2.1 Modular curves and lattices

We begin by recalling some notations about modular curves and lattices. For that purpose, fix algebraic closures $\bar {\mathbb {Q}}$ , $\bar {\mathbb {Q}}_p$ of $\mathbb {Q}$ and $\mathbb {Q}_p$ , respectively, and embeddings of $\bar {\mathbb {Q}}$ into $\bar {\mathbb {Q}}_p$ and $\mathbb {C}$ . The former singles out a prime ideal $\mathfrak p$ of ${\mathcal O}$ lying above p, and we let $\mathcal O_{\mathfrak p}$ denote the completion of $\mathcal O$ at $\mathfrak p$ . We also fix throughout an uniformizer $\varpi $ of $\mathcal O_{\mathfrak p}$ and an isomorphism $\mathbb C_p \simeq \mathbb C$ .

Given a variety $Y/\mathbb {Q}$ and a field extension $F/{\mathbb {Q}}$ , let $Y_F = Y \times F$ denote the base change of Y to F and set $\bar Y = Y_{\bar {\mathbb {Q}}}$ . Fix an integer $N\geq 3$ , and let $Y_1(N) \subset X_1(N)$ denote the canonical models over $\mathbb Q$ of the (affine and projective, respectively) modular curves classifying pairs $(A,i)$ where A is a (generalized) elliptic curve and $i: \mu _N \rightarrow A$ is an embedding of group schemes.

Let $k \geq 2$ be an integer, and let $\theta $ be a Dirichlet character. For a p-ordinary newform $f\in S_k(N,\theta )$ satisfying the congruence $f \equiv E_k(\theta ,1) \,\, \mathrm {mod} \, \mathfrak p^t$ , let $f(q)= \sum a_n(f) q^n$ denote its q-expansion at the cusp $\infty $ . Enlarge F so that it also contains the eigenvalues $\{ a_n(f)\}_{n\geq 1}$ , and still write ${\mathcal O}$ for its ring of integers and ${\mathcal O}_{{\mathfrak p}}$ for its completion at the prime ideal $\mathfrak {p}$ over p. For our further use, we denote by $\alpha _f\in {\mathcal O}_{\mathfrak p}^\times $ (resp. $\beta _f \in {\mathcal O}_{\mathfrak p}$ ) the unit root (resp. non-unit root) of the pth Hecke polynomial of f. Let $\psi _f: G_{\mathbb {Q}_p} \longrightarrow {\mathcal O}_{\mathfrak p}^\times $ denote the unramified character characterized by $\psi _f(\mathrm {Fr}_p)=\alpha _f$ .

Since f is ordinary at $\mathfrak {p}$ , it is well known (cf., e.g., [Reference Fukaya and KatoFK, Section 1.7]) that there are exact sequences of finitely generated ${\mathcal O}_{\mathfrak {p}}[G_{\mathbb Q_p}]$ -modules

(1) $$ \begin{align} 0 \rightarrow T_{f,X}^{\operatorname{\mathrm{sub}}} \rightarrow T_{f,X} \rightarrow T_{f,X}^{\operatorname{\mathrm{quo}}} \rightarrow 0, \\ \nonumber 0 \rightarrow T_{f,Y}^{\operatorname{\mathrm{sub}}} \rightarrow T_{f,Y} \rightarrow T_{f,Y}^{\operatorname{\mathrm{quo}}} \rightarrow 0 \end{align} $$

such that $T_{f,X}^{\operatorname {\mathrm {quo}}}$ and $T_{f,Y}^{\operatorname {\mathrm {quo}}}$ are unramified as $G_{\mathbb Q_p}$ -modules; and where the map $T_{f,X} \rightarrow T_{f,Y}$ induces an isomorphism $T_{f,X}^{\operatorname {\mathrm {sub}}}=T_{f,Y}^{\operatorname {\mathrm {sub}}}$ of free ${\mathcal O}_{{\mathfrak p}}$ -modules of rank $1$ on which $G_{\mathbb Q_p}$ acts through the cyclotomic character.

Let $I_f^* = (T^*_\ell -a_\ell (f)) \subset \mathbb {T}^*$ denote the ideal associated with the system of eigenvalues of f with respect to the dual Hecke operators. As in the introduction, consider the ${\mathcal O}_{{\mathfrak p}}$ -module

(2) $$ \begin{align} T_{f,Y} = H^1_{\operatorname{\mathrm{et}}}(\bar{Y},\operatorname{\mathrm{TSym}}^{k-2}(\mathscr H_{\mathbb{Z}_p})(1)) \otimes_{\mathbb Z_p} \mathcal O_{\mathfrak p}/ I^*_f, \end{align} $$

and define similarly $T_{f,X}$ , using compact support cohomology instead.

Note that the two lattices $T_{f,X}$ and $T_{f,Y}$ may give rise to different ${\mathcal O}_{{\mathfrak p}}[G_{\mathbb {Q}}]$ -modules in spite of the fact that the associated rational Galois representations

$$\begin{align*}V_f := T_{f,X} \otimes F_{{\mathfrak p}} \simeq T_{f,Y} \otimes F_{{\mathfrak p}} \end{align*}$$

are isomorphic.

