4.3 The Deligne conjecture↩︎

We review the Deligne conjecture [29], which is a special case of the “rationality" part of the Tamagawa number conjecture (see [6]).

Assume that \(M\) is critical. Then the period map is an isomorphism: \[\alpha_M: \mathbb{R}\otimes_\mathbb{Q}H_B(M)^+ \xrightarrow{\sim} \mathbb{R}\otimes_\mathbb{Q}t(M).\] Let \[\det(\alpha_M) : \mathbb{R}\otimes_\mathbb{Q}{\det}_\mathcal{F}(H_B(M)^+) \xrightarrow{\sim} \mathbb{R}\otimes_\mathbb{Q}{\det}_\mathcal{F}(t(M))\] be the induced isomorphism. Take \(\mathcal{F}\)-bases \[\gamma \in {\det}_\mathcal{F}(H_B(M)^+) \text{ and }\delta \in {\det}_\mathcal{F}(t(M)).\] We define the period \[\Omega_{\gamma,\delta} \in (\mathbb{R}\otimes_\mathbb{Q}\mathcal{F})^\times\] with respect to \(\gamma\) and \(\delta\) by \[{\det}(\alpha_M)(\gamma)=\Omega_{\gamma,\delta}\cdot \delta.\] We regard \[\Omega_{\gamma,\delta}\in \mathbb{C}^\times\] via the map \(\mathbb{R}\otimes_\mathbb{Q}\mathcal{F}\to \mathbb{C}; \;a\otimes b\mapsto ab\).

Note that the validity of Conjecture 26 is obviously independent of the choices of \(\gamma\) and \(\delta\). Note also that the conjecture is trivial when \(L(M,0)=0\).

We shall state a general version of the Deligne conjecture, which treats the case \(L(M,0)=0\). Let \[L^\ast(M,0):=\underset{s\to 0}{\lim} s^{-\mathrm{ord}_{s=0}L(M,s)} L(M,s)\] be the leading term of \(L(M,s)\) at \(s=0\). Assuming Conjecture 24, let \[\det(h_M): \mathbb{R}\otimes_\mathbb{Q}\left( {\det}_\mathcal{F}(H^1_f(K,M)) \otimes_\mathcal{F}{\det}_\mathcal{F}(H^1_f(K,M^\ast(1)))\right)\xrightarrow{\sim}\mathbb{R}\otimes_\mathbb{Q}\mathcal{F}\] be the isomorphism induced by the height pairing \(h_M\). Take \(\mathcal{F}\)-bases \[x \in {\det}_\mathcal{F}(H^1_f(K,M)) \text{ and }y \in {\det}_\mathcal{F}(H^1_f(K,M^\ast(1)))\] and define the regulator with respect to \(x\) and \(y\) by \[R_{x,y}:= {\det}(h_M)(x\otimes y) \in (\mathbb{R}\otimes_\mathbb{Q}\mathcal{F})^\times.\] We regard \[R_{x,y}\in \mathbb{C}^\times\] via the map \(\mathbb{R}\otimes_\mathbb{Q}\mathcal{F}\to \mathbb{C}; \;a\otimes b\mapsto ab\).

The validity of Conjecture 27 is independent of the choices of \(\gamma,\delta,x,y\).

4.4 The Tamagawa number conjecture in the critical case↩︎

We review the formulation of the Tamagawa number conjecture.

The following is [6].

We shall formulate the “integrality" part of the Tamagawa number conjecture (see [6]). We only treat the critical case, since we do not consider non-critical cases in this article.

Fix an odd prime number \(p\) and \(\lambda \in S_p(\mathcal{F})\). We set \[F:=\mathcal{F}_\lambda \text{ and }V:=V_\lambda(M^\ast(1)).\] Note that \(V_\lambda(M)=V^\ast(1)\). Take a finite set \(S\) of places of \(K\) containing all the infinite places, \(p\)-adic places, and the places at which \(V\) ramify. Let \(G_{K,S}\) be the Galois group of the maximal extension of \(K\) unramified outside \(S\). We fix a \(G_K\)-stable \(\mathcal{O}_F\)-lattice \(T\subset V\). Recall that we set \[P_v(M,x):= \begin{cases} \det(1-{\rm Fr}_v^{-1} x \mid V^\ast(1)^{I_v}) &\text{if v\nmid p,}\\ \det(1-\varphi x \mid D_{{\rm cris},v}(V^\ast(1))) &\text{if v\mid p.} \end{cases}\]

We formulate the Tamagawa number conjecture under the following hypothesis.

We use the Poitou-Tate exact sequence \[\begin{gather} \label{TNCPT} 0\to H^1_f(K,V)\to H^1(G_{K,S},V)\to \bigoplus_{v\in S}H^1_{/f}(K_v,V)\\ \to H^1_f(K,V^\ast(1))^\ast \to H^2(G_{K,S},V)\to \bigoplus_{v\in S_f}H^0(K_v,V^\ast(1))^\ast. \end{gather}\tag{16}\] (Here \(S_f\subset S\) denotes the subset of finite places.) Note that the last term vanishes by Hypothesis 30(ii).

We first consider the case \(H^1_f(K,V)=H^1_f(K,V^\ast(1))=0\). (According to Conjecture 28, this is the case when the “analytic rank" is zero, i.e., \(L(M,0)\neq 0\).) In this case, (16 ) implies that \(H^2(G_{K,S},V)=0\) and that the localization maps at \(p\)-adic places induce an isomorphism \[{\rm loc}_p: H^1(G_{K,S},V) \xrightarrow{\sim} \bigoplus_{v \in S_p(K)}H^1_{/f}(K_v,V),\] where \(S_p(K)\) denotes the set of \(p\)-adic places of \(K\). (Note that Hypothesis 30(ii) implies \(H^1_{/f}(K_v,V)\simeq H^1_f(K_v,V^\ast(1))^\ast=0\) for \(v\nmid p\).) Also, since Hypothesis 30(ii) implies \(D_{{\rm cris},v}(V^\ast(1))^{\varphi=1}=0\) for \(v\in S_p(K)\), the dual exponential map induces an isomorphism \[\exp_v^\ast: H^1_{/f}(K_v,V)\xrightarrow{\sim} D_{{\rm dR},v}^0(V).\] Note that we have a canonical isomorphism \[D_{{\rm dR},v}^0(V)\simeq \left(D_{{\rm dR},v}(V^\ast(1))/D^0_{{\rm dR},v}(V^\ast(1))\right)^\ast.\] Combining this with the comparison isomorphism \[F\otimes_\mathcal{F}t(M)\simeq \bigoplus_{v\in S_p(K)} D_{{\rm dR},v}(V^\ast(1))/D_{{\rm dR},v}^0(V^\ast(1)),\] we can regard \(\bigoplus_{v\in S_p(K)}\exp_v^\ast\) as an isomorphism \[\exp^\ast: \bigoplus_{v\in S_p(K)}H^1_{/f}(K_v,V)\xrightarrow{\sim} F\otimes_\mathcal{F}t(M)^\ast.\] For an \(\mathcal{F}\)-basis \(\delta \in {\det}_\mathcal{F}(t(M))\), let \(\delta^\ast \in {\det}_\mathcal{F}(t(M)^\ast)\simeq {\det}_\mathcal{F}(t(M))^\ast\) denote the dual basis.

