3.1 Weight Space

Definition 3.1.1 Let $\Lambda := \mathbb {Z}_p \left [\!\!\left [ \mathbb {Z}_p^{\times } \right ]\!\!\right ]$ . The weight space $\mathcal {W}$ is defined to be the rigid generic fibre of the formal spectrum $\operatorname {Spf}\Lambda $ . It represents the functor taking a rigid analytic space X over $\operatorname {Sp}\mathbb {Q}_p$ to the set $\operatorname {Hom}_{\text {cont}}(\mathbb {Z}_p^{\times }, \mathcal {O}_X(X)^{\times })$ . Let

$$\begin{align*}\kappa \colon \mathbb{Z}_p^{\times} \longrightarrow \Lambda^{\times} \subset \mathcal{O}_{\mathcal{W}}(\mathcal{W})^{\times} \end{align*}$$

denote the tautological character.

The space $\mathcal {W}$ is isomorphic to a union of $p-1$ wide open discs (recall that we have assumed $p>2$ ).

Definition 3.1.2 Let E be a finite extension of $\mathbb {Q}_p$ with ring of integers $\mathcal {O}_E$ , and let $\mathcal {U}$ be a wide open disc in $\mathcal {W}_E := \mathcal {W} \times _{\operatorname {Sp}(\mathbb {Q}_p)} \operatorname {Sp}(E)$ . Define $\Lambda _{\mathcal {U}} := \mathcal {O}_{\mathcal {W}}(\mathcal {U})^{\circ }$ , the subring of power-bounded elements of $\mathcal {O}_{\mathcal {W}}(\mathcal {U})$ (so $\Lambda $ is non-canonically isomorphic to $\mathcal {O}_E \left [\!\!\left [ t \right ]\!\!\right ]$ ), and write

$$\begin{align*}\kappa_{\mathcal{U}} \colon \mathbb{Z}_p^{\times} \to \Lambda_{\mathcal{U}}^{\times} \end{align*}$$

for the map induced by $\kappa $ .

Definition 3.1.3 The m-accessible part of the weight space, denoted $\mathcal {W}_m$ , is the union of wide open discs defined by the inequality

$$\begin{align*}\big| \kappa(1+p^{m+1}) - 1 \big|^{p-1} < |p|. \end{align*}$$

We will eventually restrict our attention to $\mathcal {W}_0$ .

3.2 Families of Overconvergent Modular Forms

The following definition gives a criterion for when a modular form lies in a Coleman family.

3.3 Locally Analytic Distribution Modules

Now we begin defining a family of Galois representations on $\mathcal {W}$ as in [Reference Loeffler and ZerbesLZ16, §4].

Let $Y = Y_1(N(p))$ be the modular curve at level $\Gamma _1(N) \cap \Gamma _0(p)$ , and let $\pi \colon \mathcal {E} \to Y$ be the universal elliptic curve over Y. Let

$$\begin{align*}\mathscr{H} := \mathbf{R}^1 \pi_* \mathbb{Z}_p(1) \end{align*}$$

be the relative Tate module of $\mathcal {E}$ . We will define several pro-sheaves of “functions and distributions on $\mathscr {H}$ .” By this, we mean the following. Let $Y(p^{\infty }, p^{\infty }N)$ denote the pro-scheme $\varprojlim _n Y(p^n, p^nN)$ and $t \colon Y(p^{\infty }, p^{\infty }N) \to Y$ the natural projection; it is a Galois covering, and its Galois group can be identified with the Iwahori subgroup $U_0(p) \subset \mathrm {GL}_2(\mathbb {Z}_p)$ (with respect to the standard Borel).

The pro-sheaf $t^* \mathscr {H}$ is canonically isomorphic to the constant pro-sheaf $\underline {H}$ , where $H=\mathbb {Z}_p^2$ . We will define several spaces of functions and distributions on subsets of H that are equipped with actions of $U_0(p)$ . Since a $U_0(p)$ -module determines a $U_0(p)$ -equivariant pro-sheaf on $Y(p^{\infty },p^{\infty }N)$ , these spaces will descend to pro-sheaves on Y.

First, we recall the definition of two locally analytic distribution modules following [Reference Loeffler and ZerbesLZ16, §4.2].

Definition 3.3.1 Let $T_0$ , $T_0^{\prime }$ be the subsets of H defined by

$$\begin{align*}T_0 := \mathbb{Z}_p^{\times} \times \mathbb{Z}_p \quad \text{and}\quad T_0^{\prime} := p\mathbb{Z}_p \times \mathbb{Z}_p^{\times}, \end{align*}$$

and let $\Sigma _0(p)$ , $\Sigma _0^{\prime }(p)$ be the submonoids of $M_2(\mathbb {Z}_p)$ defined by

$$\begin{align*}\Sigma_0(p) := \begin{pmatrix} \mathbb{Z}_p^{\times} & \mathbb{Z}_p \\ p\mathbb{Z}_p & \mathbb{Z}_p \end{pmatrix} \quad\text{and}\quad \Sigma_0^{\prime}(p) := \begin{pmatrix} \mathbb{Z}_p & \mathbb{Z}_p \\ p\mathbb{Z}_p & \mathbb{Z}_p^{\times} \end{pmatrix}. \end{align*}$$

The monoids $\Sigma _0(p)$ and $\Sigma _0^{\prime }(p)$ act on the right on $T_0$ and $T_0^{\prime }$ , respectively.

Let R be a complete topological $\mathbb {Z}_p$ -algebra, and let $w \colon \mathbb {Z}_p^{\times } \to R^{\times }$ be a continuous homomorphism. Suppose there exists an integer $m \geq 0$ such that the restriction of w to $1+p^{m+1} \mathbb {Z}_p$ is analytic. We are primarily interested in the following cases:

1. (a) $R = \Lambda _{\mathcal {U}}$ , $w=\kappa _{\mathcal {U}}$ for some finite extension $E/\mathbb {Q}_p$ and some $\mathcal {U} \subset (\mathcal {W}_m)_E$ .

2. (b) $R = \mathcal {O}_E$ , $w(z) = z^k$ for some finite extension $E/\mathbb {Q}_p$ and some nonnegative integer k.

Definition 3.3.2 Let T be either $T_0$ or $T_0^{\prime }$ . Let $A^{\circ }_{w,m}(T)$ denote the space of functions $f \colon T \to R$ satisfying the following properties:

1. (i) The function f is homogeneous of weight w, i.e. $f(\lambda v) = w(\lambda ) f(v)$ for any $v \in T$ , $\lambda \in \mathbb {Z}_p^{\times }$ .

2. (ii) The function f is analytic on discs of radius $p^{-m}$ , i.e. for any $v \in T$ , the restriction of f to $v+p^m T$ is given by a power series with coefficients in R.

Let

$$ \begin{align*} D^{\circ}_{w,m}(T)&:= \operatorname{\mathrm{Hom}}_{R, \mathrm{cont}}(A^{\circ}_{w,m}(T), R), \\ D_{w,m}(T)& := D^{\circ}_{w,m}(T)[ 1/p ]. \end{align*} $$

When $R=\Lambda _{\mathcal {U}}$ , $w=\kappa _{\mathcal {U}}$ , we will denote the modules by $A^{\circ }_{\mathcal {U},m}$ , $D^{\circ }_{\mathcal {U},m}$ , $D_{\mathcal {U},m}$ . When $R = \mathcal {O}_E$ , $w(z)=z^k$ , we will denote the modules by $A^{\circ }_{k,m}$ , $D^{\circ }_{k,m}$ , $D_{k,m}$ .

The modules $A^{\circ }_{w,m}(T)$ , $D^{\circ }_{w,m}(T)$ , $D_{w,m}(T)$ inherit an action of $\Sigma _0(p)$ or $\Sigma _0^{\prime }(p)$ from the action on T. If the disc $\mathcal {U}$ contains the point corresponding to the homomorphism $z \mapsto z^k$ , then the specialization map $\Lambda _{\mathcal {U}} \to \mathbb {Z}_p$ induces a homomorphism

$$\begin{align*}D^{\circ}_{\mathcal{U},m} \to D^{\circ}_{k,m} \end{align*}$$

and similarly there are specialization maps with $D^{\circ }$ replaced by $A^{\circ }$ or D.

As mentioned at the beginning of this subsection, each of the modules defined above determines a pro-sheaf on Y. We let $\mathcal {D}^{\circ }_{w,m}(\mathscr {H}_0)$ , $\mathcal {D}^{\circ }_{w,m}(\mathscr {H}_0^{\prime })$ , be the pro-sheaves corresponding to $D^{\circ }_{w,m}(T_0)$ , $D^{\circ }_{w,m}(T_0^{\prime })$ , respectively.

3.4 Galois Representations

Now we define families of Galois representations coming from the cohomology of the sheaves defined above. For a wide open disc $\mathcal {U} \subset (\mathcal {W}_0)_E$ we set $B_{\mathcal {U}} := \Lambda _{\mathcal {U}}[\frac {1}{p}]$ .

Definition 3.4.1 As before, let $Y = Y_1(N(p))$ denote the modular curve of level $\Gamma _1(N) \cap \Gamma _0(p)$ . Set

$$ \begin{align*} M^{\circ}_{w,m}(\mathscr{H}_0) & := \operatorname{H}^1_{\acute{\mathrm{e}}\mathrm{t}}(Y_{\overline{\mathbb{Q}}}, \mathcal{D}^{\circ}_{w,m}(\mathscr{H}_0))(-\kappa_{\mathcal{U}}) \\ M^{\circ}_{w,m}(\mathscr{H}_0^{\prime}) & := \operatorname{H}^1_{\acute{\mathrm{e}}\mathrm{t}}(Y_{\overline{\mathbb{Q}}}, \mathcal{D}^{\circ}_{w,m}(\mathscr{H}_0^{\prime}))(1). \end{align*} $$

Given two Coleman families ${\mathcal {F}}$ and $\mathcal {G}$ defined over $\mathcal {U}_1, \mathcal {U}_2 \subset \mathcal {W}_E$ , respectively, we will write $M := M_{\mathcal {U}_1}({\mathcal {F}})^* \hat {\otimes } M_{\mathcal {U}_2}(\mathcal {G})^*$ for the family of Galois representations on $\mathcal {U}_1 \times \mathcal {U}_2$ given by the $B_{\mathcal {U}_1} \hat {\otimes } B_{\mathcal {U}_2}$ -module

$$\begin{align*}M_{\mathcal{U}_1}({\mathcal{F}})^* \hat{\otimes}_E M_{\mathcal{U}_2}(\mathcal{G})^*, \end{align*}$$

and to ease notation, we will often omit the subscripts when the spaces $\mathcal {U}_1$ and $\mathcal {U}_2$ are clear. Furthermore, we will often restrict this representation to open affinoids $V_1 \subset \mathcal {U}_1$ and $V_2 \subset \mathcal {U}_2$ ; in this case, M is a Banach module over the affinoid algebra $\mathcal {O}_{\mathcal {W}}(V_1 \times V_2)$ that is free of rank four.