Finally, let $\Sigma _X$ and $\Sigma _Y$ denote the torsion submodules of $T_{f,X}$ and $T_{f,Y}$ , respectively, and let

(3) $$ \begin{align} T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} := T_{f,Y}^{\operatorname{\mathrm{quo}}}/\Sigma_Y \simeq {\mathcal O}_{\mathfrak p}(\psi_f) \end{align} $$

denote the free quotient.

Since $f \equiv E_k(\theta ,1) \,\mbox {mod}\,\mathfrak p^t$ , we have

(4) $$ \begin{align} T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes \mathcal O/\mathfrak p^t \simeq \mathcal O/\mathfrak p^t(\theta), \end{align} $$

which amounts to the congruence $\psi _f \equiv \theta \, (\mbox {mod}\,\mathfrak p^t)$ as unramified characters of $G_{\mathbb {Q}_p}$ .

In the cases where we deal with several modular forms of common level N, we slightly abuse notations and still write F for the finite extension of $\mathbb Q$ generated by the field of coefficients of all of them and the values of all Dirichlet characters of conductor N. Similarly, $\mathcal O$ stands for the ring of integers and $\mathfrak p \subset \mathcal O$ for a prime ideal above p.

As in our previous work, we need the following assumption to establish our main results.

In this case, we will say that $T_{f,Y}$ (or simply f, for short) satisfies the weak-Gorenstein condition. The reason is that this assumption automatically follows if the localization of the Hecke algebra acting on $M_k(\Gamma _1(N))$ at the Eisenstein ideal is Gorenstein.

Several of our results are suitably formulated in the language of families. Let $\mathcal W$ denote the weight space defined as the formal spectrum of the Iwasawa algebra $\Lambda =\mathcal O_{\mathfrak p}[[\mathbb Z_p^{\times }]]$ . The set of classical points in $\mathcal W$ is given by characters $\nu _{s,\xi }$ of the form $\xi \varepsilon _{\operatorname {\mathrm {cyc}}}^{s-1}$ where $\xi $ is a Dirichlet character of p-power conductor, $\varepsilon _{\operatorname {\mathrm {cyc}}}$ is the cyclotomic character, and s is an integer; this forms a dense subset in $\mathcal W$ for the Zariski topology. Let $\mathcal W^{\circ }$ further denote the set of those points with $\xi =1$ ; we shall often write s in place of $\nu _s = \nu _{s,1}$ . Let $\mathcal W^{\pm } \subset \mathcal W$ denote the topological closure of the set of points $\xi \varepsilon _{\operatorname {\mathrm {cyc}}}^{s-1}$ with $(-1)^{s-1} \xi (-1) = \pm 1$ . We have $\mathcal W= \mathcal W^+ \sqcup \mathcal W^-$ , and we write $\Lambda = \Lambda ^+ \oplus \Lambda ^-$ for the corresponding decomposition of the Iwasawa algebra.

2.2 Congruences between canonical periods

Recall that the de Rham Dieudonné module $D_{\operatorname {{\mathrm {dR}}}}(V_f)$ associated with the eigenform f is an $F_{\mathfrak p}$ -filtered vector space of dimension $2$ . Poincaré duality yields a perfect pairing

$$\begin{align*}\langle \, , \, \rangle: D_{\operatorname{{\mathrm{dR}}}}(V_f(-1)) \times D_{\operatorname{{\mathrm{dR}}}}(V_{f^*}) \rightarrow F_{\mathfrak p}, \end{align*}$$

and there is an exact sequence of Dieudonné modules

(5) $$ \begin{align} 0 \rightarrow D_{\operatorname{{\mathrm{dR}}}}(V_f^{\operatorname{\mathrm{sub}}}) \rightarrow D_{\operatorname{{\mathrm{dR}}}}(V_f) \rightarrow D_{\operatorname{{\mathrm{dR}}}}(V_f^{\operatorname{\mathrm{quo}}}) \rightarrow 0, \end{align} $$

where $D_{\operatorname {{\mathrm {dR}}}}(V_f^{\operatorname {\mathrm {sub}}})$ and $D_{\operatorname {{\mathrm {dR}}}}(V_f^{\operatorname {\mathrm {quo}}})$ have both dimension $1$ .

Falting’s theorem associates with f a regular differential form $\omega _f \in \mathrm {Fil}(D_{\operatorname {{\mathrm {dR}}}}(V_f))$ , which gives rise to an element in $D_{\operatorname {{\mathrm {dR}}}}(V_f^{\operatorname {\mathrm {quo}}})$ via the right-most map in (5) and in turn induces a linear form

(6) $$ \begin{align} \omega_f \, : \, D_{\operatorname{{\mathrm{dR}}}}(V_{f^*}^{\operatorname{\mathrm{sub}}}(-1)) \rightarrow F_{\mathfrak p}, \quad \eta \mapsto \langle \omega_f, \eta \rangle \end{align} $$

that we continue to denote with same symbol by a slight abuse of notation.