Note that by \(H^2(G_{K,S},V)=0\) and Hypothesis 30(iii) we have an identification \[{\det}_F^{-1}(\mathbf{R}\Gamma(G_{K,S},V))= {\det}_F(H^1(G_{K,S}, V)) .\] Consider the composition map \[\vartheta_0: {\det}_F^{-1}(\mathbf{R}\Gamma(G_{K,S},V))= {\det}_F(H^1(G_{K,S}, V)) \stackrel{{\rm loc}_p}{\simeq} {\det}_F\left(\bigoplus_{v\in S_p(K)} H^1_{/f}(K_v,V)\right) \stackrel{{\rm exp^\ast}}{\simeq} F\otimes_\mathcal{F}{\det}_\mathcal{F}(t(M)^\ast).\]

We set \[L_S(M,s):= \left(\prod_{v\in S}P_v(M, {\mathop{\mathrm{N}}}v^{-s})\right) L(M,s).\] Note that the comparison isomorphism \(F\otimes_\mathcal{F}H_B(M)\simeq \bigoplus_{\sigma:K\hookrightarrow \mathbb{C}} V^\ast(1)\) induces an isomorphism \[F\otimes_\mathcal{F}H_B(M)^+\simeq \bigoplus_{v\in S_\infty(K)} H^0(K_v, V^\ast(1)),\] where \(S_\infty(K)\) denotes the set of infinite places of \(K\).

We shall next formulate the Tamagawa number conjecture in arbitrary analytic rank.

Under Hypothesis 30, the Poitou-Tate exact sequence (16 ) and the dual exponential map \(\exp^\ast\) induce an isomorphism \[{\det}_F^{-1}(\mathbf{R}\Gamma(G_{K,S},V)) \simeq {\det}_F(H^1_f(K,V^\ast(1))) \otimes_F {\det}_F(H^1_f(K,V)) \otimes_\mathcal{F}{\det}_\mathcal{F}(t(M)^\ast).\] Combining this with the isomorphisms in Conjecture 23, we obtain an isomorphism \[\vartheta: {\det}_F^{-1}(\mathbf{R}\Gamma(G_{K,S},V)) \simeq F\otimes_\mathcal{F}\left({\det}_\mathcal{F}(H^1_f(K,M)) \otimes_\mathcal{F}{\det}_\mathcal{F}(H^1_f(K,M^\ast(1))) \otimes_\mathcal{F}{\det}_\mathcal{F}(t(M)^\ast) \right).\] Take \(\mathcal{F}\)-bases \[x \in {\det}_\mathcal{F}(H^1_f(K,M)) \text{ and }y \in {\det}_\mathcal{F}(H^1_f(K,M^\ast(1))).\]

4.5 Examples↩︎

We give some basic examples of motives.

4.5.1 Tate motives↩︎

For a number field \(K\) and an integer \(j\in \mathbb{Z}\), there is a Tate motive \(M=h^0(\mathop{\mathrm{Spec}}K)(j)\). This is a motive defined over \(K\) of rank one and weight \(-2j\) with coefficients in \(\mathbb{Q}\). The realizations are the following.

• For an embedding \(\sigma: K \hookrightarrow \mathbb{C}\), the \(\sigma\)-Betti realization is \[H_\sigma(M):=\mathbb{Q}(j):=(2\pi i)^j\mathbb{Q}.\]

• The \(p\)-adic realization is \[V_p(M):=\mathbb{Q}_p(j).\]

• The de Rham realization is \[H_{\rm dR}(M):=K\] with filtration \[{\rm Fil}^i H_{\rm dR}(M) := \begin{cases} K &\text{if i\leq -j,}\\ 0&\text{if i>-j.} \end{cases}\]

The comparison isomorphisms are naturally defined. Note that \(M^\ast(1)\) is identified with \(h^0(\mathop{\mathrm{Spec}}K)(1-j)\). Hypothesis 20 is satisfied. The \(L\)-function of \(M\) is \(L(M,s)=\zeta_K(s+j)\), where \(\zeta_K(s)\) denotes the Dedekind zeta function of \(K\). Conjecture 23 is true with \[H^0_f(K,M):=\begin{cases} \mathbb{Q}&\text{if j=0,}\\ 0&\text{if j\neq 0,} \end{cases} \text{ and }H^1_f(K,M):=\begin{cases} \mathbb{Q}\otimes_\mathbb{Z}\mathcal{O}_K^\times &\text{if j=1,}\\ \mathbb{Q}\otimes_\mathbb{Z}K_{2j-1}(K) &\text{if j>1,}\\ 0&\text{if j\leq 0.} \end{cases}\] (The case \(j>1\) is due to the Voevodsky-Rost theorem.) Conjecture 28 is true by Borel’s theorem.

\(M\) is critical if and only if \(K\) is totally real and either \(j\) is negative odd or positive even. Conjecture 24 is trivially true since \(H^1_f(K,M)=H^1_f(K,M^\ast(1))=0\) in this case. Conjecture 26 (which is the same as Conjecture 27 in this case) is true by the Klingen-Siegel theorem. Conjecture 33 (which is the same as Conjecture 35) for negative odd \(j\) is equivalent to the Lichtenbaum conjecture, which is proved by Wiles [53]. Conjecture 33 for positive even \(j\) can be proved when \(p\) is unramified in \(K\) by using the functional equation (see [54]).

When \(K\) is abelian over \(\mathbb{Q}\), the Tamagawa number conjecture (and its equivariant refinement) is proved by Burns-Greither [7], Huber-Kings [8], and Burns-Flach [55].

4.5.2 Artin motives↩︎

Let \(\chi: G_K\to \mathbb{C}^\times\) be a finite order character. Suppose that \(\chi\) takes values in a number field \(\mathcal{F}\). Then there is an Artin motive \(M=M(\chi)\) defined over \(K\) of rank one and weight zero with coefficients in \(\mathcal{F}\). The realizations are the following.

• For an embedding \(\sigma: K \hookrightarrow \mathbb{C}\), the \(\sigma\)-Betti realization is \[H_\sigma(M):=\mathcal{F}.\]

• For \(\lambda \in S_p(\mathcal{F})\), the \(\lambda\)-adic realization is \[V_\lambda(M):=\mathcal{F}_\lambda(\chi),\] i.e., \(V_\lambda(M)\) is a one dimensional \(\mathcal{F}_\lambda\)-vector space on which \(G_K\) acts via \(\chi\).