Definition 3.4.3 Let A be a $\mathbb {Q}_p$ -affinoid algebra and let M be an A-valued (continuous) representation of $G_{\mathbb {Q}}$ . Let $D^{\mathrm {la}}(\Gamma , A)$ denote the space of locally analytic distributions with values in A; this comes equipped with an action of $G_{\mathbb {Q}}$ given by

(3.4.4) $$ \begin{align} \int_{\Gamma} f \; d(g \cdot \mu) := \int_{\Gamma} f([g]^{-1}x) \; d\mu(x), \end{align} $$

where $[g]$ denotes the image of $g \in G_{\mathbb {Q}}$ in $\Gamma = \operatorname {\mathrm {Gal}}(\mathbb {Q}(\mu _{p^{\infty }})/\mathbb {Q})$ , and is isomorphic (as $A[G_{\mathbb {Q}}]$ -modules) to $\mathcal {O}_{\mathcal {W}}(\mathcal {W})^{\iota } \hat {\otimes } A$ . The cyclotomic deformation of M is defined to be

$$\begin{align*}M(-\kappa) := D^{\mathrm{la}}(\Gamma, M) := M \widehat{\otimes}_{\mathbb{Q}_p} D^{\mathrm{la}}(\Gamma, A). \end{align*}$$

with the diagonal Galois action.

Similarly, for any $\lambda \in \mathbb {R}_{\ge 0}$ , let $D_{\lambda }(\Gamma ,\mathbb {Q}_p)$ be the space of $\mathbb {Q}_p$ -valued distributions on $\Gamma $ of order $\lambda $ as in [Reference ColmezCol10, §II.3], with Galois action given by the same formula in (3.4.4). Define $D_{\lambda }(\Gamma ,M) := D_{\lambda }(\Gamma ,\mathbb {Q}_p) \widehat {\otimes }_{\mathbb {Q}_p} M$ .

3.5 Some Properties of Locally Analytic Distribution Modules

We mention some properties of the modules defined above that will be useful in Section 4.

Definition 3.5.1 Define $\Lambda (H)$ to be the space of continuous $\mathbb {Z}_p$ -valued distributions on H, and let $\Lambda (\mathscr {H})$ be the corresponding pro-sheaf on Y. This coincides with the sheaf of Iwasawa modules for $\mathscr {H}$ , i.e., $\Lambda (\mathscr {H})$ is the pro-system of étale sheaves corresonding to the inverse system $\left ( \mathbb {Z}/p^n \mathbb {Z} [\mathscr {H} /p^n \mathscr {H} ] \right )_{n \geq 1}$ with the natural transition maps.

For any nonnegative integer k, let $ \operatorname {\mathrm {TSym}}^k H$ be the space of degree k symmetric tensors over H, i.e., it is the subgroup of $H^{\otimes k}$ that is invariant under the action of the symmetric group $S_k$ . Let $ \operatorname {\mathrm {TSym}}^k \mathscr {H}$ be the corresponding pro-sheaf on Y.

For $j \geq 0$ , set $\Lambda ^{[j]}(\mathscr {H}) := \Lambda (\mathscr {H}) \otimes \operatorname {TSym}^j\mathscr {H}$ and $\Lambda ^{[j, j]} = \Lambda ^{[j]}(\mathscr {H}) \boxtimes \Lambda ^{[j]}(\mathscr {H})$ . Then there is a Clebsch–Gordon map (see [Reference Loeffler and ZerbesLZ16, §3.2])

$$\begin{align*}\operatorname{CG}^{[j]} \colon \Lambda(\mathscr{H}) \longrightarrow \big( \Lambda^{[j]}(\mathscr{H}) \hat{\otimes} \Lambda^{[j]}(\mathscr{H}) \big) (-j). \end{align*}$$

For any $\mathcal {U}$ , m, there is a natural restriction map

$$\begin{align*}\Lambda(\mathscr{H}) \longrightarrow \mathcal{D}^{\circ}_{U,m}(T) \, \end{align*}$$

and for any nonnegative integer k, $ \operatorname {\mathrm {TSym}}^k H$ can be identified with the space of distributions on homogeneous degree k polynomial functions on H. Hence, there is a natural surjection

$$\begin{align*}\mathcal{D}^{\circ}_{k,m}(T) \longrightarrow \operatorname{\mathrm{TSym}}^k \mathscr{H}. \end{align*}$$

3.6 The Three-variable p-adic L-function

Let f and g be two normalised cuspidal eigenforms of weights $k +2, k^{\prime }+2$ and levels $\Gamma _1(N_1)$ and $\Gamma _1(N_2),$ respectively, where $k> k' \geq 0$ . Let $\chi $ be a Dirichlet character of conductor $N_{\chi }$ and suppose that p does not divide $N_1 \cdot N_2$ . With this data, one has the associated (imprimitive) Rankin–Selberg L-function, defined as

$$\begin{align*}L(f, g, \chi, s) = L_{(N_1 N_2 N_{\chi})}(\varepsilon_f \varepsilon_g \chi^2, 2s - 2 - k - k^{\prime}) \cdot \sum_{\substack{n \geq 1 \\ (n, N_{\chi}) = 1}} a_n(f) a_n(g) \chi(n) n^{-s} \end{align*}$$

for $\operatorname {Re}(s)$ sufficiently large. Here the subscript $(N_1 N_2 N_{\chi })$ denotes the omission of the Euler factors at primes dividing $N_1 N_2 N_{\chi }$ . This L-function differs from the automorphic L-function attached to the representation $\pi _f \otimes \pi _g \otimes \chi $ by only finitely many Euler factors. Since we have assumed $k \neq k^{\prime }$ , the function $L(f, g, \chi , -)$ has analytic continuation to all of $\mathbb {C}$ (see [Reference LoefflerLoe18, §2.1]).

Using the theory of nearly overconvergent families of modular forms as described in [Reference Andreatta and IovitaAI17], Urban has constructed a three-variable p-adic L-function that interpolates critical values of the above Rankin–Selberg L-function. More precisely, suppose that there exist noble p-stabilisations of f and g (as in Definition 3.2.2) and let $\mathcal {F}$ and $\mathcal {G}$ be Coleman families over affinoid domains $V_1$ and $V_2$ , passing through these p-stabilisations. We can shrink $V_1$ and $V_2$ to ensure that all classical specialisations of $\mathcal {F}$ and $\mathcal {G}$ are noble (see Remark 8.2.5)—if $\mathcal {F}_{k_1}$ denotes such a specialisation, then we let $\mathcal {F}^\circ _{k_1}$ denote the associated newform, and similarly for $\mathcal {G}$ .

In the following section, we will recall the construction of the Beilinson–Flach classes in families. It turns out that this p-adic L-function $L_p$ is closely related to the images of these Beilinson–Flach classes under Perrin–Riou’s “big logarithm”. This will be important in Proposition 6.2.2 later on.

4.1 Rankin–Iwasawa Classes

Let $(\mathcal {E}, P, Q)$ denote the universal triple over the curve $Y:= Y(m, mN)$ as defined in Section 2, and recall that

$$\begin{align*}\mathscr{H} = \mathscr{H}_{\mathbb{Z}_p} := \mathbf{R}^1\pi_* \mathbb{Z}_p(1) \end{align*}$$

denotes the relative p-adic Tate module of $\mathcal {E}/Y$ . Here, $\pi \colon \mathcal {E} \to Y$ denotes the structure map, and $\mathbb {Z}_p(1)$ is the Tate twist by the cyclotomic character. This is a locally free étale pro-sheaf on Y of rank $2$ .

Let $c \geq 1$ be an integer prime to $6mN$ . In [Reference KingsKin16], Kings constructs Eisenstein classes ${{}_{c}\operatorname {Eis}^k_{\mathbb {Q}_p}}$ arising from motivic classes whose de Rham realisations recover the usual Eisenstein series of weight $k+2$ (see also [Reference Kings, Loeffler and ZerbesKLZ15, §4]). In addition to this, he constructs so-called Eisenstein–Iwasawa classes

$$\begin{align*}{{}_{c} \mathcal{EI}_{m, mN}} \in \operatorname{H}^1_{\acute{\mathrm{e}}\mathrm{t}}\big(Y(m, mN), \Lambda(\mathscr{H})(1) \big), \end{align*}$$

which interpolate ${ {}_{c}\operatorname {Eis}^k_{\mathbb {Q}_p}}$ via the “moment maps”

$$\begin{align*}\operatorname{mom}^k \colon \Lambda(\mathscr{H}) \longrightarrow \operatorname{TSym}^k \mathscr{H}. \end{align*}$$

From these classes, one obtains Rankin–Iwasawa classes in the following way.

Definition 4.1.1 Let $c \geq 1$ be an integer that is coprime to $6mN$ . We define the Rankin–Iwasawa class to be

$$\begin{align*}{}_{c} \mathcal{RI}_{m, mN, 1}^{[j]} := \big[ (u_1)_* \circ \Delta_* \circ \operatorname{CG}^{[j]} \big]({}_{c}\mathcal{EI}_{m, mN}), \end{align*}$$

which lies in the cohomology group $\operatorname {H}^3_{\acute {\mathrm {e}}\mathrm {t}}(Y(m, mN)^2, \Lambda ^{[j, j]}(2-j) )$ . Here

• • $\operatorname {CG}^{[j]}$ is the Clebsch–Gordon map described in Definition 3.5.1.

• • $\Delta \colon Y(m, mN) \rightarrow Y(m, mN)^2$ denotes the diagonal embedding where, by abuse of notation, we write $Y(m, mN)^2$ for the fibre product

  $$\begin{align*}Y(m, mN) \times_{\mu_m^{\circ}} Y(m, mN). \end{align*}$$

• • $u_1 \colon Y(m, mN)^2 \to Y(m, mN)^2$ denotes the automorphism that is the identity on the first factor and acts on the moduli interpretation as

  $$\begin{align*}(E, P, Q) \longmapsto (E, P + NQ, Q) \end{align*}$$
  on the second factor.

The Rankin–Iwasawa classes satisfy the following norm compatibility relations.

4.2 Beilinson–Flach Classes in Families

Let E be a finite extension of $\mathbb {Q}_p$ with ring of integers $\mathcal {O}_E$ , and $\mathcal {U}_1, \mathcal {U}_2 \subset (\mathcal {W}_0)_E$ two wide open discs, where $\mathcal {W}_0 \subset \mathcal {W}$ is the wide open subspace of $0$ -accessible weights (see Definition 3.1.3). Recall that $\Lambda (\mathscr {H}^{\prime }_0)$ and $\mathcal {D}_{\mathcal {U}_i}^{\circ }(\mathscr {H}^{\prime }_0) := \mathcal {D}_{\mathcal {U}_i, 0}^{\circ }(\mathscr {H}^{\prime }_0)$ are the sheaves of continuous (resp. locally analytic) $\mathbb {Z}_p$ -valued (resp. $\Lambda _{\mathcal {U}_i}$ -valued) distributions on $\mathscr {H}^{\prime }_0$ , the subsheaf of $\mathscr {H}$ which is locally isomorphic to $T^{\prime }_0$ .