There is also a differential $\eta _f$ , which is characterized by the property that it spans the line $D_{\operatorname {{\mathrm {dR}}}}(V_f^{\operatorname {\mathrm {sub}}}(-1))$ and is normalized so that

(7) $$ \begin{align} \langle \eta_f, \omega_{f^*} \rangle = 1. \end{align} $$

Again, it induces a linear form

(8) $$ \begin{align} \eta_f \, : \, D_{\operatorname{{\mathrm{dR}}}}(V_{f^*}^{\operatorname{\mathrm{quo}}}) \rightarrow F_{\mathfrak p}, \quad \omega \mapsto \langle \eta_f, \omega \rangle. \end{align} $$

Now, let T be an unramified ${\mathcal O}_{\mathfrak p}[\mathrm {Gal\,}(\bar {\mathbb {Q}}_p/\mathbb {Q}_p)]$ -module and set $V=T\otimes F_{\mathfrak p}$ . Let $\hat {\mathbb {Z}}_p^{\mathrm {ur}}$ denote the completion of the ring of integers of the maximal unramified extension of $\mathbb {Q}_p$ , and define the integral Dieudonné module

$$ \begin{align*}D(T) := (T \, \hat{\otimes}_{\mathbb{Z}_p} \, \hat{\mathbb{Z}}_p^{\mathrm{ur}})^{\mathrm{Fr}_p=1}. \end{align*} $$

As shown in [Reference OhtaOh1, Theorem 2.1.11], we have $D_{\operatorname {{\mathrm {dR}}}}(V) = D(T)\otimes F_{\mathfrak p}$ .

As explained, e.g., in [Reference Fukaya and KatoFK, Proposition 1.7.6], there is a functorial isomorphism of ${\mathcal O}_{\mathfrak p}$ -modules (forgetting the Galois structure) given by

(9) $$ \begin{align} T \, \stackrel{\sim}{\longrightarrow } D(T). \end{align} $$

This map is not canonical as it depends on a choice of root of unity; for $T={\mathcal O}_{\mathfrak p}(\chi )$ , we take it to be given by the rule $1 \mapsto \, \mathfrak g(\chi )$ .

Recall the free ${\mathcal O}_{\mathfrak {p}}$ -quotient

$$ \begin{align*}T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \simeq {\mathcal O}_{\mathfrak p}(\psi_f); \end{align*} $$

note in particular that $T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}}$ is unramified.

Recall the uniformizer $\varpi $ fixed at the outset in Section 2.1. Let $C_{f^*} =(\varpi ^r)\subseteq O_{\mathfrak p}$ denote the congruence ideal attached with $f^*$ as defined, e.g., in [Reference OhtaOh2]. In Hida’s terminology, $\varpi ^r$ is sometimes called a congruence divisor. According to [Reference Kings, Loeffler and ZerbesKLZ2, Section 10.1.2], the image of $\eta _{f^*|D(T_{f,Y}^{\operatorname {\mathrm {quo}}})}$ is precisely the inverse $C_{f^*}^{-1}$ of the congruence ideal, and hence there is an isomorphism

$$\begin{align*}\eta_{f^*} \, : \, D(T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}}) \longrightarrow C_{f^*}^{-1}, \quad \omega \mapsto \langle \eta_{f^*},\omega \rangle. \end{align*}$$

Setting $\tilde {\eta} _{f^*} := \varpi ^r \cdot {\eta} _{f^*}$ , the above map gives rise to an isomorphism of $O_{\mathfrak p}$ -modules

(10) $$ \begin{align} \tilde {\eta}_{f^*} \, : \, D(T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}}) \longrightarrow \mathcal O_{\mathfrak p}, \quad \omega \mapsto \langle \tilde {\eta}_{f^*},\omega \rangle. \end{align} $$

Choose an isomorphism of $G_{\mathbb Q_p}$ -modules.

(11) $$ \begin{align} \bar\iota_1: \bar{T}_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \stackrel{\sim}{\longrightarrow} {\mathcal O}/\mathfrak{p}^t(\theta). \end{align} $$

Fixing such a map amounts to choosing the class $(\mathrm {mod}\,{\mathfrak p^t})$ of an isomorphism of local modules $\iota _1: T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}} \simeq \mathcal O_{\mathfrak p}(\psi _f)$ . In light of the functoriality provided by (9), this determines and is determined by the class $(\mathrm {mod}\,{\mathfrak p^t})$ of an isomorphism $D(\iota _1): D(T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}}) \simeq D({\mathcal O}_{\mathfrak p}(\psi _f))$ .

We choose $\iota _1$ as the single isomorphism making the following diagram commutative:

(12)

Indeed, since both $ \langle \, , \tilde {\eta} _{f^*}\rangle $ and $\cdot 1/ \mathfrak g(\theta )$ are isomorphisms, it follows that such a map $D(\iota _1)$ exists and is unique, and this in turn pins down $\iota _1$ and $\bar \iota _1$ in light of (9).

