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An \(l \ne p\)interpolation of genuine padic Lfunctions
 Ashay A. Burungale^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4068701600602
© Burungale. 2016
 Received: 12 May 2015
 Accepted: 17 February 2016
 Published: 1 July 2016
Abstract
Let \(\mathcal {F}\) be a totally real field, l and p distinct odd prime unramified in \(\mathcal {F}\) and \({\mathfrak {l}}\) a prime above l. Let \(\mathcal {K}/\mathcal {F}\) be a pordinary CM quadratic extension and \(\lambda \) an arithmetic Hecke character over \(\mathcal {K}\). Hida constructed a measure on the \({\mathfrak {l}}\)anticyclotomic class group of \(\mathcal {K}\) interpolating the normalised Hecke Lvalues \(L^{\mathrm{alg},{\mathfrak {l}}}(0,\lambda \nu )\), as \(\nu \) varies over the finite order \({\mathfrak {l}}\)power conductor anticyclotomic characters. In this article, we interpolate the measures as \(\lambda \) varies in a padic family. In particular, this gives padic deformation of the measures. An analogue holds in the case of selfdual Rankin–Selberg convolution of a Hilbert modular form and a theta series. In the case of root number \(1\), we describe an upcoming analogous interpolation of the padic Abel–Jacobi image of generalised Heegner cycles associated with the convolution.
Keywords
 padic Lfunctions
 Modular measure
 Hilbert modular Shimura variety
 Toric modular forms
Mathematics Subject Classification
 Primary 11F33
 11F41
 11G18
 11R23
1 Background
The variation of arithmetic invariants in a family seems to be a natural phenomena to explore. One expects the existence of a commutative padic Lfunction associated with a critical motive \(\mathcal M\) over a tower of number fields with the Galois group being a commutative padic Lie group \(\Gamma \), characterised by an interpolation of the pstabilised critical Lvalues of \(\mathcal M\) twisted by a dense subset of characters of \(\Gamma \). One also expects that these padic Lfunctions can be interpolated as \(\mathcal M\) itself varies in a padic family. In [6], Hida refers to the padic Lfunction along the family variable as the genuine padic Lfunction of the family. One thus expects padic interpolation of the genuine padic Lfunction of the family over the characters of \(\Gamma \). When the family is selfdual with root number \(1\), the Lvalues identically vanish. In such a situation, the Bloch–Beilinson conjectures predict the existence of nontorsion cycles associated with the family which are homologically trivial. If a candidate for such cycles is available, one can investigate the existence of the interpolation for arithmetic invariants associated with the cycles, for example their padic Abel–Jacobi image.
This article concerns ladic anticyclotomic Iwasawa theory of a pordinary CM field in a padic Hida family with \(l \ne p\). Strictly speaking, we consider arithmetic Hecke characters over the CM field in a padic Hida family. A rich source of such characters arises from CM abelian varieties. In [7], Hida constructed an \(l\ne p\)analogue of anticyclotomic padic Lfunction for arithmetic Hecke characters over a CM field. Let \(\mathcal {F}\) be a totally real field and \({\mathfrak {l}}\) a prime above l. Let \(\mathcal {K}/\mathcal {F}\) a CM quadratic extension and \(\lambda \) an arithmetic Hecke character over \(\mathcal {K}\). More precisely, Hida constructed a \(\overline{\mathbf {Z}}_{p}\)valued measure \({\hbox {d}}\varphi _\lambda \) on the \({\mathfrak {l}}\)anticyclotomic Galois group \(\Gamma _{{\mathfrak {l}}}^{}\) over \(\mathcal {K}\) interpolating the \({\mathfrak {l}}\)stabilised Hecke Lvalues \(L^{\mathrm{alg},{\mathfrak {l}}}(0,\lambda \nu )\), as \(\nu \) varies over the finite order \({\mathfrak {l}}\)power conductor characters of \(\Gamma _{{\mathfrak {l}}}^{}\). In this article, we interpolate the measures \({\hbox {d}}\varphi _\lambda \) as \(\lambda \) varies in a padic family, namely in a \(\Lambda \)adic family \(\Psi \). This gives padic deformation of the measures. Strictly speaking, we construct a measure \({\hbox {d}}\varphi _{\Psi }\) interpolating a modification of \({\hbox {d}}\varphi _\lambda \) which interpolates \({\mathfrak {l}}p\)stabilised Hecke Lvalues \(L^{\mathrm{alg},{\mathfrak {l}}p}(0,\lambda \nu )\). For a fixed \(\nu \), the integral \(\int _{\Gamma _{{\mathfrak {l}}}^{}}\nu {\hbox {d}}\varphi _\Psi \) gives rise to the genuine padic Lfunction of the \(\Lambda \)adic Hecke character \(\Psi \nu \) in the sense of Hida introduced in [6]. We thus obtain a genuine padic Lfunction as a byproduct of our \(l\ne p\) consideration. Just to rephrase, the measure \({\hbox {d}}\varphi _\Psi \) gives an \(l\ne p\)interpolation of the genuine padic Lfunction of the \(\Lambda \)adic Hecke character \(\Psi \) over the finite order characters of \(\Gamma _{{\mathfrak {l}}}^{}\). An analogue of the results holds in the case of selfdual Rankin–Selberg convolution of a Hilbert modular form and a theta series. In the case of root number \(1\) over the rationals, a candidate for the cycles alluded to above is generalised Heegner cycles. The construction is due to Bertolini–Darmon–Prasanna and generalises the one of classical Heegner cycles (cf. [1, 2]). The cycle lives in a middledimensional Chow group of a fibre product of a Kuga–Sato variety arising from a modular curve and a selfproduct of a CM elliptic curve. In the case of weight two, the cycle coincides with a Heegner point and the padic Abel–Jacobi image with the padic formal group logarithm. In an ongoing work, we construct an analogous interpolation of the padic Abel–Jacobi image of the cycles. In this manner, the phenomena of variation seem to prevail even in selfdual situations with root number \(1\).

(ord) Every prime of \(\mathcal {F}\) above p splits in \(\mathcal {K}\).
In this article, we consider the variation of \({\hbox {d}}\varphi _\lambda \) as \(\lambda \) varies in a padic family. A rather systematic source of a padic family of Hecke characters is given by Hida’s notion of a \(\Lambda \)adic Hecke character. To briefly recall the notion, we introduce more notation. Let \({{\varvec{\Gamma }}}=1+p\mathbf {Z}_p\) be the maximal torsionfree subgroup of \(\mathbf {Z}_{p}^\times \) and \(\gamma =(1+p)\) a topological generator of \({{\varvec{\Gamma }}}\). Let W be a discrete valuation ring finite flat over \(\mathbf {Z}_p\) and \(\Lambda _{W}=W\llbracket {{\varvec{\Gamma }}}\rrbracket \). We identify \(\Lambda _{W}=W\llbracket {T}\rrbracket \) via \(\gamma \leftrightarrow 1+T\). We say that \(P \in \mathrm {Spec}\,(\Lambda _{W})(\overline{\mathbf {Q}}_p)\) is arithmetic of weight k(P) with character \(\epsilon _{P}:{{\varvec{\Gamma }}}\rightarrow \mu _{p^\infty }(\overline{\mathbf {Q}}_p)\) if \(P(1+T\epsilon _{P}(\gamma )\gamma ^{k(P)1})=0\) and \(k(P)\ge 2\). Let \({\mathbb {I}}\) be a domain of finite rank over \(\Lambda _W\). Let Q be the quotient field of \(\Lambda \), \(\overline{Q}\) an algebraic closure and \(\overline{{\mathbb {I}}}\) an integral closure of \({\mathbb {I}}\) in \(\overline{Q}\). We say that \(P \in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) is arithmetic of weight k(P) with character \(\epsilon _{P}:{{\varvec{\Gamma }}}\rightarrow \mu _{p^\infty }(\overline{\mathbf {Q}}_p)\) if it lies above an arithmetic (still denoted by) \(P \in \mathrm {Spec}\,(\Lambda _{W})(\overline{\mathbf {Q}}_p)\) of weight k(P) and character \(\epsilon _P\). An \({\mathbb {I}}\)adic Hecke character is a Galois character \(\Psi :\mathrm{Gal}({\overline{\mathbf {Q}}}/\mathcal {K})\rightarrow {\mathbb {I}}^\times \) such that the arithmetic specialisation \(\Psi _P\) is the padic avatar of an arithmetic Hecke character of infinity type \((k(P)1){\varSigma }\). For the notion of the primetop conductor of \(\Psi \), we refer to Sect. 4.2. For \(\nu \in \mathfrak {X_l}^\) , let \({\mathbb {I}}_\nu \) be the finite flat extension of \({\mathbb {I}}\) obtained by adjoining values of the finite order character \(\nu \).