Consider the map of sheaves

(4.2.1) $$ \begin{align} \Lambda^{[j, j]}(\mathscr{H}_0^{\prime}) \longrightarrow \mathcal{D}_{[\mathcal{U}_1, \mathcal{U}_2]}(\mathscr{H}_0^{\prime}) := \big( \mathcal{D}^{\circ}_{\mathcal{U}_i}(\mathscr{H}_0^{\prime}) \boxtimes \mathcal{D}^{\circ}_{\mathcal{U}_2}(\mathscr{H}_0') \big) \Big[ \frac{1}{p} \Big] \end{align} $$

induced from the composition

(4.2.2) $$ \begin{align} \Lambda(\mathscr{H}_0^{\prime}) \otimes \operatorname{TSym}^j\mathscr{H} \longrightarrow \mathcal{D}^{\circ}_{\mathcal{U}_i - j}(\mathscr{H}_0^{\prime}) \otimes \operatorname{TSym}^j\mathscr{H} \xrightarrow{\delta^*_j} \mathcal{D}_{\mathcal{U}_i}(\mathscr{H}_0^{\prime}) \end{align} $$

described in [Reference Loeffler and ZerbesLZ16, Definition 5.3.1]. We will not need an explicit description of these maps, but we do note that we have the following commutative diagram.

Lemma 4.2.3 We have the following commutative diagram of sheaves

where the horizontal arrows are the composition in (4.2.2) and, as usual, l is a prime not dividing $Np$ and $\kappa _i$ is the universal character of $\mathcal {U}_i$ .

We are now in a position to define the Beilinson–Flach classes. Consider the following composition, which we will denote by $\tau _m$

$$\begin{align*}\tau_m\colon Y(m, mNp)^2 \xrightarrow{t_m \times t_m} Y_1(Np)^2 \times \mu_m^{\circ} \rightarrow Y_1(N_1(p)) \times Y_1(N_2(p)) \times \mu_m^{\circ} \end{align*}$$

where $t_m \colon Y(m, mNp) \to Y_1(Np) \times \mu _m^{\circ }$ is the map defined in (2.0.3), and the second map is induced from the maps $s_{N_i}\colon Y_1(Np) \to Y_1(N_i(p))$ as defined in (2.0.4).

Definition 4.2.4 We define the Beilinson–Flach class

$$\begin{align*}{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{U}_1, \mathcal{U}_2, j]} \in \operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}\big(Y_1(N_1(p)) \times Y_1(N_2(p)) \times \mu_m^{\circ}, \mathcal{D}_{[\mathcal{U}_1, \mathcal{U}_2]}(\mathscr{H}_0^{\prime})(2-j) \big) \end{align*}$$

to be the pushforward of ${}_{c} \mathcal {RI}_{m, mNp, 1}^{[j]}$ under $\tau _m$ composed with the map in (4.2.1).

Let $\mathcal {F}$ and $\mathcal {G}$ be Coleman families over $\mathcal {U}_1$ and $\mathcal {U}_2$ respectively. To specialise the Beilinson–Flach classes at $\mathcal {F}$ and $\mathcal {G}$ one introduces the following differential operators ${\nabla _i \choose j}$ on $\Lambda _{\mathcal {U}_i}[1/p]$ given by the formula

$$\begin{align*}\binom{\nabla_i}{j} := \frac{1}{j!}\prod_{k = 0}^{j-1}{(\nabla_i - k)}, \end{align*}$$

where $\nabla _i$ is given by $(\nabla _if)(x) = \frac {d}{dt}f(tx) \mid _{t=1}$ (see [Reference Loeffler and ZerbesLZ16, Proposition 5.1.2] for more details). Let $V_1$ and $V_2$ be open affinoid subdomains in $\mathcal {U}_1$ and $\mathcal {U}_2$ respectively and $j \geq 0$ an integer not contained in either $V_1$ or $V_2$ . Then the operators ${\nabla _1 \choose j}$ and $\binom {\nabla _2}{ j}$ are injective and there exist unique classes

$$\begin{align*}{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}, j]} \in \operatorname{H}^1\left(\mathbb{Q}(\mu_m), [M_{V_1}(\mathcal{F})^* \hat{\otimes} M_{V_2}(\mathcal{G})^*](-j) \right) \end{align*}$$

such that ${\nabla _1 \choose j}{\nabla _2 \choose j} {{}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}, j]}}$ equals the image of ${}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {U}_1, \mathcal {U}_2, j]}$ under the Abel–Jacobi map $\operatorname {AJ}_{\mathcal {F}, \mathcal {G}}$ defined below.

Definition 4.2.5 The Abel–Jacobi map $\operatorname {AJ}_{\mathcal {F}, \mathcal {G}}$ is defined to be the composition

$$\begin{align*} & \operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}\big(Y_1(N_1(p)) \times Y_1(N_2(p) ) \times \mu_m^{\circ}, \mathcal{D}_{[\mathcal{U}_1, \mathcal{U}_2]}(\mathscr{H}_0^{\prime})(2-j)\big) \\ & \quad \longrightarrow \operatorname{H}^1\big(\mathbb{Q}(\mu_m), \operatorname{H}^2_{\acute{\mathrm{e}}\mathrm{t}}(Y_1(N_1(p))_{\bar{\mathbb{Q}}} \times Y_1(N_2(p))_{\bar{\mathbb{Q}}}, \mathcal{D}_{[\mathcal{U}_1, \mathcal{U}_2]}(\mathscr{H}_0^{\prime})(2-j) \big) \\ & \quad \xrightarrow{\sim} \operatorname{H}^1(\mathbb{Q}(\mu_m), \operatorname{H}^1_{\acute{\mathrm{e}}\mathrm{t}}(Y_1(N_1(p))_{\bar{\mathbb{Q}}}, \mathcal{D}_{\mathcal{U}_1}(\mathscr{H}_0^{\prime})(1)) \\ & \qquad \hat{\otimes} \operatorname{H}^1_{\acute{\mathrm{e}}\mathrm{t}}(Y_1(N_2(p))_{\bar{\mathbb{Q}}}, \mathcal{D}_{\mathcal{U}_2}(\mathscr{H}_0^{\prime})(1))(-j)) \\ & \quad \longrightarrow \operatorname{H}^1\big(\mathbb{Q}(\mu_m), \big[ M_{V_1}(\mathcal{F})^* \hat{\otimes} M_{V_2}(\mathcal{G})^* \big] (-j) \big), \end{align*}$$

where the first map arises from the Leray spectral sequence (using the fact that $Y_1(N_1(p))_{\bar {\mathbb {Q}}} \times Y_1(N_2(p))_{\bar {\mathbb {Q}}}$ is an affine scheme, so its étale cohomology vanishes in degree $3$ and above); the second isomorphism is the Künneth formula (again using the fact that $Y_1(N_i(p))_{\bar {\mathbb {Q}}}$ is affine), and the third map is the projection down to the Galois representations associated with $\mathcal {F}$ and $\mathcal {G}$ .

The Beilinson–Flach classes associated with $\mathcal {F}$ and $\mathcal {G}$ satisfy norm compatibility relations similar to those for the Rankin–Iwasawa classes.

Proof Consider the composition

$$ \begin{align*} &\operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}\big(Y(m, mNp)^2, \Lambda^{[j, j]}(2-j) \big) \\ &\qquad \xrightarrow{\tau_{m, *}} \operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}(Y_1(N_1(p)) \times Y_1(N_2(p)) \times \mu_m^{\circ}, \Lambda^{[j, j]}(\mathscr{H}_0^{\prime}) ) \\ &\qquad \longrightarrow \operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}\big(Y_1(N_1(p)) \times Y_1(N_2(p)) \times \mu_m^{\circ}, \mathcal{D}_{[\mathcal{U}_1, \mathcal{U}_2]}(\mathscr{H}_0^{\prime})\big). \end{align*} $$

By applying Lemma 2.0.5, the morphisms $(\Xi _l)_*$ and $\operatorname {cores}^{\mathbb {Q}(\mu _{lm})}_{\mathbb {Q}(\mu _m)}$ are compatible under the map

(4.2.7) $$ \begin{align} {\operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}(Y(m, mN)^2, \Lambda^{[j, j]}(2-j))} \longrightarrow {\operatorname{H}^1(\mathbb{Q}(\mu_m), [M(\mathcal{F})^* \hat{\otimes} M(\mathcal{G})^*](-j))} \end{align} $$

obtained by composing the above map with $\operatorname {AJ}_{\mathcal {F}, \mathcal {G}}$ .

Immediately we see that if l divides m, then

$$ \begin{align*} &\binom{\nabla_1}{ j} \binom{\nabla_2}{ j} \operatorname{cores}^{\mathbb{Q}(\mu_{ml})}_{\mathbb{Q}(\mu_m)} {{}_{c} \mathcal{BF}_{lm, 1}^{[\mathcal{F}, \mathcal{G}, j]}}\\ &\qquad = \operatorname{cores}^{\mathbb{Q}(\mu_{ml})}_{\mathbb{Q}(\mu_m)} \binom{\nabla_1}{ j} \binom{\nabla_2}{ j} {{}_{c} \mathcal{BF}_{lm, 1}^{[\mathcal{F}, \mathcal{G}, j]}} \\ &\qquad = (a_l(\mathcal{F})a_l(\mathcal{G}))\binom{\nabla_1}{ j} \binom{\nabla_2}{ j} {{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}, j]}}, \end{align*} $$

where the second equality follows from Proposition 4.1.2(i) and the fact that $T_l^{\prime }$ acts as multiplication by $a_l(\mathcal {F})$ (resp. $a_l(\mathcal {G})$ ) on $M(\mathcal {F})^*$ (resp. $M(\mathcal {G})^*$ ). Note that under the morphism $t_m$ the operators $U_l'$ and $T_l'$ are compatible. Since j is not contained in $V_1$ or $V_2$ , the operator $\binom {\nabla _1 }{ j} \binom {\nabla _2}{ j}$ is invertible, and we have the required relation.

For the second part, recall that by Proposition 4.1.2(ii), the Rankin–Iwasawa classes satisfy $(\Xi _l \times \Xi _l)_*({}_{c} \mathcal {RI}_{lm, lmN, 1}^{[j]}) = \widetilde {Q}_l \cdot {{}_{c} \mathcal {RI}_{m, mN, 1}^{[j]}}$ . We have the following commutative diagram:

where the horizontal arrows are the maps in (4.2.7). Indeed, $M(\mathcal {F})^*$ can be described as the quotient of

$$\begin{align*}\operatorname{H}^3_{\acute{\mathrm{e}}\mathrm{t}}\big(Y_1(N_1(p))_{\bar{\mathbb{Q}}}, \mathcal{D}_{V_1}(1)\big) \end{align*}$$

such that $T^{\prime }_l$ acts as multiplication by $a_l(\mathcal {F})$ and $\langle l \rangle $ acts as multiplication by $\varepsilon _{\mathcal {F}}(l)^{-1}$ . We have a similar description for $M(\mathcal {G})^*$ . Furthermore, the action of $\sigma _l$ becomes the natural action of $\sigma _l$ under the horizontal map in the above diagram (this is an application of the push-pull lemma for étale cohomology). Finally, by Lemma 4.2.3, $[l]_*$ becomes multiplication by $l^{\kappa _i - j}$ . This shows that the above diagram is commutative and completes the proof of the proposition. ▪

4.3 Interpolation in the Cyclotomic Variable

In this section, we recall how to interpolate the Beilinson–Flach classes ${}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}, j]}$ in the cyclotomic variable j. As a consequence, we show that the three-variable Beilinson–Flach classes satisfy a norm-compatibility relation closely related to the Euler system relations.