There are similar results for the differential $\omega _f$ . Using again [Reference Kings, Loeffler and ZerbesKLZ2, Section 10.1.2], there is an isomorphism

$$\begin{align*}\omega_{f^*} \, : \, D(T_{f,Y}^{\operatorname{\mathrm{sub}}}) \longrightarrow \mathcal O_{\mathfrak p}, \quad {\eta} \mapsto \langle \omega_{f^*}, {\eta} \rangle. \end{align*}$$

Hence, we may choose an isomorphism of $G_{\mathbb Q_p}$ -modules

(13) $$ \begin{align} \bar\iota_2: \bar{T}_{f,Y}^{\operatorname{\mathrm{sub}}} \stackrel{\sim}{\longrightarrow} {\mathcal O}/\mathfrak{p}^t(1). \end{align} $$

Fixing such a map amounts to choosing the class $(\mathrm {mod}\,{\mathfrak p^t})$ of an isomorphism of local modules $\iota _2: T_{f,Y}^{\operatorname {\mathrm {sub}}} \simeq \mathcal O_{\mathfrak p}(k-1)$ . This determines and is determined by the class $(\mathrm {mod}\,{\mathfrak p^t})$ of an isomorphism $D(\iota _2): D(T_{f,Y}^{\operatorname {\mathrm {sub}}}) \simeq D({\mathcal O}_{\mathfrak p})$ .

We choose $\iota _2$ as the single isomorphism making the following diagram commutative:

(14)

Indeed, since both $ \langle \, , \omega _{f^*}\rangle $ and $\cdot 1$ are isomorphisms, it follows that such a map $D(\iota _2)$ exists and is unique, and this in turn pins down $\iota _2$ and $\bar \iota _2$ .

2.3 The Beilinson–Kato Euler system

In this section, we recall the reciprocity law for Beilinson–Kato elements of arbitrary weight. For details on the constructions, see, e.g., [Reference KatoKa] or [Reference Bertolini and DarmonBD]. As in the introduction, let $f \in S_k(N,\chi _f)$ , let $\psi $ stand for a Dirichlet character, and consider two auxiliary positive integers $(\ell ,m)$ , with $\ell ,m \geq 2$ and the triple $(k,\ell ,m)$ balanced. The Beilinson–Kato class is an element

(15) $$ \begin{align} \kappa_f \in H^1(\mathbb Q, T_{f,Y}(\psi)(1)) \end{align} $$

obtained as the cup product of two Eisenstein classes $\delta _g$ and $\delta _h$ of weights $\ell -2$ and $m-2$ , respectively. The class is then obtained by applying the Hochschild–Serre map and projecting to the f-isotypical component. In particular,

$$\begin{align*}\delta_g \in H_{\operatorname{\mathrm{et}}}^1(\bar Y, \operatorname{\mathrm{TSym}}^{\ell-2}(\mathscr H_{\mathbb Z_p})(\psi)(1)) \quad \text{and} \quad \delta_h \in H_{\operatorname{\mathrm{et}}}^1(\bar Y, \operatorname{\mathrm{TSym}}^{m-2}(\mathscr H_{\mathbb Z_p})(1)). \end{align*}$$

As a piece of notation, we write $c = (k+\ell +m-2)/2$ and $q = (\ell +m-k)/2> 0$ . The classes also depend on an auxiliary pair of Dirichlet character $(\chi _1,\chi _2)$ , and $\chi _1/\chi _2$ is assumed to be nontrivial modulo p and of nebentype relatively prime to p. If the choices of $\ell $ , m, and $\chi $ are clear from the setting, we drop them from the notation.

Further, we emphasize that in Kato’s construction, in order to get integral classes, one has to choose a pair of integers $c,d$ and the class actually depends on this choice; here, we content ourselves with having p-integral classes. Under the running assumptions, the class of equation (15) is integral at p provided that $p \nmid N$ and that $\chi $ is nontrivial modulo p, as it follows from the discussions of, e.g., [Reference KingsKings, Section 5.4] and [Reference Kings, Loeffler and ZerbesKLZ, Sections 4, 7, and 9].

It may be instructive to keep in mind the parallelism with the weight two scenario considered in [Reference Rivero and RotgerRiRo, Section 2]. In that case, where $k=\ell =m=2$ , the Beilinson–Kato class is obtained as the cup product of the modular units

$$\begin{align*}u_{\chi_1,\chi_2} \cup u_{1,\xi} \in H_{\operatorname{\mathrm{et}}}^2(\bar Y,\mathbb Z_p(\chi)(2)), \end{align*}$$

where $u_{\chi _1,\chi _2} \in H^1(\mathbb Q, \mathbb Z_p(\chi _1)(1))$ and $u_{1,\xi } \in H^1(\mathbb Q, \mathbb Z_p(1))$ . Applying now the Hochschild–Serre map and projecting to the f-isotypical component, we get the desired class.