Our result regarding the variation is the following.
Theorem A
Here \(\nu \) is a finite order character of \(\Gamma _{\mathfrak {l}}^\), \(P \in \mathrm {Spec}\,({\mathbb {I}}_\nu )(\overline{\mathbf {Q}}_p)\) an arithmetic prime with weight k(P), \({\mathbb {I}}_{\nu }\) the finite flat extension of \({\mathbb {I}}\) and CM periods \((\Omega _{p},\Omega _{\infty })\) as above. In particular, \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _{\Psi } \in {\mathbb {I}}_\nu \) equals the genuine padic Lfunction of the \({\mathbb {I}}_\nu \)adic Hecke character \(\Psi \nu \).
In particular, we can consider \({\hbox {d}}\varphi _\Psi \) as a padic deformation of modular measures constructed in [7] and [16]. For nontriviality of the measure \({\hbox {d}}\varphi _\Psi \), we refer to Sect. 5.3. The nontriviality is based on the nontriviality results in [7] and [16].
We now give a sketch of the proof. Some of the notation used here is not followed in the rest of the article. In [7] and [16], the measure \({\hbox {d}}\varphi _\lambda \) on \(\Gamma _{\mathfrak {l}}^\) is constructed via essentially a weighted sum of evaluation of a toric Eisenstein series \(f_\lambda \) at the \({\mathfrak {l}}\)power conductor CM points on an underlying Hilbert modular Shimura variety. The Hecke Lvalues in consideration essentially equal such an evaluation. In these articles, the results are under the hypothesis that the conductor of \(\lambda \) is primetop. To remove it, we make an appropriate choice for the local sections corresponding to the toric Eisenstein series at the places dividing p. The latter is indeed technical novelty of the article and allows to treat the case of conductor being divisible by places above p. In consideration of \(\Lambda \)adic family of Hecke characters, the conductor of the corresponding arithmetic Hecke characters is indeed divisible by places above p. As \(P \in \mathrm {Spec}\,({\mathbb {I}}_\nu )(\overline{\mathbf {Q}}_p)\) varies over arithmetic primes, we prove that the toric Eisenstein series \(f_{\Psi _P}\) are interpolated by an \({\mathbb {I}}_\nu \)adic toric Eisenstein series \(F_\Psi \). The \({\mathbb {I}}_\nu \)adic Eisenstein series is constructed based on an analysis of the qexpansion coefficients of the toric Eisenstein series. The \(\Lambda \)adic measure \({\hbox {d}}\varphi _\Psi \) is then constructed by a similar evaluation of \(F_\Psi \) at CM points on an underlying pordinary Igusa tower as the measure \({\hbox {d}}\varphi _{\lambda }\). Summarising, padic deformation of the toric Eisenstein series gives rise to the padic deformation of modular measures.
Let us briefly recall the history of “primetop Iwasawa theory”. It first arose in Washington’s work [26] as \(l \ne p\)variation of Iwasawa’s theorem on the class numbers of the padic cyclotomic towers of number fields. This result has an application to prove the nonvanishing of the corresponding Dirichlet Lvalues modulo p. In [23], Sinnott found a different way to prove this result based on the Zariski density of certain roots of unity on selfproducts of the multiplicative group \({\mathbb {G}}_{m}\) in characteristic p. In [7], Hida adapted this idea and generalised it to Hecke Lfunctions over CM fields based on the mod p geometry of Hilbert modular Shimura varieties. Hida’s work crucially relies on Chai’s theory of Heckestable subvarieties of a mod p Shimura variety. In [16], Hsieh refined computational aspects of Hida’s strategy. For related results, we refer to Vatsal [25], Finis [4] and Sun [24].
The existence of genuine padic Lfunctions for an \({\mathbb {I}}\)adic Hecke character is proven in [5, §8]. Our byproduct construction is slightly different and seems to give additional information, for example regarding the ring of definition. Moreover, our construction also works in the case of selfdual Rankin–Selberg convolution, and the byproduct construction of the genuine padic Lfunction is perhaps new in this case. More precisely, an analogue of Theorem A holds for selfdual Rankin–Selberg convolution of a parallel weight \(\Lambda \)adic Hilbert modular form and a parallel weight \(\Lambda \)adic theta series. Based on the Waldspurger formula, the Lvalue in consideration is again essentially a sum of evaluation of a toric Hilbert modular form at the \({\mathfrak {l}}\)power conductor CM points (cf. [18]). The construction of the relevant \(\overline{{\mathbb {I}}}\)valued measure is accordingly similar and we skip the details. In this article, we only consider \(\Lambda \)adic Hecke character for the onevariable Iwasawa algebra \(\Lambda \). We can also consider \(\Lambda _{\mathcal {F}}\)adic Hecke character for the \((d+1)\)variable Iwasawa algebra \(\Lambda _{\mathcal {F}}\). An analogue of Theorem A holds in this case as well.
Based on the padic Waldspurger formula in [1] and [2], we can show the existence of an analogous interpolation of the padic Abel–Jacobi image of the generalised Heegner cycles alluded to above. This is an ongoing work in progress. The construction is quite similar to that of the measure \({\hbox {d}}\varphi _{\Psi }\). We refer to Sect. 6 for a brief sketch in the case of weight two.
The \({\mathbb {I}}_\nu \)adic Eisenstein series would perhaps have other arithmetic applications. It essentially appears in [20] and is a key input to show a congruence between Katz padic Lfunctions over certain CM extensions of a fixed totally real field.
Finer questions about the nontriviality of the \({\mathbb {I}}\)adic measure seem to require new ideas (cf. the remark following Proposition 4.5). It would be interesting to see whether there is an Iwasawa main conjecture in the setting of the measure \({\hbox {d}}\varphi _\Psi \) i.e. as \(\nu \in \mathfrak {X_l}^\) varies. Also, it would be interesting to see whether the \(l\ne p\)interpolation of genuine padic Lfunction of a padic family consisting the critical motive \(\mathcal M\) exists in other contexts. We are tempted to believe the answer to be affirmative in the case of pordinary family of Hilbert modular forms.
The article is organised as follows. In Sect. 2, we recall certain generalities about Hilbert modular Shimura variety. In Sect. 3, we recall certain generalities about Hilbert modular forms. In Sect. 4, we describe anticyclotomic modular measures associated with a class of classical and \(\Lambda \)adic Hilbert modular forms. This section is perhaps the technical heart of the article. In Sect. 4.1, we describe anticyclotomic modular measures associated with a class of classical Hilbert modular forms. In Sect. 4.2, we describe certain generalities about Hida’s \(\Lambda \)adic Hecke character. In Sect. 4.3, we describe anticyclotomic modular measures associated with a class of \(\Lambda \)adic Hilbert modular forms. In Sect. 5, we prove Theorem A. In Sect. 5.1, we describe construction of a toric Eisenstein series following Hsieh. In Sect. 5.2, we describe a \(\Lambda \)adic interpolation of the toric Eisenstein series in Sect. 5.1, thereby obtaining a \(\Lambda \)adic toric Eisenstein series. In Sect. 5.3, we prove Theorem A. In Sect. 6, we briefly describe the upcoming analogous interpolation of the padic Abel–Jacobi image of generalised Heegner cycles associated with selfdual Rankin–Selberg convolution of an elliptic modular form and a theta series with root number \(1\).
Notation We use the following notation unless otherwise stated.
Let \(\mathcal {F}_+\) denote the totally positive elements in \(\mathcal {F}\). We sometime use O to denote the ring of integers \(\mathcal O_\mathcal {F}\). Let \(\mathcal {D}_\mathcal {F}\) (resp. \(D_\mathcal {F}\)) be the different (resp. discriminant) of \(\mathcal {F}/\mathbf {Q}\). Let \(\mathcal {D}_{\mathcal {K}/\mathcal {F}}\) (resp. \(D_{\mathcal {K}/\mathcal {F}}\)) be the different (resp. discriminant) of \(\mathcal {K}/\mathcal {F}\). Let h (resp. \({\mathbf {h}}_\mathcal {K}\)) be the set of finite places of \(\mathcal {F}\) (resp. \(\mathcal {K}\)). Let v be a place of \(\mathcal {F}\) and w be a place of \(\mathcal {K}\) above v. Let \(\mathcal {F}_v\) be the completion of \(\mathcal {F}\) at v, \(\varpi _v\) an uniformiser and \(\mathcal {K}_{v} = \mathcal {F}_{v} \otimes _{\mathcal {F}} \mathcal {K}\). For the pordinary CM type \({\varSigma }\) of \(\mathcal {K}\) as above, let \({\varSigma }_p = \big \{ w \in {{\mathbf {h}}}_\mathcal {K} wp\) and w induced by \(\iota _p \circ \sigma \), for \(\sigma \in {\varSigma }\big \}\).