• The de Rham realization is \[H_{\rm dR}(M):=\mathcal{F}\otimes_\mathbb{Q}K\] with filtration \[{\rm Fil}^i H_{\rm dR}(M) := \begin{cases} \mathcal{F}\otimes_\mathbb{Q}K &\text{if i\leq 0,}\\ 0&\text{if i>0.} \end{cases}\]

If \(\chi\) is the trivial character and \(\mathcal{F}=\mathbb{Q}\), then we have \(M=h^0(\mathop{\mathrm{Spec}}K)\). The \(L\)-function of \(M\) is the Artin \(L\)-function \(L(M,s)=L(\chi^{-1},s)\). When \(K=\mathbb{Q}\), the Tamagawa number conjecture for \(M(j)\) (for any \(j\in \mathbb{Z}\)) is proved by Huber-Kings [8].

4.5.3 Elliptic curves↩︎

Let \(E\) be an elliptic curve defined over a number field \(K\). Then we can consider the critical motive \(M=h^1(E/K)(1)\) of rank two and weight \(-1\) with coefficients in \(\mathbb{Q}\). The realizations are the following.

• For an embedding \(\sigma: K \hookrightarrow \mathbb{C}\), the \(\sigma\)-Betti realization is \[H_\sigma(M):=H_1(E^\sigma(\mathbb{C}),\mathbb{Q})\simeq H^1(E^\sigma(\mathbb{C}),\mathbb{Q}(1)).\] Here we set \(E^\sigma:= E\times_{K,\sigma}\mathbb{C}\).

• The \(p\)-adic realization is \[V_p(M):=H^1_{\text{\'et}}(E\times_K \overline{\mathbb{Q}}, \mathbb{Q}_p(1)).\] (This is canonically isomorphic to \(V_p(E):=\mathbb{Q}_p\otimes_{\mathbb{Z}_p} T_p(E)\).)

• The de Rham realization is \[H_{\rm dR}(M):=H^1_{\rm dR}(E/K)\] with filtration \[{\rm Fil}^i H_{\rm dR}(M) := {\rm Fil}^{i+1} H^1_{\rm dR}(E/K)=\begin{cases} H^1_{\rm dR}(E/K) &\text{if i<0,}\\ \Gamma(E,\Omega_{E/K}^1) &\text{if i=0,}\\ 0 &\text{if i>0.} \end{cases}\]

The comparison isomorphisms are well-known. Hypothesis 20(i) and (ii) are satisfied. The \(L\)-function of \(M\) is \(L(M,s)=L(E/K,s+1)\), where \(L(E/K,s)\) is the Hasse-Weil \(L\)-function for \(E/K\). Hypothesis 20(iii) is not known in general: when \(K=\mathbb{Q}\), it is a consequence of the Shimura-Taniyama conjecture proved by Wiles et al. Conjecture 23 is true with \[H^0_f(K,M):=0\text{ and }H^1_f(K,M):=\mathbb{Q}\otimes_\mathbb{Z}E(K)\] if the \(p\)-part of the Tate-Shafarevich group \(\russ\char 88(E/K)\) is finite. Conjecture 24 is satisfied with the Néron-Tate height pairing. It is known that the Tamagawa number conjecture for \((h^1(E/K)(1),\mathbb{Z}_p)\) (Conjecture 35) is equivalent to the \(p\)-part of the Birch and Swinnerton-Dyer formula. (See Proposition 46 in the case of analytic rank one.)

We remark that \(M\) is self-dual, i.e., \(M^\ast(1)=M\). The meaning of this is that each realization of \(M^\ast(1)\) is canonically isomorphic to that of \(M\).

4.5.4 Algebraic varieties↩︎

Let \(X\) be a smooth projective variety defined over \(K\). For \(n, j \in \mathbb{Z}\) with \(n\geq 0\), there is a motive \(M=h^n(X)(j)\) defined over \(K\) of weight \(n-2j\) with coefficients in \(\mathbb{Q}\). The realizations are the following.

• For an embedding \(\sigma: K \hookrightarrow \mathbb{C}\), the \(\sigma\)-Betti realization is \[H_\sigma(M):=H^n(X^\sigma(\mathbb{C}),\mathbb{Q}(j)).\]

• The \(p\)-adic realization is \[V_p(M):=H^n_{\text{\'et}}(X\times_K \overline{\mathbb{Q}}, \mathbb{Q}_p(j)).\]

• The de Rham realization is \[H_{\rm dR}(M):=H^n_{\rm dR}(X/K)\] with filtration \[{\rm Fil}^i H_{\rm dR}(M) := {\rm Fil}^{i+j}H^n_{\rm dR}(X/K).\]

These realizations satisfy the axioms of Weil cohomology and have the well-known comparison isomorphisms. Properties of the \(L\)-function \(L(M,s)\) are highly conjectural. For possible definitions of the motivic cohomology, see [4], [6] for example.

The dual of \(M\) is described as follows. Let \(d:= \dim X\) and suppose \(0\leq n\leq 2d\). Then we have \(h^{2d}(X)(d) = h^0(\mathop{\mathrm{Spec}}K)\) and the Poincaré duality pairing \[h^n(X) \times h^{2d-n}(X) \to h^{2d}(X)=h^0(\mathop{\mathrm{Spec}}K)(-d)\] induces an identification \(h^n(X)^\ast= h^{2d-n}(X)(d)\). Hence we have \(M^\ast(1)= h^{2d-n}(X)(d+1-j)\). Also, by the hard Lefschetz theorem, we have \(M^\ast(1)=h^n(X)(n+1-j)\).

4.5.5 Modular forms↩︎

Let \(f=\sum_{n=1}^\infty a_n q^n \in S_{2r}(\Gamma_0(N))\) be a normalized newform of weight \(2r\) and level \(N\). We set \(\mathcal{F}:=\mathbb{Q}(\{a_n\}_n)\), which is a totally real number field. Then there is a motive \(M_f\) attached to \(f\), which is defined over \(\mathbb{Q}\) of rank two and weight \(2r-1\) with coefficients in \(\mathcal{F}\). The motive \(M_f\) was first constructed by Scholl [56]. We shall describe its realizations, following [57].

Let \(\mathcal{E}_N \to X(N)\) be the universal generalized elliptic curve. The Kuga-Sato variety \(X:=\widetilde{\mathcal{E}}_N^{2r-2}\) is defined to be the canonical desingularization of \(\mathcal{E}_N^{2r-2}\) described in [58]. Scholl constructed a certain projector \(\Pi\) which acts on the cohomology of \(X\) (see [57]). The motive \(M_f\) is defined to be \(\Pi \cdot h^{2r-1}(X)\), i.e., the realizations are the following.