As a piece of notation, let

$$ \begin{align*}\underline{\varepsilon}_{\operatorname{\mathrm{cyc}}}: G_{\mathbb Q} \rightarrow \Lambda^{\times} \end{align*} $$

denote the $\Lambda $ -adic cyclotomic character which sends a Galois element $\sigma $ to the group-like element $[\varepsilon _{\operatorname {\mathrm {cyc}}}(\sigma )]$ . It interpolates the powers of the $\mathbb Z_p$ -cyclotomic character, in the sense that for any arithmetic point of the form $\nu _{s,\xi }\in \mathcal W$ ,

(16) $$ \begin{align} \nu_{s,\xi} \circ \underline{\varepsilon}_{\operatorname{\mathrm{cyc}}} = \xi \cdot \varepsilon_{\operatorname{\mathrm{cyc}}}^{s-1}. \end{align} $$

Finally, let t stand for Fontaine’s p-adic analogue of $2\pi i$ .

Proposition 2.2 Assume that $\alpha _f \psi (p) \not \equiv 1\ \ \pmod {\mathfrak p}$ . Then there exists a homomorphism of $\Lambda $ -modules

$$\begin{align*}\mathcal L_{f}^-: H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes \Lambda(\psi)(\varepsilon_{\operatorname{\mathrm{cyc}}}^q \underline{\varepsilon}_{\operatorname{\mathrm{cyc}}})) \longrightarrow \Lambda \end{align*}$$

satisfying the following interpolation property: for $s \in \mathcal W^{\circ }$ , the specialization of $\mathcal L_{f}^-$ at s is the homomorphism

$$\begin{align*}\mathcal L_{f,s}^-: H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}}(\psi)(s+q-1)) \longrightarrow \mathcal O_{\mathfrak p} \end{align*}$$

given by

$$\begin{align*}\mathcal L_{f,s}^- = \frac{1- \bar \psi(p) \alpha_{f}^{-1} p^{-s}}{1-\psi(p)\alpha_f p^{s-1}} \times \begin{cases} \frac{(-1)^s}{(s-1)!} \times \langle \log_{\operatorname{\mathrm{BK}}}, t^s \tilde {\eta}_{f^*} \rangle, & \text{ if } s \geq 2-q, \\ (-s)! \times \langle \exp_{\operatorname{\mathrm{BK}}}^*, t^s \tilde {\eta}_{f^*} \rangle, & \text{ if } s < 2-q, \end{cases} \end{align*}$$

where $\log _{\operatorname {\mathrm {BK}}}$ is the Bloch–Kato logarithm and $\exp _{\operatorname {\mathrm {BK}}}^*$ is the dual exponential map.

Proof This follows from Coleman and Perrin-Riou’s theory of $\Lambda $ -adic logarithm maps as extended by Loeffler and Zerbes in [Reference Loeffler and ZerbesLZ]. In particular, [Reference Kings, Loeffler and ZerbesKLZ2, Theorem 8.2.3] yields an injective map

$$ \begin{align*}H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes \Lambda(\psi)(\varepsilon_{\operatorname{\mathrm{cyc}}}^q \underline{\varepsilon}_{\operatorname{\mathrm{cyc}}})) \longrightarrow D(T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}}) \otimes \Lambda, \end{align*} $$

since $H^0(\mathbb Q_p,T_{f,\circ }^{\operatorname {\mathrm {quo}}}(\psi )) = 0$ because of the assumption that $\alpha _f \psi (p) \not \equiv 1$ modulo $\mathfrak p$ . This map is characterized by the interpolation property formulated in [Reference Loeffler and ZerbesLZ, Appendix B]. Next, we apply the pairing of (10) and the result follows.

In particular, for $q=0$ , we recover the homomorphism introduced in [Reference Rivero and RotgerRiRo, Section 3.4]. The assumption $\alpha _f \psi (p) \not \equiv 1$ modulo $\mathfrak p$ is necessary to have image in $\Lambda $ , but it may be still possible that for a fixed value of s the image is integral even though the condition is not satisfied (see [Reference Kings, Loeffler and ZerbesKLZ2, Section 8.2] for a precise description of the kernel and cokernel of that map).

The following result, essentially due to Kato (and reformulated in our language by Bertolini and Darmon), shows that there exists a $\Lambda $ -adic cohomology class whose bottom layer is precisely $\kappa _f$ and whose image under a suitable regulator map recovers the p-adic L-function attached to $f^*$ . To fix notations, let $L_p(f^*,\psi ,s)$ stand for the Mazur–Swinnerton-Dyer p-adic L-function attached to a modular form $f^*$ and a Dirichlet character $\psi $ , with the conventions used in Section 3 of loc. cit. Let $L_p(f^*,E_k(\chi _1,\chi _2),s)$ be the Hida–Rankin p-adic L-function attached to the two modular forms $f^*$ and $E_k(\chi _1,\chi _2)$ . The result admits different variations according to our precise choice of twists, but all of them follow from this one in an easy way.