Let $p^{-\lambda _1} = \|a_p(\mathcal {F})\|$ (resp. $p^{-\lambda _2} = \|a_p(\mathcal {G})\|$ ) where $\|\cdot \|$ denotes the canonical supremum norm on $\mathcal {O}(V_i)$ (which exists because we have restricted our Coleman families to reduced affinoid subdomains). Let $\lambda = \lambda _1 + \lambda _2$ and $h \geq \lambda $ a positive integer. Define the following elements

$$\begin{align*}x_{n, j} := \big(a_p(\mathcal{F})a_p(\mathcal{G})\big)^{-n} \frac{{}_{c} \mathcal{BF}_{mp^n, 1}^{[\mathcal{F}, \mathcal{G}, j]}}{(-1)^jj!} \end{align*}$$

for $0 \leq j \leq h$ , $n \geq 1$ , and set

$$\begin{align*}x_{0, j} := \Big(1 - \frac{p^j}{a_p(\mathcal{F})a_p(\mathcal{G})} \Big) \frac{{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}, j]}}{(-1)^jj!} \end{align*}$$

for $0 \leq j \leq h$ . These elements are compatible under corestriction and satisfy a certain growth bound (see [Reference Loeffler and ZerbesLZ16, Proposition 5.4.1]), so by Proposition 2.3.3 in op. cit. there exists a unique element

$$\begin{align*}{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}]} \in \operatorname{H}^1 \big(\mathbb{Q}(\mu_{mp^{\infty}}), D_{\lambda}(\Gamma, M)\big)^{\Gamma} \cong \operatorname{H}^1\big(\mathbb{Q}(\mu_m), D_{\lambda}(\Gamma, M)\big) \end{align*}$$

satisfying

$$\begin{align*}\int_{\Gamma_n} \chi_{\mathrm{cycl}}^j \; {{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}]}} = x_{n, j} \end{align*}$$

for all $n, j$ . Here, $M = M_{V_1}(\mathcal {F})^* \hat {\otimes } M_{V_2}(\mathcal {G})^*$ and $\Gamma _n \subset \Gamma $ is the unique subgroup of index $p^{n-1}(p-1)$ (we set $\Gamma _0 = \Gamma $ ). The class ${}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}]}$ is independent of the choice of h.

Proof Let $\mathcal {Q}_l(X)$ denote the polynomial appearing in Proposition 4.2.6 and let l be a prime not dividing $6mpNc$ . Set

$$\begin{align*}\nu_m = \mathcal{Q}_l(l^{-\mathbf{j}}\sigma_l^{-1}) \cdot {{}_{c} \mathcal{BF}_{m, 1}^{[\mathcal{F}, \mathcal{G}]}} - \operatorname{cores}^{\mathbb{Q}(\mu_{lm})}_{\mathbb{Q}(\mu_m)} \left( {}_{c} \mathcal{BF}_{lm, 1}^{[\mathcal{F}, \mathcal{G}]} \right). \end{align*}$$

Then for all $n,j \geq 0$ the specialisation $\int _{\Gamma _n}{\chi ^j \nu _m}$ is zero; so $\nu _m$ interpolates only zero classes. By uniqueness, this implies that $\nu _m = 0$ . A similar argument works for $l | m$ .▪

4.4 Euler System Relations in Families

In Proposition 4.3.1, we showed that the Beilinson–Flach classes satisfy norm compatible relations. It turns out that we can adjust these classes so that we obtain cohomology classes satisfying the Euler system relations.

As before, let $M = M_{V_1}(\mathcal {F})^* \hat {\otimes } M_{V_2}(\mathcal {G})^*$ and let $\mathcal {P}_l(X)$ denote the polynomial $\operatorname {det}\left (1 - \operatorname {Frob}_l^{-1} X | M^*(1) \right )$ , where $\operatorname {Frob}_l$ denotes any lift of the arithmetic Frobenius at l. Then one can observe that

$$\begin{align*}\mathcal{Q}_l(X) = X^{-1}\big( (l-1)(1-l^{\kappa_1 + \kappa_2 + 2}\varepsilon_{\mathcal{F}}(l)\varepsilon_{\mathcal{G}}(l) X^2 ) - l \mathcal{P}_l(X) \big), \end{align*}$$

so in particular, $\mathcal {Q}_l(X) \equiv -X^{-1}\mathcal {P}_l(X)$ modulo $l-1$ . Such a congruence allows us to adjust the classes ${}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}]}$ so that we obtain Euler system relations.

Unfortunately, in general, there is no way to force these classes to lie in a Galois stable lattice inside $D^{\mathrm {la}}(\Gamma , M)$ , so we do not get an Euler system for this representation. However, this is possible after specialisation so long as we use a weaker notion of an Euler system.

Corollary 4.4.2 Let $\mathbf {x} = (k_1,\, k_2, \eta ) \in V_1 \times V_2 \times \mathcal {W}$ defined over a finite extension $E/\mathbb {Q}_p$ , and let $T(\eta ^{-1})$ be a Galois stable lattice inside $M_E(\mathcal {F}_{k_1})^* \otimes M_E(\mathcal {G}_{k_2})^* (\eta ^{-1})$ . Assume that $k_1 \neq k_2$ . Let $c \geq 1$ be an integer prime to $6Np$ and let $\mathcal {N}$ be a finite product of primes containing all primes dividing $6cNp$ . Let S denote the set of positive integers divisible only by primes not dividing $\mathcal {N}$ . Then for $m \in S$ and $V_1$ and $V_2$ small enough, there exist cohomology classes

$$\begin{align*}c_m \in \operatorname{H}^1\big(\mathbb{Q}(\mu_m), T(\eta^{-1}) \big) \end{align*}$$

that satisfy

$$\begin{align*}\operatorname{cores}^{\mathbb{Q}(\mu_{lm})}_{\mathbb{Q}(\mu_m)} {c_{lm}} = \begin{cases} c_m & \text{if } l | m, \\ P_l(\eta^{-1}(l)\sigma_l^{-1}) \cdot c_m & \text{if } l \nmid m, \end{cases} \end{align*}$$

where l is a prime not dividing $\mathcal {N}$ and $P_l(\eta ^{-1}(l)X)$ is the specialisation of $\mathcal {P}_l(l^{-\mathbf {j}}X)$ at $(k_1,\, k_2, \eta )$ . Furthermore, the bottom class $c_1$ is a non-zero multiple of ${{}_{c} \mathcal {BF}_{1, 1}^{[\mathcal {F}, \mathcal {G}]}}$ .

Proof First, note that $\operatorname {H}^0(\mathbb {Q}(\mu _{mp^{\infty }}), M_{\mathbf {x}}) = 0$ for all $m \in S$ , where

$$\begin{align*}M_{\mathbf{x}} = M_E(\mathcal{F}_{k_1})^* \otimes M_E(\mathcal{G}_{k_2})^* (\eta^{-1}). \end{align*}$$

Indeed this is true, because we have assumed $k_1 \neq k_2$ , for the following reason. Shrinking $V_1$ and $V_2$ if necessary, we can assume that $M_{\mathbf {x}}$ is (absolutely) irreducible. Hence, any twist of $M_{\mathbf {x}}$ by a character is also irreducible. But if $M_{\mathbf {x}}$ has any non-trivial invariants under the group $G_{\mathbb {Q}^{ab}}$ then there is a one-dimensional submodule of $M_{\mathbf {x}}$ on which $G_{\mathbb {Q}}$ acts via a character. This is a contradiction to irreducibility.

Therefore, by applying [Reference Loeffler and ZerbesLZ16, Proposition 2.4.7], there exists a constant $R> 0$ independent of m such that $R \cdot {{}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}]}}$ specialised at $\mathbf {x}$ lands in the cohomology of the Galois stable lattice $T(\eta ^{-1})$ .

But since ${{}_{c}\mathcal {Z}_m^{[\mathcal {F}, \mathcal {G}]}}$ differs from ${{}_{c} \mathcal {BF}_{m, 1}^{[\mathcal {F}, \mathcal {G}]}}$ by an element of $\mathcal {O}(V_1 \times V_2 \times \mathcal {W})^{\circ }[(\mathbb {Z}/m\mathbb {Z})^{\times }]$ , this implies that the specialisation of $R \cdot {{}_{c}\mathcal {Z}_m^{[\mathcal {F}, \mathcal {G}]}}$ lands in the cohomology of the Galois stable lattice $T(\eta ^{-1})$ . We set $c_m$ to be the specialisation of $R \cdot {{}_{c}\mathcal {Z}_m^{[\mathcal {F}, \mathcal {G}]}}$ at $\mathbf {x}$ .

By Proposition 4.4.1, we obtain the Euler system relations for the classes $c_m$ , and this proves the corollary for all $m \in A$ , where A is the subset of S consisting of all square-free integers.

But we can extend the classes $\{c_m : m \in A\}$ to a collection of classes indexed over the set S by defining $c_m$ to be

$$\begin{align*}c_m := \prod_{l | m}\frac{1}{l^{v_l(m) - 1}} \operatorname{res}^{\mathbb{Q}(\mu_m)}_{\mathbb{Q}(\mu_{m^{\prime}})} c_{m^{\prime},} \end{align*}$$

where $m^{\prime }$ is the radical of m (i.e., the product of all prime factors that divide m) and $v_l(m)$ is the l-adic valuation of m. We do not lose integrality of the classes because all integers in S are coprime to p.▪

5.1 Period Rings

In this section, we (briefly) recall the period rings that will be used throughout the paper. Since we only work with representations of $G_{\mathbb {Q}_p}$ , we specialise immediately to this case and refer the reader to [Reference BergerBer03] for the definitions over more general p-adic fields; proofs and their corresponding references for all of the assertions in this section can also be found in op. cit..

Let $\mathbb {C}_p$ denote a fixed completed algebraic closure of $\mathbb {Q}_p$ and let $v_p$ denote the unique valuation on $\mathbb {C}_p$ such that $v_p(p) = 1$ . Set $\mathcal {O}_{\mathbb {C}_p}^{\flat } := \varprojlim \mathcal {O}_{\mathbb {C}_p}/p$ , where the inverse limit is over the p-th power map, and fix $\varepsilon := (\varepsilon _n) \in \mathcal {O}_{\mathbb {C}_p}^{\flat }$ a compatible system of p-th roots of unity, i.e., $\varepsilon _n$ is a $p^n$ -th root of unity such that $\varepsilon _{n+1}^p = \varepsilon _n$ .

Let $\widetilde {\mathbf {A}}^+$ denote the (p-typical) Witt vectors of $\mathcal {O}_{\mathbb {C}_p}^{\flat }$ and let $\widetilde {\mathbf {B}}^+ = \widetilde {\mathbf {A}}^+[\frac {1}{p}]$ . One has a canonical homomorphism $ \theta \colon \widetilde {\mathbf {B}}^+ \rightarrow \mathbb {C}_p $ whose kernel is a principal ideal of $\widetilde {\mathbf {B}}^+$ generated by $\omega := ([\varepsilon ] - 1)/([\varepsilon ^{\prime }] - 1)$ where $\varepsilon ^{\prime } = (\varepsilon _{n+1})_{n \geq 0}$ and $[\cdot ]$ denotes the Teichmüller lift. We set $t := \operatorname {log}([\varepsilon ])$ to be the period of Fontaine.