For a number field \(\mathcal {L}\), let \({\mathbf {A}}_{\mathcal {L}}\) be the adele ring, \({\mathbf {A}}_{\mathcal {L},f}\) the finite adeles and \({\mathbf {A}}_{\mathcal {L},f}^{\Box }\) the finite adeles away from a finite set of places \(\Box \) of \(\mathcal {L}\). For \(a \in \mathcal {L}\), let \(\mathfrak {il}_{\mathcal {L}}(a)=a(\mathcal O_\mathcal {L}\otimes _{\mathbf {Z}} \widehat{\mathbf {Z}}) \cap \mathcal {L}\). For a fractional ideal \({\mathfrak {a}}\), let \(\widehat{{\mathfrak {a}}}={\mathfrak {a}}\otimes \widehat{\mathbf {Z}}\). Let \(G_\mathcal {L}\) be the absolute Galois group of L and \(\mathrm{rec}_{\mathcal {L}}: {\mathbf {A}}_{\mathcal {L}}^\times \rightarrow G_{\mathcal {L}}^{ab}\) the geometrically normalised reciprocity law. Let \(\psi _\mathbf {Q}\) be the standard additive character of \({\mathbf {A}}_\mathbf {Q}\) such that \(\psi _\mathbf {Q}(x_\infty )=\exp (2\pi i x_{\infty })\), for \(x_\infty \in \mathbf {R}\). Let \(\psi _\mathcal {L}: {\mathbf {A}}_{\mathcal {L}}/ \mathcal {L}\rightarrow \mathbf {C}\) be given by \(\psi _{\mathcal {L}}(y)=\psi _{\mathbf {Q}}\circ (\mathrm{Tr}_{\mathcal {L}/\mathbf {Q}}(y))\), for \(y \in {\mathbf {A}}_{\mathcal {L}}\). We denote \(\psi _\mathcal {F}\) by \(\psi \).
2 Hilbert modular Shimura variety
In this section, we recall certain generalities about Hilbert modular Shimura variety.
In regard to the article, the section is preliminary. It briefly recalls the geometric theory of Hilbert modular Shimura variety, pordinary Igusa tower and CM points on them. The CM points inherently give rise to pintegral structure. The theory plays foundational role in the geometric theory of Hilbert modular forms and padic modular forms (cf. Sect. 3). Roughly speaking, classical modular forms (resp. padic modular forms) are functions on the Hilbert modular Shimura variety (resp. Igusa tower). The latter in turn play an underlying role in the construction of modular measures (cf. Sect. 4). Evaluation of padic modular forms at well chosen CM points is central to the construction.
2.1 Setup
In this subsection, we recall a basic setup regarding Hilbert modular Shimura variety. We refer to [7, §2.2] and [16, §2.1] for details.
Let us introduce some notation. Consider \(V = \mathcal {F}^2\) as a twodimensional vector space over \(\mathcal {F}\). Let \(e_{1} = (1,0)\) and \(e_{2}=(0,1)\). Let \(\langle . , . \rangle : V \times V \rightarrow \mathcal {F}\) be the \(\mathcal {F}\) bilinear pairing defined by \(\langle e_{1},e_{2}\rangle =1\). Let \( \mathcal {L}= Oe_{1} \oplus O^{*}e_{2}\) be the standard O lattice in V. For a fractional ideal \(\mathfrak {b}\) of O, \(\mathfrak {b}^{*} := \mathfrak {b}^{1} \mathfrak {d}_{\mathcal {F}}^{1}\). Here \(\mathfrak {d}_{?}\) denotes the different of \(?/ \mathbf {Q}\), where ? equals \(\mathcal {F}\) or \(\mathcal {K}\). Sometimes, we denote \(\mathfrak {d}_{\mathcal {F}}\) by \(\mathfrak {d}\). For \(g \in G(\mathbf {Q})\), \(g':= {\hbox {det}}(g) g^{1}\). Note that \(G(\mathbf {Q})\) has a natural right action on \(\mathcal {F}^2\). For \(x \in V\), consider the left action \(gx := xg'\).
2.2 pIntegral model
In this subsection, we briefly recall generalities about a canonical pintegral smooth model of a Hilbert modular Shimura variety. The description gives rise to the pintegral structure on the space of Hilbert modular forms. We refer to [8, §4.2] for details.
Let the notation and hypotheses be as in the introduction and Sect. 2.1. Hilbert modular Shimura variety \(Sh_{/\mathbf {Q}}\) represents a functor classifying abelian schemes having multiplication by O along with additional structure, where O is the ring of integers of \(\mathcal {F}\) (cf. [8, §4.2] and [21]). When the prime p is unramified in \(\mathcal {F}\), a pintegral interpretation of the functor leads to a pintegral smooth model of \(Sh/G(\mathbf {Z}_p)_{/\mathbf {Q}}\).
 (PM1)
A is abelian scheme over S of dimension of d.
 (PM2)
\(\iota : O \hookrightarrow \mathrm{End}_{S}A\) is an algebra embedding.
 (PM3)\(\bar{\lambda }\) is the polarisation class of a homogeneous polarisation \(\lambda \) of degree primetop up to scalar multiplication by \(\iota (O^{\times }_{(p),+})\), where \(O_{(p),+} := \{ a \in O_{(p)} \sigma (a) > 0, \forall \sigma \in I\}\). Let \(^{t}\) denote the Rosati involution induced by the polarisation class on \(\mathrm{End}_{S}A \otimes \mathbf {Z}_{(p)}\). We then have \(\iota (l)^{t}=\iota (l)\) for \(l \in O\). Let us fix an isomorphism \(\zeta : \mathbf {A}_{\mathbf {Q},f} \simeq \mathbf {A}_{\mathbf {Q},f}(1)\) with the Tate twist. We can thus regard the Weil pairing \(e^\lambda \) induced by \(\lambda \) as an \(\mathcal {F}\)alternate formHere \(V^{(p)}(A)=\mathcal {T}^{(p)}(A)\otimes \mathbf {Q}\), where \(\mathcal {T}^{(p)}(A)\) is as in (PM4). Let \(e^\eta \) denote the \(\mathcal {F}\)alternate form \(e^{\eta }(x,x'):= \langle x\eta , x'\eta \rangle \). Then,$$\begin{aligned} e^\lambda : V^{(p)}(A) \times V^{(p)}(A) \rightarrow O^{*} \otimes _{\mathbf {Z}} \mathbf {A}_{\mathbf {Q},f}^{(p)}. \end{aligned}$$for some \(u \in \mathbf {A}_{\mathcal {F},f}^{(p)}\).$$\begin{aligned} e^\lambda = ue^\eta \end{aligned}$$
 (PM4)
Let \(\mathcal {T}^{(p)}(A)\) be the primetop Tate module \(\varprojlim _{p \not \mid N} A[N]\). \(\eta ^{(p)}\) is a primetop level structure given by an Olinear isomorphism \(\eta ^{(p)}: O^{2} \otimes _\mathbf {Z}\widehat{\mathbf {Z}}^{(p)} \simeq \mathcal {T}^{(p)}(A)\), where \(\widehat{\mathbf {Z}}^{(p)}= \prod _{l\ne p} \mathbf {Z}_l\).
 (PM5)Let \(\mathrm{Lie}_{S}(A)\) be the relative Lie algebra of A. There exists an \( O \otimes _{\mathbf {Z}} \mathcal O_S\)module isomorphismlocally under the Zariski topology of S.$$\begin{aligned} \mathrm{Lie}_{S}(A) \simeq O \otimes _{\mathbf {Z}} \mathcal O_S \end{aligned}$$
Theorem 2.1
2.3 Igusa tower
In this subsection, we briefly recall the notion of pordinary Igusa tower over the pintegral Hilbert modular Shimura variety. The Igusa tower underlies the geometric theory of padic Hilbert modular forms. We refer to [8, Ch. 8] for details.
Let the notation and hypotheses be as in the introduction and Sect. 2.2. In particular, \(\mathcal {W}\) is the strict henselisation inside \(\overline{\mathbf {Q}}\) of the local ring of \(\mathbf {Z}_{(p)}\) corresponding to \(\iota _p\), \(W(\mathbb {F})\) is the Witt ring, and \(\mathbb {F}\) is the residue field of \(\mathcal {W}\).
Let \({\hbox {Sh}}^{(p)}_{/\mathcal {W}} = {\hbox {Sh}}^{(p)}(G,X) \times _{\mathbf {Z}_{(p)}} \mathcal {W}\) and \({\hbox {Sh}}^{(p)}_{/ \mathbb {F}}= Sh^{(p)}_{/\mathcal {W}} \times _{\mathcal {W}} \mathbb {F}\).