Proposition 2.3 Under the assumptions of Proposition 2.2, there exists a $\Lambda $ -adic cohomology class

$$\begin{align*}\kappa_{f,\infty} \in H^1(\mathbb Q, T_{f,Y} \otimes \Lambda^-(\varepsilon_{\operatorname{\mathrm{cyc}}}^q \underline{\varepsilon}_{\operatorname{\mathrm{cyc}}})) \end{align*}$$

such that:

1. (a) There is an explicit reciprocity law

   $$\begin{align*}\mathcal L_{f}^-(\mathrm{res}_p(\kappa_{f,\infty})^-) = L_p(f^*, E_{\ell}(1,\chi),q+s), \end{align*}$$
   and there is a factorization
   $$\begin{align*}L_p(f^*, E_{\ell}(1,\chi),q+s) = \frac{-L_p(f^*,\chi,\frac{k+\ell-m}{2})}{\mathfrak g(\bar \chi)} \times L_p(f^*,q+s). \end{align*}$$
   Here, $\mathrm {res}_p$ stands for the map corresponding to localization at p and $\mathrm {res}_p(\kappa _{f,\infty })^-$ is the image under the map induced in cohomology from the projection map $T_{f,Y} \rightarrow T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}}$ .

2. (b) The bottom layer $\kappa _{f}(q)$ lies in $H_{\operatorname {\mathrm {f}}}^1(\mathbb Q, T_{f}(q))$ and satisfies

   $$\begin{align*}\kappa_{f}(q) = \mathcal E_f \cdot \kappa_f, \end{align*}$$
   where $\mathcal E_f$ is the Euler factor
   $$\begin{align*}\mathcal E_f = (1- \bar \chi(p) \beta_{f} p^{m-c-1})(1-\bar \chi_f \chi(p) \beta_{f} p^{\ell-c-1})(1- \alpha_{f}p^{-1})(1-\beta_{f}p^{-1}). \end{align*}$$

Note that [Reference Bertolini and DarmonBD] does not keep track of integrality issues, but the running assumptions are enough to have integral classes at p; in particular, the factors depending on c and d are invertible modulo p because of the nontriviality of the auxiliary character $\chi _1/\chi _2$ and may be absorbed inside the cohomology classes (see, e.g., [Reference Kings, Loeffler and ZerbesKLZ2, Section 9.4] for a discussion on the shape of the factors, where the role of $\chi $ is played by the product of $\chi $ and $\nu $ in their notations).

2.4 The Beilinson–Flach Euler system

We now introduce the Beilinson–Flach classes and a reciprocity law for them. See [Reference Kings, Loeffler and ZerbesKLZ, Reference Kings, Loeffler and ZerbesKLZ2] for details on the construction. Let $f \in S_k(N,\chi _f)$ and $g \in S_{\ell }(N,\chi _g)$ be two cuspidal eigenforms with $\chi _f \chi _g \not \equiv 1\ \ \pmod {\mathfrak p}$ , and let s be an integer such that $1 \leq s < \min (k,\ell )$ . The Beilinson–Flach element constructed in [Reference Kings, Loeffler and ZerbesKLZ] is a class

(17) $$ \begin{align} \kappa_{f,g} \in H^1(\mathbb Q, T_{f,Y} \otimes T_{g,Y}(1-s)), \end{align} $$

obtained via the pushforward of a class

$$\begin{align*}\delta_h \in H_{\operatorname{\mathrm{et}}}^1(\bar Y, \operatorname{\mathrm{TSym}}^{m-2}(\mathscr H_{\mathbb Z_p})(1)) \end{align*}$$

along the diagonal $Y \hookrightarrow Y^2$ ; then one applies the Hochschild–Serre map and projects to the $(f,g)$ -isotypic component. Here, we can see the cyclotomic twist s as a shortcut for $c-m+1$ . Observe once more that these classes have no denominators at p; see [Reference Kings, Loeffler and ZerbesKLZ, Remark 7.1.2] and [Reference Kings, Loeffler and ZerbesKLZ2, Section 3.2].

In the case where $k=\ell =2$ and $s=1$ , the construction can be summarized as follows: take the Eisenstein element $u_{1, \chi _f \chi _g} \in H_{\operatorname {\mathrm {et}}}^1(Y, \mathbb Z_p(1))$ and consider the Gysin map (push-forward of appropriate sheaves) along the diagonal $Y \hookrightarrow Y \times Y$ to get a class in $H_{\operatorname {\mathrm {et}}}^3(Y^2,\mathbb Z_p(2))$ . Applying the Hochschild–Serre map and projecting to the $(f,g)$ -istypical component, we get the desired class. See, e.g., [Reference Kings, Loeffler and ZerbesKLZ, Section 4]. (It must be pointed out that we are allowed to use the Hochschild–Serre identification since the $H^3$ of a surface over $\bar {\mathbb Q}$ vanishes.)