• The Betti realization is \[H_B(M_f):=\Pi\cdot H^{2r-1}(X(\mathbb{C}), \mathcal{F}).\]

• For \(\lambda \in S_p(\mathcal{F})\), the \(\lambda\)-adic realization is \[V_\lambda(M_f):= \Pi \cdot H_{\text{\'et}}^{2r-1}(X\times_\mathbb{Q}\overline{\mathbb{Q}}, \mathcal{F}_\lambda).\]

• The de Rham realization is \[H_{\rm dR}(M_f):=\Pi\cdot H_{\rm dR}^{2r-1}(X/\mathbb{Q})\otimes_\mathbb{Q}\mathcal{F}\] with the usual filtration.

The comparison isomorphisms are induced by those for \(h^{2r-1}(X)\). Hypothesis 20 is satisfied. The \(L\)-function of \(M_f\) is \(L(M_f,s)=L(f,s):=\sum_{n=1}^\infty a_n n^{-s}\). The motive \(M_f(j)\) is critical when \(1\leq j\leq 2r-1\). In this article, we are interested in the “central critical twist" \(M:=M_f(r)\), which is self-dual. Conjectures 23 and 24 for \(M\) are not known in general: see [57]. Conjecture 26 for \(M\) is known modulo comparison of periods (due to Shimura [42], [59]). Kato essentially proves in [60] that Conjecture 33 for \(M\) is implied by the Iwasawa main conjecture for \(f\) (see [60]). Conjecture 35 for \(M\) in analytic rank one is studied by Longo-Vigni [57].

4.5.6 Hecke motives↩︎

We first review the definition and basic properties of Hecke characters.

Let \(K\) be an imaginary quadratic field. Let \(\mathbb{A}_K^\times\) be the idèle group of \(K\). Let \(\widehat K^\times:= (\widehat \mathbb{Z}\otimes_\mathbb{Z}K)^\times\) be the finite idèle group. Note that \(\mathbb{A}_K^\times= \mathbb{C}^\times \times \widehat K^\times\).

Let \(k,\ell \in \mathbb{Z}\). A Hecke character of \(K\) of infinity type \((k,\ell)\) is a continuous homomorphism \(\chi: \mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times\) such that its restriction on \(\mathbb{C}^\times\) is given by \(z\mapsto z^{-k}\overline{z}^{-\ell}\). (Note that our sign convention is opposite to [60], [15], but agrees with [49], [22].) We regard \(\chi\) as a map \(\mathbb{A}_K^\times \to \mathbb{C}^\times\) which is trivial on \(K^\times\).

\(\chi\) is called anticyclotomic if \(\chi\) is trivial on \(\mathbb{A}_\mathbb{Q}^\times\). Note that, if \(\chi\) is anticyclotomic, then its infinity type is of the form \((k,-k)\).

We say that \(\chi\) takes values in a number field \(\mathcal{F}\) if \(\chi(\widehat K^\times)\subset \mathcal{F}\). We set \(\widehat \mathcal{O}_K^\times:=\prod_{v<\infty}\mathcal{O}_{K_v}^\times\). The conductor of \(\chi\) is defined to be the largest ideal \(\mathfrak{f}\) of \(\mathcal{O}_K\) such that the restriction of \(\chi\) on \(\widehat \mathcal{O}_K^\times\) factors through \((\mathcal{O}_K/\mathfrak{f})^\times\). (Note that “\(\mathfrak{f}\) is larger than \(\mathfrak{g}\)" means \(\mathfrak{f}\mid \mathfrak{g}\).)

Let \(I_K\) be the group of fractional ideals of \(K\). Let \(i_K: \widehat K^\times \to I_K; \;z \mapsto \prod_\mathfrak{p}\mathfrak{p}^{\mathrm{ord}_\mathfrak{p}(z_\mathfrak{p})}\) be the natural surjection. Let \(I_{K,\mathfrak{f}}\subset I_K\) be the group of fractional ideals of \(K\) prime to \(\mathfrak{f}\). For a Hecke character \(\chi\) of conductor \(\mathfrak{f}\) which takes values in \(\mathcal{F}\), we define \[\widetilde{\chi}: I_{K,\mathfrak{f}} \to \mathcal{F}^\times\] in the following way: for \(\mathfrak{a} \in I_{K,\mathfrak{f}}\), choose \(z \in \widehat K^\times\) such that \(i_K(z)=\mathfrak{a}\) and \(z_\mathfrak{p}\equiv 1\) (mod \(\mathfrak{p}^{\mathrm{ord}_\mathfrak{p}(\mathfrak{f})}\)) for any \(\mathfrak{p}\mid \mathfrak{f}\), and define \(\widetilde{\chi}(\mathfrak{a}):=\chi(z)\). (One checks that this is well-defined.) Note that, for a principal ideal \((a) \in I_{K,\mathfrak{f}}\) such that \(a \equiv 1\) (mod \(\mathfrak{f}\)), we have \[\widetilde{\chi} ((a))= \chi|_{\widehat K^\times}(a) = \chi|_{\mathbb{C}^\times}(a)^{-1}=a^k\overline{a}^{\ell}.\] Conversely, for a given character \(\widetilde{\chi}: I_{K,\mathfrak{f}}\to \mathcal{F}^\times\) such that \(\widetilde{\chi}((a)) = a^k\overline{a}^\ell\) for \(a\in K^\times\) with \(a\equiv 1\) (mod \(\mathfrak{f}\)), one can naturally construct a Hecke character \(\chi\) of infinity type \((k,\ell)\) (see [61]). Via this correspondence, we often identify \(\widetilde{\chi}\) with \(\chi\). In particular, we write \(\chi(\mathfrak{a})\) instead of \(\widetilde{\chi}(\mathfrak{a})\) for \(\mathfrak{a}\in I_{K,\mathfrak{f}}\). As usual, we set \(\chi(\mathfrak{a}):=0\) if \(\mathfrak{a}\) is not prime to the conductor \(\mathfrak{f}\).

The Hecke \(L\)-function for \(\chi\) is defined by \[L(\chi,s):=\sum_{\mathfrak{a}}\frac{ \chi(\mathfrak{a})}{{\mathop{\mathrm{N}}}\mathfrak{a}^s},\] where \(\mathfrak{a}\) runs over all non-zero integral ideals of \(K\) and we set \({\mathop{\mathrm{N}}}\mathfrak{a}:=\#(\mathcal{O}_K/\mathfrak{a})\). We have an expression by the Euler product \[L(\chi,s)= \prod_\mathfrak{p}(1-\chi(\mathfrak{p}){\mathop{\mathrm{N}}}\mathfrak{p}^{-s})^{-1},\] where \(\mathfrak{p}\) runs over all primes of \(K\).