From now, let Sh (resp. Sh\(_{K}\)) denote \({\hbox {Sh}}^{(p)}_{/ W}\) or \({\hbox {Sh}}^{(p)}_{/ \mathbb {F}}\) (resp. \({\hbox {Sh}}^{(p)}_{K/ W}\) or \({\hbox {Sh}}^{(p)}_{K/ \mathbb {F}}\)). The base will be clear from the context. Let \(\mathcal {A}\) be the universal abelian scheme over Sh.
Let \({\hbox {Sh}}^\mathrm{ord}\) be the subscheme of Sh on which the Hasse invariant does not vanish. It is an open dense subscheme. Over \({\hbox {Sh}}^{\mathrm{ord}}\), the connected part \(\mathcal {A}[p^m]^{\circ }\) of \(\mathcal {A}[p^m]\) is étale locally isomorphic to \(\mu _{p^{m}} \otimes _{\mathbf {Z}_p} O^*\) as an \(O_{p}\)module for \(O_{p}=O \otimes \mathbf {Z}_{p}\).

(PL) Let \(\eta _{p}^\circ \) be a level \(p^\infty \)structure on \(A_x[p^\infty ]^\circ \). For the primes \(\mathfrak {p}\) in O dividing p, it is a collection of level \(\mathfrak {p}^\infty \)structures \(\eta _{\mathfrak {p}}^\circ \), given by isomorphisms \(\eta _{\mathfrak {p}}^\circ : O_{\mathfrak {p}}^{*}\simeq A_x[\mathfrak {p}^\infty ]^\circ \), where \(O_{\mathfrak {p}}^{*}=O^{*}\otimes O_{\mathfrak {p}}\). The Cartier duality and the polarisation \(\bar{\lambda }_x\) induces an isomorphism \(\eta _{\mathfrak {p}}^{\acute{e}t}: O_{\mathfrak {p}}\simeq A_x[\mathfrak {p}^\infty ]^{\acute{e}t}\). Thus, we get a level \(p^\infty \)structure \(\eta _{p}^{\acute{e}t}\) on \(A_x[p^\infty ]^{\acute{e}t}\) from \(\eta _{p}^\circ \).

(Ir) I is an irreducible component of Ig.
2.4 CM points
In this subsection, we briefly recall some notation regarding CM points on the Igusa tower. The notion is originally due to Shimura and underlies the theory of Shimura varieties. We refer to [22] for details.
Let the notation and hypotheses be as in the introduction and Sect. 2.3. In particular, \(\mathcal {K}/\mathcal {F}\) is a pordinary CM quadratic extension. Let \({\mathfrak {a}} \subset \mathcal O_{\mathcal {K}}\) be an Olattice. Let \(R({\mathfrak {a}})=\left\{ \alpha \in \mathcal O_{\mathcal {K}} \alpha {\mathfrak {a}}\subset {\mathfrak {a}}\right\} \) be the corresponding order of \(\mathcal {K}\) and \(\mathfrak {f}({\mathfrak {a}})\subset O\) the corresponding conductor ideal. Recall that \(R({\mathfrak {a}})=O+\mathfrak {f}({\mathfrak {a}})\mathcal O_{\mathcal {K}}\).
Let \({\varSigma }\) be a pordinary CM type as in the introduction. By CM theory of Shimura–Taniyama–Weil (cf. [22]), the complex torus \(X({\mathfrak {a}})(\mathbf {C})=\mathbf {C}^{{\varSigma }}/{\varSigma }({\mathfrak {a}})\) is algebraisable to a CM abelian variety of CM type \((\mathcal {K},{\varSigma })\). Here \({\varSigma }({\mathfrak {a}})=\left\{ \iota _{\infty }(\sigma (a))_{\sigma }a \in {\mathfrak {a}}\right\} \). When the conductor \(\mathfrak {f}({\mathfrak {a}})\) is primetop, the abelian variety \(X({\mathfrak {a}})\) extends to an abelian scheme over \(\mathcal {W}\). We denote it by \(X({\mathfrak {a}})_{/\mathcal {W}}\).
2.5 Tate objects
In this subsection, we briefly recall some notation regarding Tate objects on the Hilbert modular Shimura variety. They naturally arise during the construction of toroidal compactifications of the Hilbert modular Shimura variety. In turn, they give rise to the key notion of qexpansion of Hilbert modular forms. We refer to [19, § 1.1] and [8, § 4.1.5] for details.
3 Hilbert modular forms
In this section, we recall certain generalities about geometric theory of Hilbert modular forms. We also briefly recall the adelic notion. The former plays a key role in the construction of modular measures in Sect. 4, while the latter in the construction of toric Eisenstein series in Sect. 5.
3.1 Classical Hilbert modular forms
In this subsection, we recall the notion of classical Hilbert modular forms. We refer to [8, §4.2] and [16, §2.5] for details.
3.1.1 Geometric definition
We first recall the geometric definition of classical Hilbert modular forms of parallel weight.
 (Gc1)
If \(x \simeq x'\), then \(f(x)=f(x')\in S\).
 (Gc2)
\(f(x \otimes S') = \rho (f(x))\) for any Ralgebra homomorphism \(\rho : S \rightarrow S'\).
 (Gc3)
\(f(\underline{\mathcal {A}},s\omega )=\underline{k}(s)^{1}f(\underline{\mathcal {A}},\omega )\), where \(s \in T(S)\).
 (Gc4)
\(f_{\mathfrak {a,b}}(q):=f(\mathrm{{Tate}}_{{\mathfrak {a}},\mathfrak {b}}(q), \iota _{{\hbox {can}}}, \lambda _{{\hbox {can}}}, \eta _{{\hbox {can}}}^{(p)},\omega _{{\hbox {can}}}) \in R\llbracket { q^{\mathfrak {ab}_{\ge 0}}}\rrbracket \), where \(\mathfrak {ab}_{\ge 0}=(\mathfrak {ab}\cap \mathcal {F}_{+}) \cup \{0\}\). We say that \(f_{\mathfrak {a,b}}(q)\) is the qexpansion of f at the cusp \((\mathfrak {a,b})\).

(qexp) The qexpansion map \(f \mapsto f_{\mathfrak {a,b}}(q) \in R\llbracket { q^{\mathfrak {ab}_{\ge 0}}}\rrbracket \) determines f uniquely.
3.1.2 Adelic definition
We now briefly describe relation of the geometric definition of Hilbert modular forms to the adelic counterpart.
Let the notation and assumptions be as in Sect. 3.1.1. In particular, k is a positive integer.
Via the above identity, we can also start with an adelic Hilbert modular form and obtain a classical Hilbert modular form. Representation theoretic methods can often be used to construct adelic Hilbert modular forms. In Sect. 5, we adopt this approach to construct toric Eisenstein series.
3.2 pAdic Hilbert modular forms
In this subsection, we recall the geometric definition of padic Hilbert modular forms. The theory is due to Katz and Hida. We refer to [8, §8] for details.
 (Gp1)
If \(x \simeq x'\), then \(f(x)=f(x')\in S\).
 (Gp2)
\(f(x \otimes S') = \rho (f(x))\) for any padic Ralgebra homomorphism \(\rho : S \rightarrow S'\).
 (Gp3)
\(f_{\mathfrak {a,b}}(q):=f(x(\mathrm{{Tate}}_{\mathfrak {a,b}}(q))) \in R\llbracket { q^{\mathfrak {ab}_{\ge 0}}}\rrbracket \). We say that \(f_{\mathfrak {a,b}}(q)\) is the qexpansion of f at the cusp \((\mathfrak {a,b})\).
For \(f \in V(K,R)\), we have the following fundamental qexpansion principle (cf. [8, Thm. 4.21])
(qexp)’ The qexpansion map \(f \mapsto f_{\mathfrak {a,b}}(q) \in R\llbracket { q^{\mathfrak {ab}_{\ge 0}}}\rrbracket \) determines f uniquely.
3.3 Hecke operators
In this subsection, we recall the definition of certain Hecke operators on the space of padic Hilbert modular forms.
3.4 \(\Lambda \)Adic Hilbert modular forms
In this subsection, we recall the definition of \(\Lambda \)adic Hilbert modular forms of parallel weight due to Hida. We refer to [9] for details.