Proposition 2.5 Assume that $\alpha _f \alpha _g^{-1} \chi _g(p) \not \equiv 1\ \ \pmod {\mathfrak p}$ . Then there exists a homomorphism of $\Lambda $ -modules

$$\begin{align*}\mathcal L_{f,g}^{-+}: H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes T_{g,Y}^{\operatorname{\mathrm{sub}}} \otimes \Lambda(\varepsilon_{\operatorname{\mathrm{cyc}}}\underline{\varepsilon}_{\operatorname{\mathrm{cyc}}})) \longrightarrow \Lambda \end{align*}$$

satisfying the following interpolation property: for $r \in \mathcal W^{\circ }$ , the specialization of $\mathcal L_{f,g}^-$ at s is the homomorphism

$$\begin{align*}\mathcal L_{f,g,s}^{-+}: H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes T_{g,Y}^{\operatorname{\mathrm{sub}}} (s)) \longrightarrow \mathcal O_{\mathfrak p} \end{align*}$$

given by

$$\begin{align*}\mathcal L_{f,g,s}^{-+} = \frac{1-\alpha_f^{-1} \beta_g^{-1} p^{-s}}{1-\alpha_f \beta_g p^{s-1}} \times \begin{cases} \frac{(-1)^{s+1}}{s!} \times \langle \log_{\operatorname{\mathrm{BK}}}, t^{s+1} \tilde {\eta}_{f^*} \otimes \omega_{g^*} \rangle, & \text{ if } s \geq 0, \\ (-s-1)! \times \langle \exp_{\operatorname{\mathrm{BK}}}^*, t^{s+1} \tilde {\eta}_{f^*} \otimes \omega_{g^*} \rangle, & \text{ if } s < 0. \end{cases} \end{align*}$$

For the following result, let $L_p(f^*, g^*,s)$ stand for the Hida–Rankin p-adic L-function attached to the convolution of two modular forms. Note that this function is not symmetric on the two modular forms. Let $\chi = \bar \chi _f \bar \chi _g$ . The following result, established by Kings, Loeffler, and Zerbes, shows that there exists a $\Lambda $ -adic class whose bottom layer is $\kappa _{f,g}$ (up to a suitable Euler factor) and which recovers the Hida–Rankin p-adic L-functions.

Proposition 2.6 Under the assumptions of Proposition 2.5, there exists a $\Lambda $ -adic cohomology class

$$\begin{align*}\kappa_{f,g,\infty} \in H^1(\mathbb Q, T_{f,Y} \otimes T_{g,Y} \otimes \Lambda(\varepsilon_{\operatorname{\mathrm{cyc}}} \underline{\varepsilon}_{\operatorname{\mathrm{cyc}}})) \end{align*}$$

such that:

1. (a) There is an explicit reciprocity law

   $$\begin{align*}\mathcal L_{f}^-(\mathrm{res}_p(\kappa_{f,g,\infty})^{-+}) = L_p(f^*, g^*,1+s), \end{align*}$$
   where $\mathrm {res}_p$ stands for the map corresponding to localization at p and $\mathrm {res}_p(\kappa _{f,g,\infty })^{-+}$ is the image under the map induced in cohomology from the projection map $T_{f,Y} \otimes T_{g,Y} \rightarrow T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}} \otimes T_{g,Y}^{\operatorname {\mathrm {sub}}}$ .

2. (b) For $1 \leq s < \min \{k,\ell \}$ , the bottom layer $\kappa _{f,g}(s) \in H^1(\mathbb Q, T_{f,Y} \otimes T_{g,Y}(s))$ satisfies

   $$\begin{align*}\kappa_{f,g}(s) = \mathcal E_{f,g} \cdot \kappa_{f,g}, \end{align*}$$
   where $\mathcal E_{f,g}$ is the Euler factor
   $$\begin{align*}\mathcal E_{f,g} = (1-\beta_f \alpha_g p^{-s}) (1-\beta_f \beta_g \chi(p) p^{s+1-k-\ell})(1-\beta_g \alpha_f p^{-s}) (1-\beta_g \beta_f p^{-s}). \end{align*}$$

Note that the condition $\chi _f\chi _g \not \equiv 1\ \ \pmod {\mathfrak p}$ assures that the c-factor appearing in the reciprocity law is integral and can be absorbed inside the cohomology class. Further, for our applications, we do not need to consider the whole cyclotomic variation, and it is enough to focus our attention to geometric twists, that is, values of s for which certain inequalities are satisfied (and in particular, the results of [Reference Kings, Loeffler and ZerbesKLZ] suffice). As anticipated before, this means that $1 \leq s < \min (k,\ell )$ .