Let \({\rm rec}_K: \mathbb{A}_K^\times/K^\times \to G_K^{\rm ab}:=\mathop{\mathrm{Gal}}(K^{\rm ab}/K)\) denote the (arithmetically normalized) global reciprocity map. (\(K^{\rm ab}\) denotes the maximal abelian extension of \(K\).) A Hecke character of infinity type \((0,0)\) is identified with a finite order character \(G_K\to \mathbb{C}^\times\) via \({\rm rec}_K\). For a fixed embedding \(\iota_p: \overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p\), we define the \(p\)-adic avatar \(\chi_p:G_K \to \overline{\mathbb{Q}}_p^\times\) of \(\chi\) by \[\chi_p({\rm rec}_K(z)) = \iota_p(\chi(z))z_\mathfrak{p}^{-k}z_{\overline{\mathfrak{p}}}^{-\ell},\] where \(z \in \widehat K^\times\) and \(\mathfrak{p}\) denotes the prime of \(K\) corresponding to \(\iota_p\). By an abuse of notation, we often denote \(\chi_p\) by \(\chi\).

The most basic example of Hecke characters is the norm Hecke character \[\mathop{\mathrm{N}}: \mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times; \;z \mapsto \prod_v |z_v|_v^{-1},\] where \(v\) runs over all places of \(K\) and \(|\cdot|_v: K_v^\times \to \mathbb{R}_{>0}\) denotes the normalized absolute value. The infinity type of \(\mathop{\mathrm{N}}\) is \((1,1)\). The conductor of \(\mathop{\mathrm{N}}\) is \((1)\), and the corresponding character of \(I_K\) is given by \[\widetilde{\mathop{\mathrm{N}}}: I_K \to \mathbb{Q}^\times; \;\mathfrak{a} \mapsto {\mathop{\mathrm{N}}}\mathfrak{a}.\] The \(p\)-adic avatar of \(\mathop{\mathrm{N}}\) is the cyclotomic character \(\chi_{\rm cyc}: G_K \to \mathbb{Z}_p^\times\).

Assuming that \(K\) has class number one, we fix an elliptic curve \(A\) defined over \(K\) with complex multiplication by \(\mathcal{O}_K\). Let \(\psi=\psi_A\) be the associated Hecke character of infinity type \((1,0)\), which takes values in \(K\) (see [62]). A prime \(\mathfrak{p}\) of \(K\) divides the conductor of \(\psi\) if and only if \(A\) has bad reduction at \(\mathfrak{p}\) (see [49]). We have \(\psi \overline{\psi} = \mathop{\mathrm{N}}\). If \(\mathfrak{p}\) is the prime of \(K\) corresponding to \(\iota_p\), then \(G_K\) acts on the \(\mathfrak{p}\)-adic Tate module \(T_\mathfrak{p}(A):=\varprojlim_n A[\mathfrak{p}^n]\) via \(\psi\).

We now construct a motive. For a Hecke character \(\chi\) of infinity type \((k,\ell)\) which takes values in \(\mathcal{F}\), there is a motive \(M(\chi)\) attached to \(\chi\) defined over \(K\) of rank one and weight \(-k-\ell\) with coefficients in \(\mathcal{F}\). We shall describe \(M(\chi)\).

By the observations above, it is natural to define \(M(\mathop{\mathrm{N}}):= h^0(\mathop{\mathrm{Spec}}K)(1)\) and \(M(\psi):=h^1(A)(1)\). For the general case, we write \(\chi=\chi_0 \psi^k \overline{\psi}^\ell = \chi_0 \psi^{k-\ell} \mathop{\mathrm{N}}^\ell\) with a finite order character \(\chi_0\). Then we define \[M(\chi):= M(\chi_0)\otimes (h^1(A)(1))^{\otimes (k-\ell)}(\ell),\] where \(M(\chi_0)\) is the Artin motive (see §4.5.2). By \(\chi^{-1}\mathop{\mathrm{N}}=\overline{\chi} \mathop{\mathrm{N}}^{1-k-\ell}\), we see that \(M(\chi)^\ast(1)=M(\overline{\chi})(1-k-\ell)\).

In this article, we mainly consider Hecke characters of the form \(\chi= \psi^k\overline{\psi}^\ell\) with \(k>\ell\). In this case, \(M(\chi)\) has coefficients in \(K\), and the realizations are given as follows.

• For an embedding \(\sigma: K \hookrightarrow \mathbb{C}\), the \(\sigma\)-Betti realization is \[H_\sigma(M(\chi)):=H_1(A^\sigma(\mathbb{C}),\mathbb{Q})^{\otimes(k-\ell)}(\ell).\] (\(H_1(A^\sigma(\mathbb{C}),\mathbb{Q})\) is a one-dimensional \(K\)-vector space and the tensor product is taken over \(K\).)

• For \(\mathfrak{p}\in S_p(K)\), the \(\mathfrak{p}\)-adic realization is \[V_\mathfrak{p}(M(\chi)):=V_\mathfrak{p}(A)^{\otimes (k-\ell)}(\ell).\] (The tensor product is taken over \(K_\mathfrak{p}\).) Here we set \(V_\mathfrak{p}(A):= K_\mathfrak{p}\otimes_{\mathcal{O}_{K_\mathfrak{p}}} T_\mathfrak{p}(A)\).

• The de Rham realization is \[H_{\rm dR}(M(\chi)):=H^1_{\rm dR}(A/K)^{\otimes (k-\ell)}.\] (\(H^1_{\rm dR}(A/K)\) is a free \(K\otimes_\mathbb{Q}K\)-module of rank one and the tensor product is taken over \(K\otimes_\mathbb{Q}K\).) The filtration is given by \[{\rm Fil}^i H_{\rm dR}(M(\chi)) := \begin{cases} H^1_{\rm dR}(A/K)^{\otimes (k-\ell)} &\text{if i\leq -k,}\\ \Gamma(A,\Omega_{A/K}^1)^{\otimes (k-\ell)} &\text{if -k<i \leq -\ell,}\\ 0 &\text{if i>-\ell.} \end{cases}\]

Hypothesis 20 is satisfied for the motive \(M(\chi)\). Note that the \(L\)-function of \(M(\chi)\) is \[L(M(\chi),s)=L(\chi^{-1},s) = L(\overline{\psi}^{k-\ell},s+k) = L(\psi^{\ell-k},s+\ell).\] We know that \(M(\chi)\) is critical (i.e., “\(\chi^{-1}\) is critical" in the sense of [49]) if and only if \(\ell \leq 0 <k\). (Note that we suppose \(k>\ell\).) The Deligne conjecture (Conjecture 26) is proved by Goldstein-Schappacher [63] (see also [22]). A large part of the Tamagawa number conjecture in the critical case (Conjecture 33) is proved by Kato [3], [60], Guo [64], Han [65] and Tsuji [22] as an application of explicit reciprocity laws and the Iwasawa main conjecture proved by Rubin [66]. For results in the non-critical case, see [21], [23].