We define the de Rham period rings $\mathbf {B}_{\mathrm {dR}}^+$ and $\mathbf {B}_{\mathrm {dR}}$ to be the $(\operatorname {ker}\theta )$ -adic completion of $\widetilde {\mathbf {B}}^+$ and the fraction field of $\mathbf {B}_{\mathrm {dR}}^+$ , respectively. Similarly, we define the crystalline period rings $\mathbf {B}_{\mathrm {cris}}^+$ and $\mathbf {B}_{\mathrm {cris}}$ to be the divided-power envelope of $\widetilde {\mathbf {B}}^+$ (with respect to $\omega $ ) and $\mathbf {B}_{\mathrm {cris}} = \mathbf {B}_{\mathrm {cris}}^+[t^{-1}],$ respectively. The field $\mathbf {B}_{\mathrm {dR}}$ comes equipped with a decreasing filtration given by $\operatorname {Fil}^i\mathbf {B}_{\mathrm {dR}} = t^i\mathbf {B}_{\mathrm {dR}}^+$ , and since $\mathbf {B}_{\mathrm {cris}} \subset \mathbf {B}_{\mathrm {dR}}$ , the ring $\mathbf {B}_{\mathrm {cris}}$ also inherits a filtration from $\mathbf {B}_{\mathrm {dR}}$ . Furthermore, $\mathbf {B}_{\mathrm {cris}}$ comes equipped with a Frobenius endomorphism $\varphi $ extending the natural Frobenius on $\widetilde {\mathbf {B}}^+$ .

In addition to these constructions, we define overconvergent period rings as follows. For $r, s$ two positive rational numbers, let $\widetilde {\mathbf {A}}^{[r, s]}$ denote the p-adic completion of $\widetilde {\mathbf {A}}^+\big [\frac {p}{[\varepsilon - 1]^r}, \frac {[\varepsilon - 1]^s}{p}\big ]$ and set $\widetilde {\mathbf {B}}^{[r, s]} = \widetilde {\mathbf {A}}^{[r, s]}[1/p]$ . We define $\widetilde {\mathbf {B}}^{\dagger , r}_{\text {rig}}$ to be the intersection $\widetilde {\mathbf {B}}^{\dagger , r}_{\text {rig}} := \cap _{r \leq s < +\infty }\widetilde {\mathbf {B}}^{[r, s]}$ , and if we set $\pi = [\varepsilon ] -1$ , then this ring contains the following subring:

$$\begin{align*}\mathbf{B}^{\dagger, r}_{\text{rig}, \mathbb{Q}_p} := \bigg\{ f(\pi) = \sum_{k = -\infty}^{\infty}{a_k \pi^k} : \begin{array}{c} a_k \in \mathbb{Q}_p \\ f(x) \text{ converges for } 0 < v_p(x) \leq \frac{1}{r} \end{array} \!\!\!\bigg\}. \end{align*}$$

Note that the period t defined previously is equal to $\operatorname {log}(1+\pi )$ .

The union of these rings, namely $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} := \varinjlim {\mathbf {B}^{\dagger , r}_{\mathrm {rig}, \mathbb {Q}_p}}$ , is the Robba ring associated to the p-adic field $\mathbb {Q}_p$ and can be identified with all power series in $\mathbb {Q}_p\left [\!\!\left [ \pi , \pi ^{-1} \right ]\!\!\right ]$ that converge on an annulus of the form $\{x \in \mathbb {C}_p : 0 < v_p(x) \leq 1/r \}$ for some positive rational number r. We let $\mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p} \subset \mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p}$ denote the subring of power series which converge on the whole open unit disc, i.e., $f(x)$ converges for all $v_p(x)> 0$ . Both of these rings come equipped with an action of Frobenius given by the formula

$$\begin{align*}(\varphi \cdot f) (\pi) = f\big((1+\pi)^p - 1\big) \end{align*}$$

and $\Gamma := \operatorname {\mathrm {Gal}}(\mathbb {Q}_p(\mu _{p^{\infty }})/\mathbb {Q}_p)$ given by the formula

$$\begin{align*}(\gamma \cdot f) (\pi) = f\big((1+\pi)^{\chi_{\mathrm{cycl}}(\gamma)} - 1\big), \end{align*}$$

where $\gamma \in \Gamma $ and $\chi _{\mathrm {cycl}}\colon \Gamma \to \mathbb {Z}_p^{\times }$ is the cyclotomic character. The morphism $\varphi $ has a left inverse, denoted by $\psi $ , which satisfies the following relation:

$$\begin{align*}(\varphi \psi \cdot f)(\pi) = \frac{1}{p} \sum_{\xi^p = 1}{f\big((1+\pi)\xi - 1\big)}, \end{align*}$$

where the sum is over all p-th roots of unity.

If A is an affinoid $\mathbb {Q}_p$ -algebra, we define the Robba ring over A to be the completed tensor product $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } A$ , and similarly, we denote the subring of bounded power series by $\mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } A$ . As above, both of these rings come equipped with an action of $\varphi $ , $\psi ,$ and $\Gamma $ by the exact same formulae.

5.2 Overconvergent $(\varphi , \Gamma )$ -modules Over Affinoid Algebras

If M is a Galois representation over an affinoid algebra A (i.e., a finite projective A-module with a continuous A-linear action of $G_{\mathbb {Q}}$ ), then Berger and Colmez [Reference Berger and ColmezBC08] (and more generally Kedlaya and Liu [Reference Kedlaya and LiuKL10]) have constructed a functor

$$\begin{align*}M \mapsto \mathbf{D}^{\dagger}_{\mathrm{rig}}(M) \end{align*}$$

that associates with a Galois representation a $(\varphi , \Gamma )$ -module over the Robba ring $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } A$ . This agrees with the usual functor as constructed by Berger [Reference BergerBer02] when A is a finite field extension of $\mathbb {Q}_p$ .

For the rest of this section let $A=E$ be a finite field extension of $\mathbb {Q}_p$ and let D be a $(\varphi , \Gamma )$ -module over $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } E$ . By taking the “stalk at $\zeta _{p^n} - 1$ ”, one can define a $\mathbb {Q}_p(\mu _{p^{\infty }})(\!(t)\!)$ -module $D_{\mathrm {dif}}$ with a semilinear action of $\Gamma $ (see [Reference NakamuraNak14]). We set

$$\begin{align*}D_{\mathrm{dR}} := D_{\mathrm{dif}}^{\Gamma} \quad \text{ and } \quad D_{\mathrm{cris}} := D[t^{-1}]^{\Gamma} \end{align*}$$

and note that both $D_{\mathrm {dR}}$ and $D_{\mathrm {cris}}$ are finite-dimensional vector spaces over E.

If D is a crystalline $(\varphi , \Gamma )$ -module, then one can associate a sub- $(\mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p} \widehat {\otimes } E)$ -module of D, denoted $\mathbf {N}_{\mathrm {rig}}(D)$ , that is free of rank equal to the rank of D and is stable under $\Gamma $ . Furthermore, $\varphi $ restricts to a morphism

$$\begin{align*}\varphi \colon \mathbf{N}_{\mathrm{rig}}(D) \longrightarrow \mathbf{N}_{\mathrm{rig}}(D)[q^{-1}], \end{align*}$$

where $q = \varphi (\pi )/\pi $ ; and if D has non-negative Hodge–Tate weights, then $\mathbf {N}_{\mathrm {rig}}(D)$ is in fact stable under $\varphi $ . This submodule is called the Wach module associated with D and will be important in Section 6 when we recall the construction of the Perrin–Riou logarithm. For more details on the construction and properties of this module, see [Reference PottharstPot12, §3]. Note that if D comes from a crystalline p-adic representation V and $\mathbf {N}(V)$ denotes the usual Wach module (over $\mathbf {B}_{\mathbb {Q}_p}^+$ ) associated with V, then $\mathbf {N}_{\mathrm {rig}}(D)$ satisfies the relation

$$\begin{align*}\mathbf{N}_{\mathrm{rig}}(D) = \mathbf{N}(V) \otimes_{\mathbf{B}^+_{\mathbb{Q}_p}} \mathbf{B}^{+}_{\mathrm{rig}, \mathbb{Q}_p} . \end{align*}$$

Here, $\mathbf {B}^+_{\mathbb {Q}_p}$ is a period ring that can be identified with the subring $\mathbb {Z}_p \left [\!\!\left [ \pi \right ]\!\!\right ][1/p] \subset \mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p}$ .

5.3 Cohomology of $(\varphi , \Gamma )$ -modules

If D is a $(\varphi ,\Gamma )$ -module over $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \widehat {\otimes } A$ , then we define the Herr complex

$$\begin{align*}C^{\bullet}_{\varphi, \gamma}(D) := D \xrightarrow{\varphi-1,\gamma-1} D \oplus D \xrightarrow{1-\gamma,\varphi-1} D \, \end{align*}$$

concentrated in degrees $0, 1, 2$ , where $\gamma $ is a choice of topological generator for $\Gamma $ (such an element exists because we have assumed $p> 2$ ). We define $\mathbf {R}\Gamma _{\mathrm {cont}}(\mathbb {Q}_p, D)$ to be the corresponding object in the derived category of bounded complexes of (continuous) A-modules, and we denote the cohomology of this complex by $\operatorname {H}^i(\mathbb {Q}_p, D) := \operatorname {H}^i(C^{\bullet }_{\varphi , \gamma }(D))$ .

By [Reference PottharstPot13, Thm. 2.8], for any A-representation M of $G_{\mathbb {Q}_p}$ there is a canonical quasi-isomorphism

$$\begin{align*}\mathbf{R}\Gamma_{\mathrm{cont}}(\mathbb{Q}_p, M) \cong \mathbf{R}\Gamma_{\mathrm{cont}}(\mathbb{Q}_p, \mathbf{D}^{\dagger}_{\mathrm{rig}}(M)) \,. \end{align*}$$

Since $\varphi (D)$ generates D as an $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } A$ -module, the map $\varphi \colon D \to D$ has a unique semilinear left inverse $\psi $ . As the following lemma shows, the cohomology of D can be computed with $\psi $ in place of $\varphi $ .

Lemma 5.3.1 [Reference Kedlaya, Pottharst and XiaoKPX14, Proposition 2.3.6]

Let $C^{\bullet }_{\psi ,\gamma }(D)$ be the bottom row of the commutative diagram

(here the top row is $C^{\bullet }_{\varphi , \gamma }(D)$ ). Then the map $C^{\bullet }_{\varphi , \gamma }(D) \to C^{\bullet }_{\psi ,\gamma }(D)$ defined above is a quasi-isomorphism.

We finish this section by stating an Euler–Poincaré characteristic formula for $(\varphi , \Gamma )$ -modules, which will be used in the proof of Theorem 7.3.3.