2.5 Diagonal cycles

We finally introduce the cohomology classes obtained via diagonal cycles and their reciprocity laws, following [Reference Bertolini, Seveso and VenerucciBSV1, Reference Darmon and RotgerDR4] and [Reference Bertolini, Seveso and VenerucciBSV2, Section 3]. We also assume that the classes we are going to deal with have no denominators at p, and following the discussion of [Reference Bertolini, Seveso and VenerucciBSV2, Remark 3.3], this is always the case provided that p is greater than the weights. Consider a triple of cuspidal eigenforms $f \in S_k(N,\chi _f)$ , $g \in S_{\ell }(N,\chi _g)$ and $h \in S_m(N,\chi _h)$ , with $k+\ell +m$ even and $\chi _f \chi _g \chi _h = 1$ ; we also assume that the Galois representation attached to f is absolutely irreducible. Let $c=(k+\ell +m-2)/2$ . The diagonal cycle class is an element

(18) $$ \begin{align} \kappa_{f,g,h} \in H^1(\mathbb Q, T_{f,Y} \otimes T_{g,Y} \otimes T_{h,Y}(1-c)). \end{align} $$

When the weights are all two, the class is constructed by pushing forward the class $1 \in H_{\operatorname {\mathrm {et}}}^0(Y, \mathbb Z_p)$ along the diagonal inclusion $Y \hookrightarrow Y^3$ ; this gives an element in $H^4(Y^3,\mathbb Z_p(2))$ , which gives the desired class after applying the Hochschild–Serre map following by projection to the $(f,g,h)$ -isotypic component. Following [Reference Darmon and RotgerDR1], one can also view the diagonal class directly over the closed modular curve, which in our case is a consequence of the congruence relations we will develop (at least modulo p).

Contrary to the previous settings, the absence of a cyclotomic variable makes the situation slightly different. Fortunately, we only need the reciprocity law pointwise, at a point of weight $(k,\ell ,m)$ . If a greater level of generality was needed, we could move all three modular forms $(f,g,h)$ along a Hida (or Coleman) family, but restricting to the central twist. Although the Perrin-Riou map makes sense for the whole family of twists, since the classes are only available for the central ones, we restrict to those ones from now on.

Definition 2.7 Assume that $\alpha _f \chi _f^{-1}(p) \alpha _g^{-1} \alpha _h^{-1} \not \equiv 1$ modulo $\mathfrak p$ . Then we define $\mathcal L_{f,g,h}^{-++}$ as the morphism of $\mathbb Z_p$ -modules

$$\begin{align*}\mathcal L_{f,g,h}^{-++}: H^1(\mathbb Q_p, T_{f,Y,\circ}^{\operatorname{\mathrm{quo}}} \otimes T_{g,Y}^{\operatorname{\mathrm{sub}}} \otimes T_{h,Y}^{\operatorname{\mathrm{sub}}}(1-c)) \longrightarrow \mathcal O_{\mathfrak p} \end{align*}$$

given by

$$\begin{align*}\mathcal L_{f,g,h}^{-++} = \frac{1-\alpha_f^{-1} \beta_g^{-1} \beta_h^{-1} p^{-1}}{1-\alpha_f \beta_g \beta_h} \times \langle \log_{\operatorname{\mathrm{BK}}}, \tilde {\eta}_{f^*} \otimes \omega_{g^*} \otimes \omega_{h^*} \rangle. \end{align*}$$

For the following result, let $L_p(f^*, g^*, h^*,c)$ stand for the special value at weights $(k,\ell ,m)$ of the triple product p-adic L-function associated with the Hida families passing through $(f,g,h)$ , as introduced in [Reference Darmon and RotgerDR1, Section 4]; in particular, this corresponds to the central value of the triple product at the point $s=\frac {k+\ell +m}{2}-1=c$ . As recalled in loc. cit., we can also interpret these values in terms of a suitable pairing, from which it is clear the parallelism with the Hida–Rankin one used in previous sections.

When dealing with diagonal cycles, we implicitly fix a choice of test vectors, for instance, as in [Reference HsiehHs] (note that we can do it since the dominant p-adic L-function will never be the Eisenstein one). To keep the parallelism with the previous settings, let

$$\begin{align*}\kappa_{f,g,h,\operatorname{\mathrm{fam}}} = \mathcal E_{f,g,h} \cdot \kappa_{f,g,h}, \end{align*}$$

where $\mathcal E_{f,g,h}$ is the Euler factor

$$\begin{align*}\mathcal E_{f,g,h} = (1-\beta_f \alpha_g \beta_h p^{-c}) (1-\beta_f \beta_g \alpha_h p^{-c})(1-\alpha_f \beta_g \beta_h p^{-c}) (1-\beta_f \beta_g \beta_h p^{-c}). \end{align*}$$

We can understand this class as the result of the specialization of a class varying in Hida families.

Proposition 2.8 Under the conditions of Definition 2.7, there is an equality

$$\begin{align*}\mathcal L_{f,g,h}^{-++}(\mathrm{res}_p(\kappa_{f,g,h,\operatorname{\mathrm{fam}}})^{-++}) = L_p(f^*, g^*, h^*,c), \end{align*}$$

where $\mathrm {res}_p$ stands for the map corresponding to localization at p and $\mathrm {res}_p(\kappa _{f,g,h,\operatorname {\mathrm {fam}}})^{-++}$ is the image under the map induced in cohomology from the projection map $T_{f,Y} \otimes T_{g,Y} \otimes T_{h,Y}(1-c) \rightarrow T_{f,Y,\circ }^{\operatorname {\mathrm {quo}}} \otimes T_{g,Y}^{\operatorname {\mathrm {sub}}} \otimes T_{h,Y}^{\operatorname {\mathrm {sub}}}(1-c)$ .