When \((k,\ell)=(1,0)\) (i.e., \(M(\chi)=h^1(A)(1)\)), Conjecture 28 in analytic rank zero is the well-known result due to Coates-Wiles [67] and Rubin [68]. A large part of Conjecture 33 is due to Rubin [66], and it has recently been solved by Burungale-Flach [69] completely. (See [69] for an explicit interpretation of Conjecture 33.) In analytic rank one, Conjecture 35 is essentially proved by Rubin [66] when \(A\) is defined over \(\mathbb{Q}\), and in a more general case it has recently been proved by Castella [35].

We shall give an explicit description of the period of \(M:=M(\chi)\), which is used in this article. Assume that \(M\) is critical and let \[\alpha_{M}: \mathbb{R}\otimes_\mathbb{Q}H_B(M)^+\xrightarrow{\sim} \mathbb{R}\otimes_\mathbb{Q}t(M)\simeq \mathbb{R}\otimes_\mathbb{Q}(\Gamma(A,\Omega_{A/K}^1)^{\otimes(k-\ell)} )^\ast\] be the period isomorphism. Note that \(H_B(M)^+\) is identified with \(H_1(A(\mathbb{C}),\mathbb{Q})^{\otimes (k-\ell)}(\ell)\). Let \(\omega_A \in \Gamma(A,\Omega_{A/K}^1)\) be a Néron differential and \(\gamma_A \in H_1(A(\mathbb{C}),\mathbb{Z})\) an \(\mathcal{O}_K\)-basis. We define the complex CM period by \[\Omega_\infty:=\int_{\gamma_A}\omega_A.\] We set \[\gamma:= (2\pi i )^\ell \gamma_A^{\otimes(k-\ell)} \in H_1(A(\mathbb{C}),\mathbb{Q})^{\otimes(k-\ell)}(\ell)= H_B(M)^+\] and \[\delta^\ast := \omega_A^{\otimes (k-\ell)} \in \Gamma(A,\Omega_{A/K}^1)^{\otimes (k-\ell)}.\] Let \(\delta \in t(M)\) be the dual basis of \(\delta^\ast\). Then the period of \(M\) \[\Omega_{\gamma,\delta} \in (\mathbb{R}\otimes_\mathbb{Q}K)^\times =\mathbb{C}^\times\] with respect to \(\gamma\) and \(\delta\) is defined by \[\alpha_{M}(\gamma)=\Omega_{\gamma,\delta}\cdot \delta.\]

5 The Birch and Swinnerton-Dyer formula in analytic rank one↩︎

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with conductor \(N\). Let \(K\) be an imaginary quadratic field with odd discriminant \(-D_K <-3\). (We do not assume \(K\) has class number one.) We assume the Heegner hypothesis: every prime divisor of \(N\) splits in \(K\). Let \(p\) be an odd prime number which does not divide \(ND_K\). (Namely, \(p\) is unramified in \(K\) and \(E\) has good reduction at \(p\).)

In this appendix, we give a proof of the following result.

The proof is given in §5.3.

We also have the following result, which we prove in §5.4.

Note that in this result we do not need to assume that \(p\) splits in \(K\), but the “ordinary" assumption is imposed.

We set some notations used in this appendix. Let \(S\) be the finite set of places of \(K\) consisting of the infinite place and the primes dividing \(pN\). We set \(T:=T_p(E)\) and \(V:=\mathbb{Q}_p\otimes_{\mathbb{Z}_p} T\). Let \(K_\infty/K\) be the anticyclotomic \(\mathbb{Z}_p\)-extension. We set \(\Gamma:=\mathop{\mathrm{Gal}}(K_\infty/K)\), \(\Lambda := \mathbb{Z}_p[[\Gamma]]\) and \(\mathbb{T}:= \Lambda \otimes_{\mathbb{Z}_p}T\).

Our idea of the proofs of Theorems 39 and 42 is to construct a \(\Lambda\)-basis \[\mathfrak{z}_S \in {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\] which interpolates the value \(L'(E/K,1)\). In the case of Theorem 39 (i.e., when \(p\) splits in \(K\)), this is the basis constructed in Proposition 14. In the case of Theorem 42, we construct \(\mathfrak{z}_S\) by using Heegner points (see §5.4 below).

5.1 The Tamagawa number conjecture for elliptic curves↩︎

We review the formulation of the Tamagawa number conjecture for the pair \((h^1(E/K)(1),\mathbb{Z}_p)\) in the case \(\mathrm{ord}_{s=1}L(E/K,s)=1\). This is a special case of Conjecture 35 for the motive given in §4.5.3.

In the following, we assume \(\mathrm{ord}_{s=1}L(E/K,s)=1\). Then by Lemma 7 we have a canonical isomorphism \[\label{ks32isom} \vartheta: {\det}_{\mathbb{Q}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},V)) \xrightarrow{\sim} \mathbb{Q}_p\otimes_\mathbb{Q}\left(E(K)\otimes_\mathbb{Z}E(K) \otimes_\mathbb{Z}{\bigwedge}_\mathbb{Q}^2 \Gamma(E,\Omega_{E/K}^1)\right).\tag{18}\]

Take a \(\mathbb{Z}\)-basis \(\gamma \in {\bigwedge}_\mathbb{Z}^2 H_1(E(\mathbb{C}),\mathbb{Z})\). Take also a non-zero element \(\delta \in {\bigwedge}_\mathbb{Q}^2 \Gamma(E,\Omega_{E/K}^1)^\ast\). Let \[\alpha: \mathbb{R}\otimes_\mathbb{Q}{\bigwedge}_\mathbb{Q}^2 H_1(E(\mathbb{C}),\mathbb{Q})\xrightarrow{\sim} \mathbb{R}\otimes_\mathbb{Q}{\bigwedge}_\mathbb{Q}^2 \Gamma(E,\Omega_{E/K}^1)^\ast\] be the period map. We define a period \(\Omega_{E,\gamma,\delta}\in \mathbb{R}^\times\) with respect to \(\gamma\) and \(\delta\) by \[\alpha(\gamma)=\Omega_{E,\gamma,\delta}\cdot \delta.\]

Take any non-torsion element \(x\in E(K)\). Let \[\langle -,-\rangle_\infty: E(K) \times E(K) \to \mathbb{R}\] be the Néron-Tate height pairing. By the Gross-Zagier formula [70], one can show that \[\frac{L'(E/K,1)}{\Omega_{E,\gamma,\delta}\langle x,x \rangle_\infty} \in \mathbb{Q}.\] (This means that the generalized Deligne conjecture (Conjecture 27) is true in this case.) The Tamagawa number conjecture is stated as follows.

The following is well-known.

For the reader’s convenience, we give a proof of Proposition 46 in §5.2.