Proposition 5.3.2 [Reference LiuLiu08, Theorem 5.3]

Let $A=E$ be a finite field extension of $\mathbb {Q}_p$ and let D be a $(\varphi , \Gamma )$ -module over $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } E$ . Then, for $i=0, 1, 2$ , $\operatorname {H}^i(\mathbb {Q}_p, D)$ are finite-dimensional vector spaces over E and

$$\begin{align*}\chi(D) := \sum_{i=0}^{2}(-1)^i\operatorname{dim}_E\operatorname{H}^i(\mathbb{Q}_p, D) = -\operatorname{rank}D, \end{align*}$$

where $\operatorname {rank}D$ is the rank of D as a $(\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } E)$ -module.

5.4 Iwasawa Cohomology

Recall that for an affinoid algebra A, we denote its unit ball by $A^\circ $ . Let M be a Galois representation over A and let T be a Galois stable lattice inside M (i.e., a sub- $A^\circ $ -module that is stable under the action of $G_{\mathbb {Q}}$ and satisfies $T[1/p] = M$ ). The classical Iwasawa cohomology of M is defined to be

$$\begin{align*}\operatorname{H}^i_{\mathrm{cl.Iw}}(\mathbb{Q}_{\infty}, M) := \big( \varprojlim_n \operatorname{H}^i(\mathbb{Q}(\mu_{p^n}), T)\big) \Big[\frac{1}{p}\Big], \end{align*}$$

where the inverse limit is over the corestriction maps, and the (analytic) Iwasawa cohomology of M is

$$\begin{align*}\operatorname{H}^i_{\mathrm{Iw}}(\mathbb{Q}_{\infty}, M) := \operatorname{H}^i(\mathbb{Q}, D^{\mathrm{la}}(\Gamma, M) ), \end{align*}$$

where $D^{\mathrm {la}}(\Gamma , M)$ denotes the space of locally analytic distributions on $\Gamma $ , valued in M. For a finite place v of $\mathbb {Q}$ , the Iwasawa cohomology groups $\operatorname {H}^i_{\mathrm {cl.Iw}}(\mathbb {Q}_{v, \infty }, M)$ and $\operatorname {H}^i_{\mathrm {Iw}}(\mathbb {Q}_{v, \infty }, M)$ are defined analogously. These two constructions satisfy the relation

$$\begin{align*}\operatorname{H}^i_{\mathrm{Iw}}(\mathbb{Q}_{\infty}, M) = \operatorname{H}^i_{\mathrm{cl.Iw}}(\mathbb{Q}_{\infty}, M) \hat{\otimes}_{\mathbb{Z}_p\left[\!\!\left[ \Gamma \right]\!\!\right]} D^{\mathrm{la}}(\Gamma, A). \end{align*}$$

Definition 5.4.2 Let D be a $(\varphi , \Gamma )$ -module over $\mathbf {B}^{\dagger }_{\mathrm {rig}, \mathbb {Q}_p} \hat {\otimes } A$ . The Iwasawa Herr complex is defined to be

$$\begin{align*}C_{\mathrm{Iw}}^{\bullet}(D)\colon D \xrightarrow{\psi - 1} D \end{align*}$$

concentrated in degrees $1$ and $2$ , where $\psi $ is the left inverse to $\varphi $ as discussed in the previous section. We denote the cohomology of this complex by $\operatorname {H}^i_{\mathrm {Iw}}(\mathbb {Q}_p, D)$ .

We have the following relation between Iwasawa cohomology for M and Iwasawa cohomology for $\mathbf {D}^{\dagger }_{\mathrm {rig}}(M)$ .

Proposition 5.4.3 Let M be a Galois representation over an affinoid algebra A. Then one has the following isomorphism

$$\begin{align*}\operatorname{H}^i_{\mathrm{Iw}}(\mathbb{Q}_{p, \infty}, M) \cong \operatorname{H}^i_{\mathrm{Iw}}\big(\mathbb{Q}_p, \mathbf{D}^{\dagger}_{\mathrm{rig}}(M)\big). \end{align*}$$

In particular , $\operatorname {H}^i_{\mathrm {Iw}}(\mathbb {Q}_{p, \infty }, M)$ vanishes for $i \neq 1, 2$ .

6.1 Perrin-Riou’s Big Logarithm

Let $\Lambda _{\infty }$ denote the global sections of $\mathcal {W}_E$ ; this can be identified with a subring of the ring of power series $E[\Delta ]\left [\!\!\left [ \gamma - 1 \right ]\!\!\right ]$ , where $\Delta $ is the torsion subgroup of $\Gamma $ and $\gamma $ is a topological generator of $\Gamma /\Delta $ . The ring $(\mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p} \widehat {\otimes } E)^{\psi = 0}$ has an action of $\Gamma $ that extends to an action of $\Lambda _{\infty }$ via the Mellin transform:

$$ \begin{align*} \mathfrak{M}\colon \Lambda_{\infty} & \xrightarrow{\sim} (\mathbf{B}^{+}_{\mathrm{rig}, \mathbb{Q}_p} \widehat{\otimes} E)^{\psi = 0} \nonumber \\ f(\gamma - 1) & \mapsto f(\gamma - 1) \cdot (1 + \pi). \nonumber \end{align*} $$

Therefore, $(\mathbf {B}^{+}_{\mathrm {rig}, \mathbb {Q}_p} \widehat {\otimes } E)^{\psi = 0}$ can be viewed as a free $\Lambda _{\infty }$ -module of rank one.

Definition 6.1.1 Suppose that D is a crystalline $(\varphi , \Gamma )$ -module satisfying hypothesis (H). The Perrin-Riou logarithm $\mathcal {L} = \mathcal {L}_D : \operatorname {H}^1_{\mathrm {Iw}}(\mathbb {Q}_{p}, D) \to D_{\mathrm {cris}} \widehat {\otimes } \Lambda _{\infty }$ is defined as the composition of the following maps:

$$ \begin{align*} \operatorname{H}^1_{\mathrm{Iw}}(\mathbb{Q}_{p}, D) \cong D^{\psi = 1} & = \mathbf{N}_{\mathrm{rig}}(D)^{\psi = 1} \nonumber \\ & \xrightarrow{1 - \varphi} (\varphi^*\mathbf{N}_{\mathrm{rig}}(D))^{\psi = 0} \nonumber \\ & \longrightarrow D_{\mathrm{cris}} \widehat{\otimes}_{E} (\mathbf{B}^{+}_{\mathrm{rig}, \mathbb{Q}_p} \widehat{\otimes} E)^{\psi = 0} \xrightarrow{\sim} D_{\mathrm{cris}} \widehat{\otimes}_{E} \Lambda_{\infty}, \nonumber \end{align*} $$

where the second, third, and fourth maps exist by Lemma 6.0.4. Here the last map is given by the inverse of the Mellin transform described above. Since all of the maps above are functorial in D, we see that $\mathcal {L}$ is a map of $\Lambda _{\infty }$ -modules that is also functorial in D.

In the case where D comes from a crystalline p-adic representation, the map $\mathcal {L}$ is a special case of a more general construction by Perrin-Riou ([Reference Perrin-RiouPR00]), which was later interpreted using $(\varphi , \Gamma )$ -modules by Lei, Loeffler, and Zerbes ([Reference Lei, Loeffler and ZerbesLLZ11]). This was generalised to potentially crystalline $(\varphi , \Gamma )$ -modules by Pottharst ([Reference PottharstPot12]) and to de Rham $(\varphi , \Gamma )$ -modules by Nakamura ([Reference NakamuraNak14]). The key property of this map is that it interpolates the Bloch–Kato logarithm and dual exponential maps at certain classical specialisations. More precisely, let $\eta \colon \Gamma \to (E^{\prime })^{\times }$ be a continuous character, where $\Gamma = \operatorname {\mathrm {Gal}}(\mathbb {Q}_p(\mu _{p^{\infty }})/\mathbb {Q}_p)$ , and consider the induced map $\Lambda _{\infty } \to E^{\prime },$ which we will also denote by $\eta $ . Then we have two specialisation maps: the first on Iwasawa cohomology

$$\begin{align*}\operatorname{H}^1_{\mathrm{Iw}}(\mathbb{Q}_{p}, D) \xrightarrow{\mathrm{pr}_{\eta}} \operatorname{H}^1\big(\mathbb{Q}_p, D_{E^{\prime}}(\eta^{-1}) \big) \end{align*}$$

induced from the map on the Herr complexes $C^{\bullet }_{Iw}(D) \to C^{\bullet }_{\psi , \gamma }(D(\eta ^{-1}))$ which in degree one is given by $x \mapsto (0, x)$ (recall that by Lemma 5.3.1 the cohomology of a $(\varphi , \Gamma )$ -module can be calculated with $\psi $ in place of $\varphi $ ). The second specialisation map is

$$\begin{align*}D_{\mathrm{cris}} \widehat{\otimes} \Lambda_{\infty} \xrightarrow{\mathrm{ev}_{\eta}} D_{\mathrm{cris}} \otimes_{\mathbb{Q}_p} E^{\prime} \end{align*}$$

induced from $\eta \colon \Lambda _{\infty } \to E^{\prime }$ . Then for classical $\eta $ (i.e., $\eta = \chi _{\mathrm {cycl}}^j$ for some integer j) there is a commutative diagram

where the dotted arrow is (up to some Euler factors) the Bloch–Kato logarithm in the range $j < h_1$ , and the dual exponential map in the range $j \geq h_1$ , where $h_1$ is the smallest Hodge–Tate weight of D (see [Reference NakamuraNak14]). We will study this map at non-classical specialisations.

Proof The first part follows from carefully tracing through the definitions.

For the second part, it is enough to show that if $\mathrm {pr}_{\eta }(x) = 0,$ then $\mathrm {ev}_{\eta }(\mathcal {L}(x)) = 0$ , because then we can just define the map by taking a lift to $\operatorname {H}^1_{\mathrm {Iw}}(\mathbb {Q}_{p}, D)$ . If $\mathrm {pr}_{\eta }(x) = 0,$ then there exists $y \in D^{\psi = 1}$ such that

$$\begin{align*}x = (\gamma - \eta(\gamma) ) y. \end{align*}$$

But $\mathcal {L}$ is $\Lambda _{\infty }$ -linear so (after base-changing D and $\Lambda _{\infty }$ to $D_{E^{\prime }}$ and $(\Lambda _{\infty })_{E^{\prime }},$ respectively)

$$\begin{align*}\mathcal{L}(x) = (\gamma - \eta(\gamma) ) \mathcal{L}(y). \end{align*}$$

But this is precisely mapped to zero under $\mathrm {ev}_{\eta }$ .▪

From the above proposition, we obtain the following corollary, which will be useful in later sections.