Let the notation and hypotheses be as in the introduction and Sect. 2. Let \(\mathfrak {N}\) be a primetop ideal of O and \(Cl_{\mathcal {F}}(\mathfrak {N}p^{\infty })\) the strict ray class group modulo \(\mathfrak {N}p^{\infty }\). Let \(N: Cl_{\mathcal {F}}(\mathfrak {N}p^{\infty }) \rightarrow \mathbf {Z}_{p}^\times \) be the norm map arising from \({\mathfrak {a}} \mapsto O/{\mathfrak {a}}\), where \({\mathfrak {a}}\) is a primeto\(\mathfrak {N}p\) ideal. Let \(\Gamma \simeq 1+p\mathbf {Z}_p\) be the maximal torsionfree quotient of \(\mathbf {Z}_{p}^\times \). Via the projection, we have \(\langle N \rangle : Cl_{\mathcal {F}}(\mathfrak {N}p^{\infty }) \rightarrow \Gamma \). Let \(\Delta = \ker ( \langle N \rangle )\) and \({{\varvec{\Gamma }}}=\mathrm{Im}(\langle N \rangle )\). As \({{\varvec{\Gamma }}}\simeq \mathbf {Z}_p\), we have a splitting \(Cl_{\mathcal {F}}(\mathfrak {N}p^{\infty })= \Delta \times {{\varvec{\Gamma }}}\).
Let \(\gamma \) be a topological generator of \({{\varvec{\Gamma }}}\). Let \(\Lambda =\mathbf {Z}_{p}\llbracket {{{\varvec{\Gamma }}}}\rrbracket \). We identify \(\Lambda \) with the Iwasawa algebra \(\mathbf {Z}_{p}\llbracket {{T}}\rrbracket \), via \( \gamma \leftrightarrow 1+T\). For a valuation ring W finite flat over \(\mathbf {Z}_p\), let \(\Lambda _{W}=\Lambda \otimes _{\mathbf {Z}_p} W\). Let \({\mathbb {I}}\) be a domain of finite rank over \(\Lambda _{W}\). We say that \( P \in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_{p})\) is arithmetic of weight k(P) with character \(\epsilon : {{\varvec{\Gamma }}} \rightarrow \mu _{p^\infty }(\mathbf {C}_p)\) if \( k(P) \ge 2\) and \(P(1+T\epsilon (\gamma ) \gamma ^{k(P)1})=0\).
Definition 3.1
Let \(M_{\Lambda }(K,{\mathbb {I}})\) be the space of \({\mathbb {I}}\)adic Hilbert modular forms of level K. We later give an example of an \({\mathbb {I}}\)adic toric Eisenstein series (cf. Sect. 5.2).
An \({\mathbb {I}}\)adic Hilbert modular form of level K is in fact a padic Hilbert modular form of level K over \({\mathbb {I}}\) (cf. [9, §4.4.1]). Thus, it has functorial interpretation as in Sect. 3.2 and satisfies (Gp1)–(Gp3). In particular, the Hecke operators as in Sect. 3.3 are defined also for \({\mathbb {I}}\)adic Hilbert modular forms.
Let \(\kappa : O_{p}^{\times }\rightarrow \Lambda ^\times \) be the universal cyclotomic character given by \( u \mapsto \langle N_p(u) \rangle \in {{\varvec{\Gamma }}} \subset \Lambda ^\times \), where \(u \in O_p^\times \).

(Gp\(\Lambda \)) \(F(\underline{\mathcal {A}},\eta _{p}^\mathrm{ord}\circ u)=\kappa (u) F(\underline{\mathcal {A}},\eta _{p}^\mathrm{ord})\), where \((\underline{\mathcal {A}},\eta _{p}^\mathrm{ord}) \in Ig(S)\), S an \({\mathbb {I}}\)algebra as in Sect. 3.2 and \(u \in O_{p}^\times \).
4 Deformation of modular measures
In this section, we describe anticyclotomic measures associated with a class of classical and \(\Lambda \)adic Hilbert modular forms.
The construction of such measures associated with classical Hilbert modular forms is due to Hida and Hsieh. The heart of this section is the construction of padic deformation of these measures. The construction builds in the geometric framework of Sects. 2 and 3.
4.1 Modular measures
In this subsection, we firstly describe an anticyclotomic measure associated with a class of classical Hilbert modular form due to Hida. The construction is perhaps formal. We then discuss its nontriviality also due to Hida. The nontriviality is quite subtle and relies on Chai’s theory of Heckestable subvarieties of a mod p Shimura variety.
 (MC1)
f is an \(U_{{\mathfrak {l}}}\)eigenform with the eigenvalue \(a_{{\mathfrak {l}}}(f)\in \overline{\mathbf {Z}}_{p}^\times \) and
 (MC2)
\(f(x_{n}(ta))= \widehat{\lambda }(a)^{1}f(x_n(t))\) for \(a \in U_{n}{\mathbf {A}}_{\mathcal {F}}^\times \).
We now suppose that f is a classical Hilbert modular defined over a number field. In [7], Hida treats the case of nearly holomorphic Hilbert modular forms. We restrict to the classical case as it suffices for the later applications.
(H) Let \(l^{r(\lambda )}\) be the order of the lSylow subgroup of \(\mathbb {F}_p[\lambda ]^\times \). There exists a strict ideal class \(\mathfrak {c} \in \mathrm{Cl}_{\mathcal {F}}\) such that \(\mathfrak {c}=\mathfrak {c(a)}\) for some Rideal \({\mathfrak {a}}\) and for every \(u \in O\) primeto\({\mathfrak {l}}\), there exists \(\beta \equiv u \,\text{ mod } {{\mathfrak {l}}^{r(\lambda )}}\) such that \({\mathbf {a}}_{\beta }(f^{\mathcal {R}},\mathfrak {c}) \in \overline{\mathbf {Z}}_{p}^\times \). Here \({\mathbf {a}}_{\beta }(f^{\mathcal {R}},\mathfrak {c})\) is the \(\beta \)th Fourier coefficient of \(f^{\mathcal {R}}\) at the cusp \((O,\mathfrak {c}^{1})\).
We have the following nontriviality of the modular measure.
Theorem 4.1
Here “almost all” means except for all but finitely many \(\chi \in \mathfrak {X}_{{\mathfrak {l}}}^\) if \(\deg {{\mathfrak {l}}}=1\) and a Zariski dense subset of \(\mathfrak {X}_{{\mathfrak {l}}}^\), otherwise.
4.2 \(\Lambda \)Adic Hecke characters
In this subsection, we describe certain generalities about Hida’s notion of \(\Lambda \)adic Hecke character.
Let the notation and hypotheses be as in Sects. 1–3. In particular, \(\mathcal {K}/\mathcal {F}\) is a CM quadratic extension. In what follows, we consider Hecke characters over \(\mathcal {K}\). We recall that \({\mathbb {I}}\) is a domain of finite rank over the Iwasawa algebra \(\Lambda _W\), where W is discrete valuation ring finite flat over \(\mathbf {Z}_p\).
Definition 4.2
(Hida) An \({\mathbb {I}}\)adic Hecke character is a continuous Galois character \(\Psi :\mathrm{Gal}({\overline{\mathbf {Q}}}/\mathcal {K})\rightarrow {\mathbb {I}}^\times \) such that for every arithmetic \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\), the arithmetic specialisation \(\Psi _P\) is the padic avatar of an arithmetic Hecke character of infinity type \((k(P)1){\varSigma }\).
We say that \(\Psi \) is selfdual if for every arithmetic \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\), the arithmetic specialisation \(\Psi _P\) is selfdual (cf. [16, §1]). This is also equivalent to the existence of an arithmetic prime \(P_{0}\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) such that the arithmetic specialisation \(\Psi _{P_0}\) is selfdual. We similarly define \(\Psi \) being not residually selfdual.
Let \(\widehat{\mathcal O}_{\mathcal {K}}^{(p)}=\prod _{l\ne p}(\mathcal O_{\mathcal {K}}\otimes \mathbf {Z}_{l})\). Note that the restriction \(\Psi :(\widehat{\mathcal O}_{\mathcal {K}}^{(p)})^{\times } \rightarrow {\mathbb {I}}^\times \) is a finite order character as \({\mathbb {I}}^\times \) is an almost pprofinite group. Thus, there exists an integral ideal \({\mathfrak {C}}^{(p)}(\Psi )\) maximal among ideals \({\mathfrak {a}}\) primetop such that \((1+{\mathfrak {a}}\widehat{\mathcal O}_{\mathcal {K}}^{(p)})\cap (\widehat{\mathcal O}_{\mathcal {K}}^{(p)})^{\times } \subset \ker {\Psi }\). We say that \({\mathfrak {C}}^{(p)}(\Psi )\) is the primetop conductor of \(\Psi \).
Lemma 4.3
Let \(\Psi \) be an \({\mathbb {I}}\)adic Hecke character. Then, the primetop conductor \({\mathfrak {C}}^{(p)}(\Psi )\) of \(\Psi \) equals the primetop conductor of the arithmetic specialisation \(\Psi _P\) for any \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\). (cf. [14, Prop. 3.2])
We now recall construction of an \({\mathbb {I}}\)adic Hecke character due to Hida (cf. [15, §1.4]). The notion naturally arises in the context of pordinary family of CM modular forms.