5.2 Proof of Proposition 46↩︎

By Remarks 45 and 43, we may assume that \(x\) is a generator of \(E(K)_{\rm tf}:=E(K)/E(K)_{\rm tors}\) so that \(R_{E/K}=\langle x,x\rangle_\infty\) and that \(\delta\) is taken so that \(\sqrt{D_K}^{-1}\Omega_{E/K}=\Omega_{E,\gamma,\delta}\). It is sufficient to show the equality \[\label{BSD32equality} \vartheta\left( {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T))\right) = \mathbb{Z}_p\cdot {\rm Eul}_S\cdot\frac{\# \russ\char 88(E/K)\cdot {\rm Tam}(E/K)}{\# E(K)_{\rm tors}^2}\cdot x\otimes x\otimes \delta.\tag{19}\] Here \({\rm Eul}_S\) is as in Remark 47. (The formula (19 ) is actually proved in [38] without assuming \({\rm rank}(E(K))=1\), but we provide a proof for the sake of completeness.)

We set \(W:=V/T=E[p^\infty]\). The Pontryagin dual of a \(\mathbb{Z}_p\)-module \(M\) is denoted by \(M^\vee\). We identify \(W\) with \(T^\vee(1)\) via the Weil pairing. By the Poitou-Tate duality, we have an exact sequence \[\begin{gather} \label{poitoutate} 0\to H^1_f(K,T)\to H^1(G_{K,S},T) \to \bigoplus_{v\in S}H^1_{/f}(K_v,T)\\ \to H^1_f(K,W)^\vee \to H^2(G_{K,S},T)\to \bigoplus_{v\in S}H^2(K_v,T)\to H^0(K,W)^\vee\to 0. \end{gather}\tag{20}\] We use the following facts.

• \(H^1_f(K,T)\simeq \mathbb{Z}_p\otimes_\mathbb{Z}E(K)\).

• \(H^1_{/f}(K_v,T)=0\) if \(v\nmid p\).

• \(H^1_{/f}(K_v,T)\simeq (\mathbb{Z}_p\widehat \otimes E(K_v) )^\ast\) if \(v\mid p\).

• There is a canonical exact sequence \[0\to \russ\char 88(E/K)[p^\infty]^\vee \to H^1_f(K,W)^\vee \to (\mathbb{Z}_p\otimes_\mathbb{Z}E(K))^\ast \to 0.\]

• \(H^2(K_v,T)\simeq H^0(K_v,W)^\vee \simeq E(K_v)[p^\infty]^\vee\) for \(v\in S\).

• \(H^0(K,W)= E(K)[p^\infty]\).

We set \(E(K_p):=\bigoplus_{v\mid p} E(K_v)\). By (20 ) and the facts above, we obtain a canonical isomorphism \[\begin{align} {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T)) &= {\det}_{\mathbb{Z}_p}(H^1(G_{K,S},T)) \otimes_{\mathbb{Z}_p} {\det}_{\mathbb{Z}_p}^{-1}(H^2(G_{K,S},T)) \\ &\simeq {\det}_{\mathbb{Z}_p}(\mathbb{Z}_p\otimes_\mathbb{Z}E(K)) \otimes_{\mathbb{Z}_p} {\det}_{\mathbb{Z}_p}((\mathbb{Z}_p\widehat \otimes E(K_p))^\ast)\otimes_{\mathbb{Z}_p} {\det}_{\mathbb{Z}_p}^{-1}(\russ\char 88(E/K)[p^\infty]^\vee )\\ & \quad \otimes_{\mathbb{Z}_p} {\det}_{\mathbb{Z}_p}^{-1}((\mathbb{Z}_p\otimes_\mathbb{Z}E(K))^\ast) \otimes_{\mathbb{Z}_p} {\det}_{\mathbb{Z}_p}^{-1}\left( \bigoplus_{v\in S}E(K_v)[p^\infty]^\vee\right) \otimes_{\mathbb{Z}_p}{\det}_{\mathbb{Z}_p}(E(K)[p^\infty]) \\ &\simeq \mathbb{Z}_p\cdot {\rm Eul}_p\cdot \frac{\#\russ\char 88(E/K)}{\# E(K)_{\rm tors}^2} \cdot \prod_{v\mid N}\# E(K_v)[p^\infty] \cdot x\otimes x\otimes \delta, \end{align}\] where the last isomorphism is due to Lemma 8 below. By definition, this isomorphism is induced by \(\vartheta\). The desired equality (19 ) follows by noting that the equality \[\prod_{v\mid N} \# E(K_v)[p^\infty] = {\rm Eul}_N\cdot {\rm Tam}(E/K)\] holds up to a \(p\)-adic unit, where \({\rm Eul}_N\) denotes the product of Euler factors at \(v\mid N\). Hence we have completed the proof of Proposition 46. 0◻

5.3 Proof of Theorem 39↩︎

In this subsection, we give a proof of Theorem 39. We assume that \(p\) splits in \(K\). Let \((p)=\mathfrak{p}\overline{\mathfrak{p}}\) be the decomposition of \(p\) in \(K\).

5.3.1 Construction of a basis↩︎

We assume the Iwasawa main conjecture for the BDP \(p\)-adic \(L\)-function (Conjecture 7). (We take \(f\) to be the modular form of weight two corresponding to \(E\).) Let \[\mathfrak{z}_S \in {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\] be the \(\Lambda\)-basis constructed in Proposition 14 by using the basis \(\mathfrak{z}_\mathfrak{p}\) in Conjecture 7 and the local epsilon element \(\varepsilon_\mathfrak{p}\) in (8 ). We define a \(\mathbb{Z}_p\)-basis \[\mathfrak{z}_E \in {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T))\] to be the image of \(\mathfrak{z}_S\) under the natural surjection \[{\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \xrightarrow{a\mapsto a\otimes 1} {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \otimes_{\Lambda}\mathbb{Z}_p \simeq {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T)).\] Here the last isomorphism follows from \[\label{control} \mathbf{R}\Gamma(G_{K,S},\mathbb{T})\otimes^{\mathbf{L}}_\Lambda \mathbb{Z}_p\simeq \mathbf{R}\Gamma(G_{K,S},T)\tag{22}\] (see [28]).

We take \(\delta\) as in Remark 43. By Remark 47, Theorem 39 is reduced to the following.

5.3.2 Proof of Theorem 48↩︎

Let \(\log=\log_{\omega,\mathfrak{p}}: \mathbb{Q}_p\otimes_\mathbb{Z}E(K) \simeq \mathbb{Q}_p\widehat \otimes E(K_\mathfrak{p}) \xrightarrow{\sim} \mathbb{Q}_p\) be the formal logarithm associated with the fixed Néron differential \(\omega\). We set \(a_p:=p+1-\# E(\mathbb{F}_p)\).