6.2 Application to Beilinson–Flach Classes

Returning to the situation in the paper, let f and g denote eigenforms satisfying the assumptions in Section 1.1, so we have Coleman families $\mathcal {F}, \mathcal {G}_{\alpha }, \mathcal {G}_{\beta }$ over $V_1, V_2^{\alpha }, V_2^{\beta }$ passing through the p-stabilisations $f_{\alpha }, g_{\alpha }, g_{\beta }$ respectively. Here $f_{\alpha }$ denotes the p-stabilisation of f that satisfies $U_p f = \alpha f$ ; and similarly for $g_{\alpha }$ and $g_{\beta }$ . Since we have assumed that the weights of f and g are not equal, we can choose $V_1$ such that it does not contain the character $k^{\prime }$ . In this subsection, $\mathcal {G}$ and $V_2$ will denote either $\mathcal {G}_{\alpha }$ and $V_2^{\alpha }$ or $\mathcal {G}_{\beta }$ and $V_2^{\beta }$ , respectively.

Let $M_{V_1}(\mathcal {F})$ and $M_{V_2}(\mathcal {G})$ denote the Galois representations associated with $\mathcal {F}$ and $\mathcal {G}$ and let $\mathbf {D}^{\dagger }_{\mathrm {rig}}(\mathcal {F})^*$ and $\mathbf {D}^{\dagger }_{\mathrm {rig}}(\mathcal {G})^*$ denote the $(\varphi , \Gamma )$ -modules of $M_{V_1}(\mathcal {F})^*$ and $M_{V_2}(\mathcal {G})^*,$ respectively. If $V_1$ and $V_2$ are small enough, both of these modules come with a canonical triangulation, which we will denote as

$$\begin{align*}0 \longrightarrow \mathscr{F}^{+}\mathbf{D}^{\dagger}_{\mathrm{rig}}(?)^* \longrightarrow \mathbf{D}^{\dagger}_{\mathrm{rig}}(?)^* = \mathscr{F}^{o}\mathbf{D}^{\dagger}_{\mathrm{rig}}(?)^* \longrightarrow \mathscr{F}^{-}\mathbf{D}^{\dagger}_{\mathrm{rig}}(?)^* \longrightarrow 0 \end{align*}$$

for $? = \mathcal {F}$ or $?=\mathcal {G}$ . In fact, there is an explicit description for both the kernel and the cokernel (see [Reference Loeffler and ZerbesLZ16, Theorem 6.3.2]).

Let $k_1$ be a (not necessarily classical) weight in $V_1$ and let $\eta $ be a character of $\Gamma = \operatorname {\mathrm {Gal}}(\mathbb {Q}_p(\mu _{p^{\infty }})/\mathbb {Q}_p)$ , and let E be a finite extension of $\mathbb {Q}_p$ that contains the fields of definition of $k_1$ and $\eta $ . Recall that $k^{\prime }+2$ denotes the weight of g. Let $M(\mathcal {F}_{k_1})$ and $M(\mathcal {G}_{k^{\prime }})$ denote the specialisations of $M_{V_1}(\mathcal {F})$ and $M_{V_2}(\mathcal {G})$ at $k_1$ and $k^{\prime }$ respectively, and note that we have isomorphisms

$$ \begin{align*} M((\mathcal{G}_{\alpha})_{k^{\prime}}) & \xrightarrow{\mathrm{pr}^{\alpha}} M_E(g), \nonumber \\ M((\mathcal{G}_{\beta})_{k^{\prime}}) & \xrightarrow{\mathrm{pr}^{\beta}} M_E(g) \nonumber \end{align*} $$

both of which follow from the fact that g is classical and the Galois representation does not change after p-stabilisation. We let $\mathbf {D}^{\dagger }_{\mathrm {rig}}(\mathcal {F}_{k_1})^*$ , $\mathbf {D}^{\dagger }_{\mathrm {rig}}(\mathcal {G}_{k^{\prime }})^*$ , and $\mathbf {D}^{\dagger }_{\mathrm {rig}}(g)^*$ denote the $(\varphi , \Gamma )$ -modules associated with $M(\mathcal {F}_{k_1})^*$ , $M(\mathcal {G}_{k^{\prime }})^*$ and $M_E(g)^*$ , respectively. By specialising the triangulation above and applying $\mathrm {pr}^{\alpha }$ or $\mathrm {pr}^{\beta }$ if necessary, we obtain triangulations for these three $(\varphi , \Gamma )$ -modules.

Let $D^-$ be the $(\varphi , \Gamma )$ -module

(6.2.1) $$ \begin{align} D^- = \mathscr{F}^{-}\mathbf{D}^{\dagger}_{\mathrm{rig}}(\mathcal{F}_{k_1})^* \otimes \mathscr{F}^{o}\mathbf{D}^{\dagger}_{\mathrm{rig}}(g)^*. \end{align} $$

Since g is classical, $D^-$ is crystalline with Hodge–Tate weights ${0, 1+k^{\prime }}$ and by the explicit description in [Reference Loeffler and ZerbesLZ16, Theorem 6.3.2], $p^n$ is not an eigenvalue for $\varphi $ on $D^-_{\mathrm {cris}}$ for any integer $n \geq 0$ (so $D^-$ satisfies hypothesis (H)). Consider the following submodules

$$ \begin{align*} D_1 = D^{\alpha} & = \mathrm{pr}^{\alpha} \big( \mathscr{F}^{-}\mathbf{D}^{\dagger}_{\mathrm{rig}}(\mathcal{F}_{k_1})^* \otimes \mathscr{F}^{+}\mathbf{D}^{\dagger}_{\mathrm{rig}}((\mathcal{G}_{\alpha})_{k^{\prime}})^* \big), \nonumber \\ D_2 = D^{\beta} & = \mathrm{pr}^{\beta} \big( \mathscr{F}^{-}\mathbf{D}^{\dagger}_{\mathrm{rig}}(\mathcal{F}_{k_1})^* \otimes \mathscr{F}^{+}\mathbf{D}^{\dagger}_{\mathrm{rig}}((\mathcal{G}_{\beta})_{k^{\prime}})^* \big), \nonumber \end{align*} $$

where $\mathrm {pr}^{\alpha }$ and $\mathrm {pr}^{\beta }$ are the isomorphisms described above. Again, by the explicit description in loc. cit., $(D_1)_{\mathrm {cris}}$ and $(D_2)_{\mathrm {cris}}$ are both rank one sub $\varphi $ -modules of $D_{\mathrm {cris}}^-$ on which $\varphi $ acts by multiplication by $\alpha _{\mathcal {F}_{k_1}}^{-1}\beta _g^{-1}$ and $\alpha _{\mathcal {F}_{k_1}}^{-1}\alpha _g^{-1}$ respectively. Since we have assumed that g is p-regular (i.e., $\alpha _g \neq \beta _g$ ), we must have $D_{\mathrm {cris}}^- = (D_1)_{\mathrm {cris}} \oplus (D_2)_{\mathrm {cris}}$ .

Let $c> 6$ be an integer that is coprime to $6Np$ and let $z_1^{\alpha }$ be the image of the Beilinson–Flach class ${}_{c} \mathcal {BF}^{[\mathcal {F}, \mathcal {G}_{\alpha }]}_{1, 1}$ (see §4.3) under the composition

$$ \begin{align*} \operatorname{H}^1\big(\mathbb{Q}, D^{\mathrm{la}}(\Gamma, M(\mathcal{F}_{k_1})^* &\otimes M((\mathcal{G}_{\alpha})_{k^{\prime}})^*)\big)\\ &\longrightarrow \operatorname{H}^1_{\mathrm{Iw}}\big(\mathbb{Q}_{p, \infty}, M(\mathcal{F}_{k_1})^* \otimes M(g)^*\big) \longrightarrow \operatorname{H}^1_{\mathrm{Iw}}(\mathbb{Q}_p, D^-) \end{align*} $$

and similarly for $z_1^{\beta }$ , where the first map is restriction to the decomposition group at p composed with the isomorphism $\mathrm {pr}^{\alpha }$ . Recall that $L_p(\mathcal {F}, g, 1 +\mathbf {j})$ is the two-variable p-adic L-function associated with the Coleman family $\mathcal {F}$ and the universal twist $\mathbf {j}$ (see [Reference UrbanUrb14] or [Reference Loeffler and ZerbesLZ16, §9]).

Proof Recall that $D^-$ is a crystalline $(\varphi , \Gamma )$ -module satisfying hypothesis (H). Therefore, we have the Perrin–Riou logarithm

$$\begin{align*}\mathcal{L}\colon \operatorname{H}^1_{\mathrm{Iw}}(\mathbb{Q}_{p}, D^-) \longrightarrow D^-_{\mathrm{cris}} \widehat{\otimes} \Lambda_{\infty}. \end{align*}$$

By Proposition 6.1.2(i), this map agrees with the map (also denoted by $\mathcal {L}$ ) constructed in [Reference Loeffler and ZerbesLZ16, Theorem 7.1.4] after specialising at $(k_1, k^{\prime })$ . Indeed, the map in loc. cit. is defined as the pull back of $\mathcal {L}_{D^-(-1-k^{\prime })}$ under the automorphism $\omega _{-1-k^{\prime }}$ .

By the “explicit reciprocity law” of Theorem 7.1.5 in op. cit., we see that $\mathrm {ev}_{\eta }(\mathcal {L}(z_1^{\alpha }))$ and $\mathrm {ev}_{\eta }(\mathcal {L}(z_1^{\beta }))$ are both non-zero if the quantity

(6.2.3) $$ \begin{align} \big( c^2 - c^{-(k_1 + k^{\prime} - 2\eta)} \varepsilon_{\mathcal{F}_{k_1}}(c)^{-1}\varepsilon_g(c)^{-1} \big) (-1)^{1+ \eta} \lambda_N(\mathcal{F}_{k_1} )^{-1} L_p(\mathcal{F}_{k_1}, g, 1+ \eta ) \end{align} $$

is non-zero, where $\lambda _N(\mathcal {F}_{k_1} )$ denotes the specialisation of the Atkin–Lehner pseudo-eigenvalue of $\mathcal {F}$ (see [Reference Kings, Loeffler and ZerbesKLZ17, §2.5] for the definition of the Atkin–Lehner operators). Since our assumption at the start of §6.2 implies that $k_1 \neq k^{\prime }$ , we can choose the integer c such that the first factor in (6.2.3) is non-zero. The result then follows by applying Corollary 6.1.3. ▪

7.1 Cohomological Preliminaries

In [Reference NekovářNek06], Nekovář defined the concept of a Selmer complex: an object in a certain derived category whose cohomology is closely related to the usual definition of Selmer groups. This construction is useful because the resulting complex has nice base-change and duality properties; attributes that one does not necessarily have for the classical Selmer groups. In [Reference PottharstPot13], Pottharst extends this construction to families of Galois representations over well-behaved rigid analytic spaces. This is the tool we will use to construct a sheaf interpolating the Bloch–Kato Selmer groups. We now summarise this construction.

Let A be an affinoid algebra over $\mathbb {Q}_p$ and let M be an A-valued representation of $G_{\mathbb {Q}} = \operatorname {\mathrm {Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})$ (i.e., a finitely generated, projective A-module with a continuous action of $G_{\mathbb {Q}}$ ). Let $\Sigma $ be a set of places of $\mathbb {Q}$ containing $p, \infty $ and all primes where M is ramified, and assume that $\Sigma $ is finite. Let $G_{\Sigma }$ denote the Galois group of the maximal algebraic unramified-outside- $\Sigma $ extension of $\mathbb {Q}$ and, for a place $v \in \Sigma $ , let $G_v$ denote a fixed decomposition group in $G_{\Sigma }$ associated with the place v.