Recall that \({\varSigma }\) is a pordinary CM type of the CM quadratic extension \(\mathcal {K}/\mathcal {F}\). For \(\mathfrak {p}p\) in \(\mathcal {F}\), let \(\mathfrak {p=PP}^c\) for \(\mathfrak {P}\in {\varSigma }_p\). Let \(\lambda \) be an arithmetic Hecke character of conductor \({\mathfrak {C}}\) with infinity type \(k{\varSigma }\) such that \(k>1\) and the conductor \({\mathfrak {C}}\) is outside \({\varSigma }_{p}^{c}\). Let \({\mathfrak {C}}^{(p)}\) be the primetop part of \({\mathfrak {C}}\). Let \(\mathfrak {N}\) be the primetop part of \(N_{\mathcal {K}/\mathcal {F}}({\mathfrak {C}})\mathcal {D}_{\mathcal {K}/\mathcal {F}}\), where \(N_{\mathcal {K}/\mathcal {F}}\) is the norm map and \(\mathcal {D}_{\mathcal {K}/\mathcal {F}}\) the discriminant of the extension \(\mathcal {K}/\mathcal {F}\). Let \(\lambda ^{}\) be the Hecke character given by \(\lambda ^{}({\mathfrak {a}})=\lambda ({\mathfrak {a}}^{c}{\mathfrak {a}}^{1})\). Let \({\mathfrak {C}}(\lambda ^{})\) be the conductor of \(\lambda ^{}\).
Let \(\mathcal O_{{\varSigma }}^\times = \varprojlim \nolimits _{n} (\mathcal O_{\mathcal {K}}/\mathfrak {S}^n )^\times = O_p^\times \). We have the following commutative diagram:
Here \({{\varvec{\Gamma }}}_{\mathcal {K}}=\mathrm{Im}(\widehat{\lambda }_{0}) \subset \mathbf {C}_p^\times \) isomorphic to \(\mathbf {Z}_p\) as a topological group and \(\langle \Phi \rangle (x)= \langle \prod _{\sigma \in {\varSigma }} \sigma (x) \rangle \in {{\varvec{\Gamma }}}\). As \({{\varvec{\Gamma }}}_{\mathcal {K}}\) is \(\mathbf {Z}_p\)free of rank 1, there exists a decomposition \(\mathrm{Cl}_{\mathcal {K}}^{}({\mathfrak {C}}(\lambda ^{})p^\infty )= \Delta _{\mathcal {K}} \times {{\varvec{\Gamma }}}_{\mathcal {K}}\) for \(\Delta _{\mathcal {K}}=\ker (\widehat{\lambda }_0)\) compatible with the decomposition \(\mathrm{Cl}_{\mathcal {F}}(\mathfrak {N}p^\infty )= \Delta \times {{\varvec{\Gamma }}}\) (cf. Sect.3.4) such that \(\Delta \) maps inside \(\Delta _\mathcal {K}\). We also note that \({{\varvec{\Gamma }}}\) is an open subgroup of \({{\varvec{\Gamma }}}_{\mathcal {K}}\). For a finite flat valuation ring W over \(\mathbf {Z}_p\), we thus conclude that \(W\llbracket {{{\varvec{\Gamma }}}_{\mathcal {K}}}\rrbracket \) is regular domain finite flat over \(\Lambda _W\). We say that \(P\in \mathrm {Spec}\,(W\llbracket {{{\varvec{\Gamma }}}_{\mathcal {K}}}\rrbracket )(\overline{\mathbf {Q}}_p)\) is arithmetic if it is above an arithmetic prime of \(\mathrm {Spec}\,(\Lambda _W)(\overline{\mathbf {Q}}_p)\). Let \(\upsilon \) be the character given by the composition:
For an arithmetic \(P\in \mathrm {Spec}\,(W\llbracket {{{\varvec{\Gamma }}}_{\mathcal {K}}}\rrbracket )(\overline{\mathbf {Q}}_p)\), the composition \(\upsilon _{P}=P\circ \upsilon \) is the padic avatar \(\widehat{\varphi }_P\) of an arithmetic Hecke character \(\varphi _p\) of infinity type \((k(P)1){\varSigma }\). Enlarging W if necessary, we suppose that \(\widehat{\lambda }\) takes values in \(W^\times \) and consider the \(\Lambda \)adic Hecke character \(\Psi \) given by the product \(\widehat{\lambda }\upsilon : \mathrm{Cl}_{\mathcal {K}}({\mathfrak {C}}^{(p)}\mathfrak {S}^\infty ) \rightarrow W\llbracket {{{\varvec{\Gamma }}}_{\mathcal {K}}}\rrbracket ^{\times }\). By a change of variable of \(\Lambda _W\), we can suppose that there exists an arithmetic \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) of weight \(k(P)=k+1\) such that the arithmetic specialisation \(\Psi _P\) equals \(\widehat{\lambda }\). We summarise the consideration as follows.
Proposition 4.4
Suppose that \(\lambda \) is an arithmetic Hecke character over \(\mathcal {K}\) with infinity type \(k{\varSigma }\) for \(k>1\) and padic avatar \(\widehat{\lambda }\). Then, there exists an \({\mathbb {I}}\)adic Hecke character \(\Psi \) and an arithmetic \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) of weight \(k(P)=k+1\) such that the arithmetic specialisation \(\Psi _P\) equals \(\widehat{\lambda }\).
4.3 Deformation of modular measures
In this subsection, we construct a modular measure associated with a class of \(\Lambda \)adic Hilbert modular forms. This gives a \(\Lambda \)adic interpolation of the modular measures associated with a class of classical Hilbert modular forms in Sect. 4.1. In particular, this gives padic deformation of the modular measures. We also discuss the nontriviality of the measure.
 (MI1)
F is an \(U_{{\mathfrak {l}}}\) eigenform with the eigenvalue \(a_{{\mathfrak {l}}}(F)\in {\mathbb {I}}^\times \) and
 (MI2)
\(F(x_{n}(ta))= \Psi (a)^{1}F(x_n(t))\) for \(a \in U_{n} \mathcal {F}^\times \).
 (MP1)
\(f_P\) is an \(U_{{\mathfrak {l}}}\) eigenform with the eigenvalue \((a_{{\mathfrak {l}}}(F))_P\in \overline{\mathbf {Z}}_{p}^\times \) and
 (MP2)
\(f_P(x_{n}(ta))= \Psi _{P}(a)^{1}f_{P}(x_n(t))\) for \(a \in U_{n} \mathcal {F}^\times \).
From now, let \(\nu \in \mathfrak {X}_{{\mathfrak {l}}}^\) and \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _F \in {\mathbb {I}}\). The following is an immediate consequence of Weierstrass preparation theorem and Theorem 4.1.
Proposition 4.5
Suppose that there exists an arithmetic \(P_{0}\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) such that the hypothesis (H) holds for the classical Hilbert modular form \(f_{P_0}\). Then, for “almost all” \(\nu \in \mathfrak {X_{l}}^\times \), we have \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _{F} \in {\mathbb {I}}^\times \). In particular, \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _{f_{P}}\in \overline{\mathbf {Z}}_{p}^\times \) for all arithmetic \(P \in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\).
This can be considered as an \({\mathbb {I}}\)adic analogue of Theorem 4.1.
Remark
 (Q)
Let \(\nu \in \mathfrak {X}_{{\mathfrak {l}}}^\) be given. When is \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _{F}\ne 0\)?
5 Interpolation of genuine padic Lfunctions
In this section, we prove the existence of an \(l\ne p\)interpolation of the genuine padic Lfunction of the \({\mathbb {I}}\)adic Hecke character over the \({\mathfrak {l}}\)power order anticyclotomic characters (cf. Theorem A). We also describe the nontriviality.
This section builds on the earlier section about deformation of modular measures. The construction of \({\mathbb {I}}\)adic family toric Eisenstein series constitutes a significant part of the section.
5.1 Toric Eisenstein series
In Sects. 5.1.1–5.1.4, we construct a classical toric Eisenstein series which will be used to construct anticyclotomic measure in Sect. 5.3.
We follow Hsieh’s construction in [17] and [16] and make an appropriate choice for the local sections at the places dividing p. The latter allows to treat the case of conductor being divisible by places above p. In consideration of \(\Lambda \)adic family of Hecke characters, the conductor of the corresponding arithmetic Hecke characters is indeed divisible by places above p. We closely follow the exposition in [17] and [16].
Let the notation and hypotheses be as in the introduction and Sect. 3.