Let \(L_\mathfrak{p}^{\rm BDP} \in \Lambda^{\rm ur}\) be the BDP \(p\)-adic \(L\)-function for \(E\) as in §2.2. The key is to use the “\(p\)-adic Gross-Zagier formula" established in [13]: \[\label{pGZ} {\boldsymbol{1}}(L_\mathfrak{p}^{\rm BDP}) = \left(\frac{1-a_p + p}{p}\right)^2 \log(z_K)^2.\tag{23}\] Here \({\boldsymbol{1}}: \Lambda^{\rm ur}\twoheadrightarrow \widehat \mathbb{Z}_p^{\rm ur}\) denotes the augmentation map.

We first need the following lemma.

It is now sufficient to prove the following.

5.4 Proof of Theorem 42↩︎

In this subsection, we give a proof of Theorem 42.

We set some notations. In this subsection, we assume that \(E\) has good ordinary reduction at \(p\) so that there is a canonical exact sequence of \(G_{\mathbb{Q}_p}\)-representations: \[0\to F^+T \to T \to F^-T\to 0.\] We set \(F^{\pm} V:= \mathbb{Q}_p\otimes_{\mathbb{Z}_p} F^{\pm}T\). We also set \(F^{\pm}\mathbb{T}:=\Lambda\otimes_{\mathbb{Z}_p} F^{\pm} T\).

Let \(\alpha \in \mathbb{Z}_p^\times\) be the unit root of \(x^2-a_px +p\), where \(a_p:=p+1-\#E(\mathbb{F}_p)\). Let \(\beta:=p/\alpha\) be the other root.

We set \(K_p:=\mathbb{Q}_p\otimes_\mathbb{Q}K = \bigoplus_{v\mid p}K_v\). We write \(\mathbf{R}\Gamma(K_p,-)\) for \(\bigoplus_{v\mid p} \mathbf{R}\Gamma(K_v,-)\). Similarly, we write \(H^i(K_p,-)\) for \(\bigoplus_{v\mid p}H^i(K_v,-)\).

We define a Selmer complex by \[\label{def32sel32greenberg} \widetilde{\mathbf{R}}\Gamma_f(K,\mathbb{T}):= {\rm Cone}\left(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})\to \mathbf{R}\Gamma(K_p,F^-\mathbb{T})\oplus \bigoplus_{v\mid N} \mathbf{R}\Gamma_{/{\rm ur}}(K_v,\mathbb{T}) \right)[-1].\tag{28}\]

5.4.1 The Heegner point main conjecture↩︎

Let \[z_\infty=(z_n)_n \in \varprojlim_n \mathbb{Z}_p\otimes_\mathbb{Z}E(K_n)\] be the system of regularized Heegner points as in [39], where \(K_n\) denotes the \(n\)-th layer of the anticyclotomic \(\mathbb{Z}_p\)-extension \(K_\infty/K\). We normalize it by multiplying \(c_\phi^{-1}\). Then we have \[\label{z0formula} z_0 = \begin{cases} (1-\alpha^{-2})z_K &\text{if p is inert,}\\ (1-\alpha^{-1})^2 z_K &\text{if p is split,} \end{cases}\tag{29}\] where \(z_K\) is the Heegner point as in Remark 47. We regard \(z_\infty \in \widetilde{H}^1_f(K,\mathbb{T}):=H^1(\widetilde{\mathbf{R}}\Gamma_f(K,\mathbb{T}))\) via the Kummer map.

If the \(\Lambda\)-rank of \(\widetilde{H}^1_f(K,\mathbb{T})\) is one, then we have a canonical isomorphism as in [11]: \[\label{ks32can32isom} Q(\Lambda) \otimes_\Lambda {\det}_\Lambda^{-1}(\mathbf{R}\Gamma_f(K,\mathbb{T})) \simeq Q(\Lambda )\otimes_\Lambda \widetilde{H}^1_f(K,\mathbb{T}) \otimes \widetilde{H}^1_f(K,\mathbb{T})\tag{30}\]

The following formulation of the Heegner point main conjecture is given in [11].

5.4.2 Local epsilon elements↩︎

Let \[\exp^\ast: {\bigwedge}_{\mathbb{Q}_p}^2 H^1(K_p, V) \to \mathbb{Q}_p\otimes_\mathbb{Q}{\bigwedge}_\mathbb{Q}^2 \Gamma(E,\Omega_{E/K}^1)\xrightarrow{\delta^\ast \mapsto 1} \mathbb{Q}_p\] be the map induced by the dual exponential map, where \(\delta\) is as in Remark 43. This induces an isomorphism \[\exp^\ast: {\det}_{\mathbb{Q}_p}^{-1}(\mathbf{R}\Gamma(K_p,F^-V))= {\bigwedge}_{\mathbb{Q}_p}^2 H^1(K_p, F^-V)\xrightarrow{\sim} \mathbb{Q}_p.\]

The basis \(\varepsilon_p\) in the following lemma plays the role of a local epsilon element (see Remark 50 below).

5.4.3 Construction of a basis↩︎

By the definition of the Selmer complex (28 ) and Lemma 2, we have a canonical isomorphism \[\label{sel32triang} {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \simeq {\det}_\Lambda^{-1}(\widetilde{\mathbf{R}}\Gamma_f(K,\mathbb{T})) \otimes_\Lambda {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(K_p,F^-\mathbb{T})).\tag{31}\] Assuming the Heegner point main conjecture (Conjecture 49), we have a \(\Lambda\)-basis \(\widetilde{\mathfrak{z}}_\infty \in {\det}_\Lambda^{-1}(\widetilde{\mathbf{R}}\Gamma_f(K,\mathbb{T}))\) and define \[\mathfrak{z}_S \in {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T}))\] to be the \(\Lambda\)-basis which corresponds to \(\widetilde{\mathfrak{z}}_\infty\otimes \varepsilon_p\) under the isomorphism (31 ), where \(\varepsilon_p\) is as in Lemma 11. We then define a \(\mathbb{Z}_p\)-basis \[\mathfrak{z}_E \in {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T))\] to be the image of \(\mathfrak{z}_S\) under the natural surjection \[{\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \xrightarrow{a\mapsto a\otimes 1} {\det}_\Lambda^{-1}(\mathbf{R}\Gamma(G_{K,S},\mathbb{T})) \otimes_{\Lambda}\mathbb{Z}_p \simeq {\det}_{\mathbb{Z}_p}^{-1}(\mathbf{R}\Gamma(G_{K,S},T)).\]

5.4.4 Completion of the proof↩︎

We now prove Theorem 42.

References↩︎

---

1. This means \(D_K \in \{7,11,19,43,67,163\}\).↩︎

2. Our convention is that the Hodge-Tate weight of \(\mathbb{Q}_p(1)\) is \(+1\).↩︎