Definition 7.1.1 A collection of local conditions $\Delta $ for M is a set of pairs $\{ (\Delta _v, \iota _v): v \in \Sigma \}$ where $\Delta _v$ is an object in the derived category of bounded complexes of (continuous) A-modules, and $\iota _v$ is a morphism

$$\begin{align*}\Delta_v \xrightarrow{\iota_v} \mathbf{R}\Gamma_{\mathrm{cont}}(G_v, M). \end{align*}$$

One defines the Selmer complex ${\mathbf {R}\widetilde {\Gamma }({G_{\Sigma }}, {M}; {\Delta })}$ in the following way.

Definition 7.1.2 Let $\Delta $ be a set of local conditions for M. Then the Selmer complex ${\mathbf {R}\widetilde {\Gamma }({G_{\Sigma }}, {M}; {\Delta })}$ is the mapping fibre

$$\begin{align*}{\mathbf{R}\widetilde{\Gamma}({G_{\Sigma}}, {M}; {\Delta})} := \operatorname{Cone}\Big( \mathbf{R}\Gamma_{\mathrm{cont}}(G_{\Sigma}, M) \oplus \bigoplus_{v \in \Sigma}{\Delta_v} \xrightarrow{\operatorname{res}_v - \iota_v} \bigoplus_{v \in \Sigma} \mathbf{R}\Gamma_{\mathrm{cont}}(G_v, M) \Big)[-1]. \end{align*}$$

We denote the i-th cohomology of this complex by $\widetilde {H}^{i}(G_{\Sigma }, M; \Delta )$ .

If $\Delta _v$ is quasi-isomorphic to a complex of finitely generated A-modules concentrates in degrees $[0, 2]$ (all our local conditions in this paper will satisfy this), then ${\mathbf {R}\widetilde {\Gamma }({G_{\Sigma }}, {M}; {\Delta })}$ is quasi-isomorphic to a complex of finitely generated A-modules, concentrated in degrees $[0, 3]$ (see [Reference PottharstPot13, §1.5]).

This construction also works in more general situations. For example, if X is a quasi-Stein rigid analytic space, then ${\mathbf {R}\widetilde {\Gamma }({G_{\Sigma }}, {M}; {\Delta })}$ is quasi-isomorphic to a complex of coherent $\mathcal {O}_X$ -modules, concentrated in degrees $[0, 3]$ (this is situation (4) described in [Reference PottharstPot13, §1.5]).

In §7.3 we will use the Poitou–Tate long exact sequence described above to show that the cohomology of the Selmer complex vanishes in degree two if the corresponding value of the p-adic L-function is non-zero. In particular we will use the following result:

Proposition 7.1.4 If the map $\xi $ in Proposition 7.1.3 (ii) is surjective, then we have an injective map

$$\begin{align*}\widetilde{H}^{2}(G_{\Sigma}, M; \Delta^{\prime}) \hookrightarrow \widetilde{H}^{2}(G_{\Sigma}, M; \Delta). \end{align*}$$

In particular, if $\widetilde {H}^{2}(G_{\Sigma }, M; \Delta )$ vanishes, then so does $\widetilde {H}^{2}(G_{\Sigma }, M; \Delta ^{\prime })$ .

7.2 Convolution of Two Coleman Families

Let $\mathbf {x} = (k_1, k^{\prime }, \eta ) \in V_1 \times \{k^{\prime }\} \times \mathcal {W}$ be a tuple of weights defined over a finite extension $E / \mathbb {Q}_p$ .

Let $\overline {M}_{\mathbf {x}}$ denote the representation $[M(\mathcal {F}_{k_1})^* \otimes M(g)^*](\eta ^{-1})$ and let $\Sigma $ be a finite set of places of $\mathbb {Q}$ that contains $p, \infty $ and all the primes where $\overline {M}_{\mathbf {x}}$ ramifies. Let $D_{\mathbf {x}} := \mathbf {D}^{\dagger }_{\mathrm {rig}}(\overline {M}_{\mathbf {x}})$ denote the $(\varphi , \Gamma )$ -module associated with $\overline {M}_{\mathbf {x}}$ and recall that we have a two dimensional quotient $\mathbf {D}^{\dagger }_{\mathrm {rig}}(\overline {M}_{\mathbf {x}}) \rightarrow D^-(\eta ^{-1}) =: D^-_{\mathbf {x}}$ , where $D^-$ is defined as in (6.2.1). We denote the kernel of this quotient by $D^+_{\mathbf {x}}$ . For $v \in \Sigma \backslash \{p\},$ we call $(\mathbf {R}\Gamma _{\mathrm {cont}}(G_v/I_v, {\overline {M}_{\mathbf {x}}}^{I_v}), \iota _v)$ the unramified local condition at v, where $\iota _v$ is the natural map induced by inflation. We are interested in the following examples of local conditions:

• • (Relaxed) For $v \in \Sigma ,$ take $\Delta _{\mathrm {rel}}$ to be the set of unramified local conditions for $v \neq p$ and

  $$\begin{align*}\Delta_{\mathrm{rel}, p} := \mathbf{R}\Gamma_{\mathrm{cont}}(G_p, \overline{M}_{\mathbf{x}}) \xrightarrow{\sim} \mathbf{R}\Gamma_{\mathrm{cont}}(G_p, \overline{M}_{\mathbf{x}}). \end{align*}$$
  We denote the cohomology of the associated Selmer complex by $\widetilde {\operatorname {H}}_{\mathrm {rel}}^{i}(\mathbb {Q}, \overline {M}_{\mathbf {x}})$ .

• • (Strict) For $v \in \Sigma ,$ take $\Delta _{\mathrm {str}}$ to be the set of unramified local conditions for $v \neq p$ and

  $$\begin{align*}\Delta_{\mathrm{str}, p} := 0 \longrightarrow \mathbf{R}\Gamma_{\mathrm{cont}}(G_p, \overline{M}_{\mathbf{x}}). \end{align*}$$
  We denote the cohomology of the associated Selmer complex by $\widetilde {\operatorname {H}}_{\mathrm {str}}^{i}(\mathbb {Q}, \overline {M}_{\mathbf {x}})$ .

• • (Panchishkin) For $v \in \Sigma $ take $\Delta _{f}$ to be the set of unramified local conditions for $v \neq p$ and

  $$\begin{align*}\Delta_{f, p} := \mathbf{R}\Gamma_{\mathrm{cont}}(G_p, D^+_{\mathbf{x}}) \longrightarrow \mathbf{R}\Gamma_{\mathrm{cont}}(\mathbb{Q}_p, \mathbf{D}^{\dagger}_{\mathrm{rig}}(\overline{M}_{\mathbf{x}})) \cong \mathbf{R}\Gamma_{\mathrm{cont}}(G_p, \overline{M}_{\mathbf{x}}). \end{align*}$$
  We denote the cohomology of the associated Selmer complex by $\widetilde {\operatorname {H}}_f^{i}(\mathbb {Q}, \overline {M}_{\mathbf {x}})$ . The reason for choosing this local condition is because it is closely related to the Bloch–Kato local condition when $\mathbf {x}$ lies in the critical range (the range where the p-adic L-function interpolates critical values of the global L-function). We will discuss this relation in §7.4.

Let $c> 6$ be an integer prime to $6Np$ and recall from §4.3 that, for an integer m coprime to $6Ncp$ , there is a Beilinson–Flach class

$$\begin{align*}{}_{c}\mathcal{BF}^{[\mathcal{F}, \mathcal{G}]}_{m, 1} \in \operatorname{H}^1 \big(\mathbb{Q}(\mu_m), D^{\mathrm{la}}(\Gamma, M) \big), \end{align*}$$

where $M = M_{V_1}(\mathcal {F})^* \widehat {\otimes } M_{V_2}(\mathcal {G})^*$ . By specialising these Beilinson–Flach classes at $\mathbf {x}$ and identifying $M((\mathcal {G}_{\alpha })_{k^{\prime }})$ and $M((\mathcal {G}_{\beta })_{k^{\prime }})$ with $M_E(g)$ as before, we obtain classes in $\operatorname {H}^1(\mathbb {Q}(\mu _m), \overline {M}_{\mathbf {x}} )$ .

In Section 4.4, we showed that these classes satisfy certain norm relations and that we could produce an Euler system from these classes. More precisely, let $\overline {T}_{\mathbf {x}}$ be a Galois stable lattice inside $\overline {M}_{\mathbf {x}}$ . Then there exist collections $\{c^{\alpha }_m \in \operatorname {H}^1(\mathbb {Q}(\mu _m), \overline {T}_{\mathbf {x}}) : m \in S \}$ and $\{c^{\beta }_m \in \operatorname {H}^1(\mathbb {Q}(\mu _m), \overline {T}_{\mathbf {x}}) : m \in S \}$ satisfying the Euler system relations, and $c_1^{\alpha }$ and $c_1^{\beta }$ are equal to non-zero multiples of the specialisations of ${{}_{c} \mathcal {BF}_{1, 1}^{[\mathcal {F}, \mathcal {G}_{\alpha }]}}$ and ${{}_{c} \mathcal {BF}_{1, 1}^{[\mathcal {F}, \mathcal {G}_{\beta }]}}$ at $\mathbf {x}$ respectively.

We choose the integer c in such a way that the conclusion of Proposition 6.2.2 holds at the point $\mathbf {x}$ (this choice may depend on $\mathbf {x}$ ). After making this choice, the classes $c_{m}^{\alpha }, c_{m}^{\beta }$ satisfy the following local conditions.

7.3 A Vanishing Result

Let $\mathcal {F}$ be a Coleman family over an affinoid domain V. For ease of notation we set $\alpha _{\mathcal {F}} := a_p(\mathcal {F})$ and similarly for specialisations of $\mathcal {F}$ . Moreover, recall that if the specialisation of $\mathcal {F}$ at $k_1$ is a noble eigenform (so it is the p-stabilisation of an eigenform h), then $\alpha _{\mathcal {F}_{k_1}} = \alpha _h$ and $\beta _h$ denote the roots of the Hecke polynomial at p associated with h, and satisfy $\alpha _h \beta _h = p^{k_1 + 1}\varepsilon _h(p)$ . In this case, we will also write $\beta _{\mathcal {F}_{k_1}} := \beta _h$ (although the notation $\beta _{\mathcal {F}}$ is, of course, meaningless).

The goal of this section is to show that if the p-adic L-function does not vanish at $\mathbf {x}$ then the Selmer group $\widetilde {\operatorname {H}}_f^{2}(\mathbb {Q}, \overline {M}_{\mathbf {x}})$ is trivial. The strategy is to combine Propositions 7.1.4 and 7.2.2 by comparing the Panchishkin and relaxed local conditions. In particular, we must show that the hypothesis in Proposition 7.1.4 is satisfied. Unfortunately this is not true in general and fails when $\overline {M}_{\mathbf {x}}$ has a “local zero”, i.e., the local Euler factor of $\overline {M}_{\mathbf {x}}$ at p vanishes at $s = 1$ . Therefore, we impose the following hypothesis on $\overline {M}_{\mathbf {x}}$ :