5.1.1 Eisenstein series on \(\mathrm{GL}_2(\mathbf {A}_\mathcal {F})\)
In this subsection, we briefly recall the construction of an Eisenstein series on \(\mathrm{GL}_2(\mathbf {A}_\mathcal {F})\) in terms of a section.
Let \(\chi \) be an arithmetic Hecke character of infinity type \(k{\varSigma }\), where \(k\ge 1\).
5.1.2 Fourier coefficients of Eisenstein series
In this subsection, we recall the formula for the Fourier coefficients of the Eisenstein series on \(\mathrm{GL}_2(\mathbf {A}_\mathcal {F})\).
5.1.3 Choice of the local sections
In this subsection, we choose the local sections which gives rise to the toric Eisenstein series (cf. [17, § 4.3]). As we consider the case of conductor possibly divisible by places above p, we choose the section at these places carefully.
We begin with some notation. Let v be a place of \(\mathcal {F}\). Let \(F=\mathcal {F}_v\) (resp. \(E=\mathcal {K}\otimes _{\mathcal {F}}\mathcal {F}_v\)). Denote by \(z\mapsto \bar{z}\) the complex conjugation. Let \({\!\cdot \!}\) be the standard absolute values on F and let \({\!\cdot \!}_E\) be the absolute value on E given by \(\left z\right _E:=\left z\bar{z}\right \). Let \(d_F=d_{\mathcal {F}_v}\) be a fixed generator of the different \(\mathfrak {d}_\mathcal {F}\) of \(\mathcal {F}/\mathbf {Q}\). Write \(\chi \) (resp. \(\chi _+\)) for \(\chi _v\) (resp. \(\chi _{+,v}\)). If \(v\in \mathbf {h}\), denote by \(\varpi _v\) a uniformiser of \(\mathcal {F}_v\). For a set Y, denote by \({\mathbb {I}}_Y\) the characteristic function of Y.
5.1.4 qExpansion of normalised Eisenstein series
In this subsection, we describe the formula for the qexpansion coefficients of the classical toric Eisenstein series arising from normalisation of the local sections in Sect. 5.1.3.
Proposition 5.1
 (1).
\(k >2\),
 (2).
\(\mathfrak {f} \ne O\),
 (3).
\(\chi _{+}=\tau _{\mathcal {K}/\mathcal {F}}\cdot _{\mathbf {A}_{\mathcal {F}}}\).
Proof
This follows from (3.1) and the calculations of local Whittaker integrals of special local sections in [16, § 4.3] (cf. [17, Prop. 4.1 and Prop. 4.4] and [16, Prop. 4.7]). \(\square \)
Let \({\mathbb {E}}^h_{\chi }=\sum _{u \in \mathcal {U}_{p}} {\mathbb {E}}^h_{\chi ,u}\) and \({\mathbf {D}}_{{\mathfrak {C}}}=\prod _{v{\mathfrak {C}}^{}\mathcal {D}_{\mathcal {K}/\mathcal {F}}} \mathcal {K}_{v}^\times \). The Eisenstein series \({\mathbb {E}}^h_{\chi }\) satisfies the following transformation properties.
Lemma 5.2
 (1).
\({\mathbb {E}}^h_{\chi } \in M_{k}(K_{0}({\mathfrak {l}}),\overline{\mathbf {Z}}_{(p)})\) is an \(U_{{\mathfrak {l}}}\)eigenform with the eigenvalues \(\chi _{+}(\varpi _{{\mathfrak {l}}})\).
 (2).
\({\mathbb {E}}^h_{\chi }[r]=\widehat{\chi }^{1}(r){\mathbb {E}}^h_{\chi }\), where \(r \in {\mathbf {D}}_{{\mathfrak {C}}}\).
 (3).
\({\mathbb {E}}^h_{\chi }(x_{n}(ta))=\widehat{\chi }^{1}(a){\mathbb {E}}^h_{\chi }(x_{n}(t))\), where \(a \in {\mathbf {D}}_{{\mathfrak {C}}}\mathbf {A}_{\mathcal {F}}^\times U_n\) (cf. [16, Prop. 4.8]).
5.2 \(\Lambda \)Adic toric Eisenstein series
In this subsection, we describe an \({\mathbb {I}}\)adic toric Eisenstein series which interpolates the classical toric Eisenstein series in the previous subsection.
Proposition 5.3
Proof
We first note that the integral in (5.9) is well defined as it is indeed a finite sum (cf. [16, (4.17)]).
We need to show that \(({\mathbf {a}}_\beta (\mathcal {E}_{\Psi ,u},{\mathfrak {c}}))_{P}={\mathbf {a}}_\beta (\mathcal {E}_{\Psi _P,u},{\mathfrak {c}})\). In view of the fact that both of the coefficients are a product of local Whittaker integrals and Lemma 4.3, it suffices to show the equality for the local Whittaker integrals. For vp, the equality from the fact that \(\widehat{\lambda }_{{\varSigma }_p}(\beta )=\beta ^{k{\varSigma }}\lambda _{{\varSigma }_p}(\beta )\), where \(\lambda _{{\varSigma }}(\beta )=\prod _{w \in {\varSigma }_p} \lambda _{w}(\beta )\). For \(v{\mathfrak {C}}^{}\mathcal {D}_{\mathcal {K}/\mathcal {F}}\), the equality follows form the formula for the local Euler factor \(L(0,\Psi _{P,v})\) (cf. [16, 4.3.6]) and (5.8). For the other v, the equality follows immediately from the definitions. \(\square \)
Let \(\mathcal {E}_{\Psi }=\sum _{u \in \mathcal {U}_{p}} \mathcal {E}_{\Psi ,u}\). The \({\mathbb {I}}\)adic toric Eisenstein series \(\mathcal {E}_{\Psi }\) satisfies the following transformation properties.
Lemma 5.4
 (1).
\(\mathcal {E}_{\Psi } \in M_{\Lambda }(K_{0}({\mathfrak {l}}),{\mathbb {I}})\) is an \(U_{{\mathfrak {l}}}\)eigenform with the eigenvalues \(\Psi _{+}(\varpi _{{\mathfrak {l}}})\).
 (2).
\(\mathcal {E}_{\Psi }[r]=\Psi ^{1}(r)\mathcal {E}_{\Psi }\), where \(r \in {\mathbf {D}}_{{\mathfrak {C}}}\).
 (3).
\(\mathcal {E}_{\Psi }(x_{n}(ta))=\Psi ^{1}(a)\mathcal {E}_{\Psi }(x_{n}(t))\), where \(a \in {\mathbf {D}}_{{\mathfrak {C}}}\mathbf {A}_{\mathcal {F}}^\times U_n\).
5.3 Interpolation
In this subsection, we prove the existence of an \(l\ne p\)interpolation of the genuine padic Lfunction of the \({\mathbb {I}}\)adic Hecke character over the \({\mathfrak {l}}\)power order anticyclotomic characters (cf. Theorem A). We also describe the nontriviality.
Let the notation and hypotheses be as in Sects. 1–4 and 5.1–5.2. We first state the following result due to Hida and Hsieh.
Theorem 5.5
Proof
We are now ready to prove Theorem A.
Theorem A
Here \(\nu \) is a finite order character of \(\Gamma _{\mathfrak {l}}^\), \(P \in \mathrm {Spec}\,({\mathbb {I}}_\nu )(\overline{\mathbf {Q}}_p)\) an arithmetic prime with weight k(P), \({\mathbb {I}}_{\nu }\) the finite flat extension of \({\mathbb {I}}\) and CM periods \((\Omega _{p},\Omega _{\infty })\) as above. In particular, \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu {\hbox {d}}\varphi _{\Psi } \in {\mathbb {I}}_\nu \) equals the genuine padic Lfunction of the \({\mathbb {I}}_\nu \)adic Hecke character \(\Psi \nu \).
Proof
In view of (ML1)–(ML2) and Lemma 5.4, we have a welldefined measure \({\hbox {d}}\varphi _{\mathcal {E}_{\Psi }}\) on \(\Gamma _{{\mathfrak {l}}}^{}\). By construction, it interpolates the measures \({\hbox {d}}\varphi _{\Psi _{P}}\) for arithmetic \(P \in \mathrm {Spec}\,({\mathbb {I}}_\nu )(\overline{\mathbf {Q}}_p)\). \(\square \)
Remark
In view of Proposition 4.4, the above theorem thus provides an \({\mathbb {I}}\)adic interpolation of Hida’s modular associated with a given Hecke character of infinity type \(k{\varSigma }\) for \(k\ge 1\). The padic deformation in “primetop Iwasawa theory” does not seem to be considered in the literature before.
We now consider the nontriviality of the measure \({\hbox {d}}\varphi _\Psi \). For simplicity, we only consider the case when \(\Psi \) is selfdual (cf. Sect. 4.2 the for the definition).
Proposition 5.6
Let \(\Psi \) be a selfdual \({\mathbb {I}}\)adic Hecke character with the primetop conductor primeto\({\mathfrak {l}}\) and the root number one. Suppose that there exists an \(P_{0} \in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) such that \(\mu _{p}(\Psi _{P_{0},v})=0\), for all \(v{\mathfrak {C}}(\Psi )^\). Then, for “almost all” \(\nu \in \mathfrak {X_{l}}^\times \), we have \(\int _{\Gamma _{{\mathfrak {l}}}^} \nu \mathrm{{d}}\varphi _{\Psi } \in {\mathbb {I}}_{\nu }^\times \).
Proof
In view of Proposition 4.5, it suffices to verify the hypothesis (H) for the toric Eisenstein series \({\mathbb {E}}^h_{\Psi _{P_0}}\). As \(({\mathbb {E}}^h_{\Psi _{P_0}})^{\mathfrak {R}}=\Delta _\mathrm{alg}{\mathbb {E}}^h_{\Psi _{P_0}}\), the hypothesis follows from [16, Prop. 6.3] and [17, Thm. 6.5]. \(\square \)
6 Interpolation of padic Abel–Jacobi image
In this section, we briefly describe the upcoming analogous interpolation of the padic Abel–Jacobi image of generalised Heegner cycles associated with a selfdual Rankin–Selberg convolution of an \({\mathbb {I}}\)adic Hida family of elliptic Hecke eigenforms and an \({\mathbb {I}}\)adic Hecke character with root number \(1\).
Under the generalised Heegner hypothesis, a selfdual Rankin–Selberg convolution of an \({\mathbb {I}}\)adic Hida family of elliptic Hecke eigenforms and an \({\mathbb {I}}\)adic Hecke character has root number \(1\). Accordingly, the centralcritical Lvalues identically vanish in the family. The Bloch–Beilinson conjectures thus predict the existence of nontorsion cycles associated with the family which are homologically trivial. A candidate for the cycles are generalised Heegner cycles. The construction is due to Bertolini–Darmon–Prasanna and generalises the one of classical Heegner cycles (cf. [1, §2] and [2, §4.1]). The cycle lives in a middledimensional Chow group of a fibre product of a Kuga–Sato variety arising from a modular curve and a selfproduct of a CM elliptic curve. In the case of weight two, the cycle coincides with a Heegner point and the padic Abel–Jacobi image with the padic formal group logarithm. In an ongoing work, we construct an analogous interpolation of the padic Abel–Jacobi image of the cycles.
For simplicity, we mostly restrict here to the case of Heegner points.
Unless otherwise stated, let the notation and hypotheses be as in the introduction. Let \(\mathcal {K}/\mathbf {Q}\) be an imaginary quadratic extension and \(\mathcal O\) the ring of integers. Let p be an odd prime split in \(\mathcal {K}\). For a positive integer n, let \(\mathcal {H}_{n}\) be the ring class field of \(\mathcal {K}\) with conductor n. Let \(\mathcal {H}\) be the Hilbert class field. Let N be a positive integer such that \(p\not \mid N\). For \(k\ge 2\), let \(S_{k}(\Gamma _{0}(N),\epsilon )\) be the space of elliptic modular forms of weight k, level \(\Gamma _{0}(N)\) and Neben character \(\epsilon \). Let \(f\in S_{2}(\Gamma _{0}(N),\epsilon )\) be a Hecke eigenform and \(\mathcal {E}_f\) the corresponding Hecke field. Let \(N_{\epsilon }N\) be the conductor of \(\epsilon \).

(Hg) \(\mathcal O\) contains an ideal \(\mathfrak {N}\) of norm N such that \(\mathcal O/\mathfrak {N} \simeq \mathbf {Z}/N\mathbf {Z}\).
Let \({\mathbf {N}}\) denote the norm Hecke character over \(\mathbf {Q}\) and \({\mathbf {N}}_{\mathcal {K}}:={\mathbf {N}}\circ N_{\mathbf {Q}}^{\mathcal {K}}\) the norm Hecke character over \(\mathcal {K}\). For a Hecke character \(\lambda \) over \(\mathcal {K}\), let \(\mathfrak {f}_\lambda \) (resp. \(\epsilon _\lambda \)) denote its conductor (resp. the restriction \(\lambda _{{\mathbf {A}}_{\mathbf {Q}}^\times }\)). We say that \(\lambda \) is centralcritical for f if it is of infinity type \((j_{1},j_{2})\) with \(j_{1}+j_{2}=2\) and \(\epsilon _{\lambda }=\epsilon {\mathbf {N}^{2}}\).
 (C1)
\(\lambda \) is centralcritical for f,
 (C2)
\(\mathfrak {f}_{\lambda }=b \cdot \mathfrak {N}_\epsilon \) and
 (C3)
The local root number \(\epsilon _{q}(f,\lambda ^{1})=1\), for all finite primes q.
Let \(X_1(N)\) be the modular curve of level \(\Gamma _1(N)\) and the cusp \(\infty \) of \(X_1(N)\). Let \(J_1(N)\) be the corresponding Jacobian. Let \(B_f\) be the abelian variety associated with f by the Eichler–Shimura correspondence and \(\Phi _{f}:J_1(N) \rightarrow B_f\) the associated surjective morphism. By possibly replacing \(B_f\) with an isogenous abelian variety, we suppose that \(B_{f}\) has endomorphisms by the integer ring \(\mathcal O_{\mathcal {E}_{f}}\). Let \(\omega _{f}\) be the differential form on \(X_{1}(N)\) corresponding to f. We use the same notation for the corresponding 1form on \(J_1(N)\). Let \(\omega _{B_{f}} \in \Omega ^{1}(B_{f}/\mathcal {E}_{f})^{\mathcal O_{\mathcal {E}_{f}}}\) be the unique 1form such that \(\Phi _{f}^{*}(\omega _{B_{f}})=\omega _{f}\). Here \(\Omega ^{1}(B_{f}/\mathcal {E}_{f})^{\mathcal O_{\mathcal {E}_{f}}}\) denotes the subspace of 1forms stable under the action of the integer ring \(\mathcal O_{\mathcal {E}_{f}}\).
Let F be an \({\mathbb {I}}\)adic Hida family of elliptic Hecke eigenforms and \(\Psi \) an \({\mathbb {I}}\)adic Hecke character over \(\mathcal {K}\) of primetop conductor \(c\mathfrak {N}_{\epsilon }\) for an integer c such that the corresponding Rankin–Selberg convolution is selfdual. As \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) varies over arithmetic primes of weight 2, we consider the variation of the padic formal group logarithm of the corresponding Heegner points.
Our result regarding the variation is the following.
Theorem B
We also study the nontriviality of the measure \({\hbox {d}}\varphi _{F, \Psi }\) based on the nontriviality results in [3].
We now briefly describe the strategy of the proof. In view of the padic Waldspurger formula in [1] and [2], the padic formal logarithms of the Heegner points associated with the weight two specialisation \(f_P\) essentially equals a weighted sum of evaluation of a toric weight zero padic modular form \(g_P\) at the lpower conductor CM points. As \(P \in \mathrm {Spec}\,({\mathbb {I}}_\nu )(\overline{\mathbf {Q}}_p)\) varies over arithmetic primes, we prove that these weight zero padic modular forms \(g_{P}\) are interpolated by an \({\mathbb {I}}_\nu \)adic elliptic modular form \(G_F\). The \({\mathbb {I}}\)adic measure \({\hbox {d}}\varphi _{F,\Psi }\) is then constructed by the technique in Sect. 4.3 applied to the \({\mathbb {I}}_\nu \)adic elliptic modular form \(G_F\) and the \({\mathbb {I}}_\nu \)adic Hecke character \(\Psi \).
As \(P\in \mathrm {Spec}\,({\mathbb {I}})(\overline{\mathbf {Q}}_p)\) varies over arithmetic primes of a fixed weight \(k\ge 2\), we also construct an analogous measure interpolating the padic Abel–Jacobi image of generalised Heegner cycles associated with the corresponding specialisations of the convolution. The details will appear elsewhere.
Declarations
Acknowledgements
We are grateful to our advisor Haruzo Hida for continuous guidance and encouragement. The topic essentially arose from conversations with him. We thank Francesc Castella, MingLun Hsieh and Chandrashekhar Khare for helpful comments and suggestions. Finally, we are indebted to the referee. The current form of the article owes much to the suggestions of the referee. Particularly, the context of the variation of arithmetic invariants and Sect. 6 arose from the suggestions.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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