Local $\varepsilon$-conjecture and $p$-adic differential equations

In [Kat93a], Kato formulated a conjecture called the generalized Iwasawa main conjecture, which is a vast generalization of the Iwasawa main conjecture and Bloch-Kato conjecture. It claims the existence of so-called zeta isomorphisms for any family of $p$-adic Galois representations of $G_{\mathbf{Q}}$, interpolating the zeta elements of geometric $p$-adic Galois representations. Note that a similar conjecture was formulated by Fontaine and Perrin-Riou in [FP94]. Since the zeta elements are conjectural bases in (the determinants of) the Galois cohomologies and closely related to the $L$-functions, it is natural to regard the zeta isomorphisms as algebraic counterparts of the $L$-functions. In [Kat93b] and [FK06], Kato’s local and global $\varepsilon$-conjectures are formulated as algebraic analogue of the functional equations of $L$-functions; the local $\varepsilon$-conjecture claims the existence of the local $\varepsilon$-isomorphisms, the algebraic analogue of local $\varepsilon$-factors for families of $p$-adic representations of $G_{\mathbf{Q}_{l}}$, and the global $\varepsilon$-conjecture states that the zeta isomorphisms satisfies the functional equations whose local factors are the local $\varepsilon$-isomorphisms.

The local $\varepsilon$-conjecture for $l\neq p$ is proved [Yas09], [Kak14]. But for the case $l=p$, which we treat in this paper, the existence of the local $\varepsilon$-isomorphisms are proved for limited families and the conjecture is still open. In particular, by generalizing the conjecture for $(\varphi,\Gamma)$-modules over relative Robba rings, the second author proves the existence of $\varepsilon$-isomorphisms for trianguline representations. The conjecture has turned out to be closely related to the Coleman isomorphisms [Kat93b] [Ven13], the Perrin-Riou maps [BB08] [LVZ13], and also the $p$-adic local Langlands correspondence [Nak17b] [RJ18].

Our main theorem compares the local $\varepsilon$-isomorphisms of the following different objects. Let $M$ be an arbitrary de Rham $(\varphi,\Gamma)$-module over a Robba ring. The first object is the cyclotomic deformation of $M$. The second one is the cyclotomic deformation of $\mathbf{N}_{{\operatorname{rig}}}(M)$, where $\mathbf{N}_{{\operatorname{rig}}}(M)$ is the $p$-adic differential equation attached to $M$ by Laurent Berger. We remark that the existences of their local $\varepsilon$-isomorphisms are still conjectural. The main theorem claims that the difference of their local $\varepsilon$-isomorphisms is written as the generalized Perrin-Riou map defined by the second author in [Nak14].

To make the statement of the main theorem more precise, we recall $(\varphi,\Gamma)$-modules over Robba rings and the local $\varepsilon$-conjecture for them.

A $(\varphi,\Gamma)$-module $D$ is a module equipped with a suitable endomorphism $\varphi:D\to D$ and a continuous group action of $\Gamma=\mathrm{Gal}({\mathbf{Q}_{p}}(\mu_{p^{\infty}})/{\mathbf{Q}_{p}})$, where $\mu_{p^{\infty}}$ is the group of $p$-power roots of unity in $\overline{\mathbf{Q}}_{p}$. There are several specific rings over which $(\varphi,\Gamma)$-modules are useful to study $p$-adic representations. An important case is the Robba rings $\mathcal{R}_{L}$ with their coefficients in local fields $L$; by results of Fontaine [Fon90], Cherbonnier and Colmez [CC99] and Kedlaya [Ked08], the category of $p$-adic representations over $L$ can be embedded fully and faithfully into the one of $(\varphi,\Gamma)$-modules over $\mathcal{R}_{L}$. A lot of important notions of $p$-adic Hodge theory can be generalized to $(\varphi,\Gamma)$-modules over $\mathcal{R}_{L}$, such as the functors $\mathbf{D}_{{\operatorname{cris}}}$ and $\mathbf{D}_{{\operatorname{dR}}}$ [Ber02], or Bloch-Kato’s exponential maps [Ber03], [Nak14]. Another important feature is that, when a $(\varphi,\Gamma)$-module $M$ is de Rham, Berger attached to $M$ a $p$-adic differential equation $\mathbf{N}_{{\operatorname{rig}}}(M)$ with Frobenius structure; as its application, one can prove the $p$-adic monodromy theorem for $p$-adic representations by reducing it to that for $p$-adic differential equations, or Colmez-Fontaine’s theorem [Ber02], [Ber08].

In [Nak17a], the second author formulated the local $\varepsilon$-conjecture for $(\varphi,\Gamma)$-modules over relative Robba rings, generalizing the Kato’s conjecture for $p$-adic representations. We recall only the conjecture for the cyclotomic deformations of de Rham $(\varphi,\Gamma)$-modules, since it is the case we treat in this paper. Let $L$ be a finite extension of ${\mathbf{Q}_{p}}$, and $M$ be a $(\varphi,\Gamma)$-module over the Robba ring $\mathcal{R}_{L}$ with coefficients in $L$. Then, one can attach to $M$ a (graded) invertible module $\Delta_{L}(M)$ over $L$ and $\Delta^{\operatorname{{Iw}}}_{L}(M)$ over ${\mathcal{R}^{+}_{L}(\Gamma)}$ for a $(\varphi,\Gamma)$-module $M$ over $\mathcal{R}_{L}$, where we put $\mathcal{R}^{+}_{L}(\Gamma)=\Gamma(\mathcal{W},\mathcal{O}_{\mathcal{W}})$ and $\mathcal{W}$ the Berthelot generic fiber of the Iwasawa algebra $\mathcal{O}_{L}[[\Gamma]]$. When $M$ is de Rham, he constructed a canonical trivialization isomorphism

Its definition involves a lot of notions of $p$-adic Hodge theory, such as the theory of local constants ($\varepsilon$-constants and $L$-constants), Bloch-Kato’s exponential and dual exponential maps, Hodge-Tate weights. Then the local $\varepsilon$-conjecture in this situation claims that, there exists a unique isomorphism

interpolating $\varepsilon^{\operatorname{dR}}_{L}(M(\delta))$ for any de Rham character $\delta:\Gamma\to L^{\times}$, i.e. any character of the form $\delta=\chi^{k}\tilde{\delta}$ for $k\in\mathbf{Z}$ and a finite character $\tilde{\delta}$, where $\chi$ is the cyclotomic character. More precisely, $\varepsilon^{{\operatorname{{Iw}}}}_{L}(M)$ is required to make the following diagram

commute for any de Rham character $\delta$ of $\Gamma$, where $f_{\delta}:{\mathcal{R}^{+}_{L}(\Gamma)}\to{\mathcal{R}^{+}_{L}(\Gamma)}$ is a continuous homomorphism of $L$-algebras given by $[g]\mapsto\delta(g)^{-1}$ and ${\mathrm{ev}}_{\delta}$ is a canonical isomorphism induced by the specialization a $f_{\delta}$. In the original article of Kato [Kat93b], he predicts the conjectural base $\varepsilon^{\operatorname{{Iw}}}_{\mathcal{O}_{L}}(T)$ of an invertible $\mathcal{O}_{L}[[\Gamma]]$-module $\Delta^{\operatorname{{Iw}}}_{\mathcal{O}_{L}}(T)$ similarly defined for any $\mathcal{O}_{L}$-representation $T$ of $G_{\mathbf{Q}_{p}}$. In [Nak17a], the second author predicts the equality $\varepsilon^{\operatorname{{Iw}}}_{\mathcal{O}_{L}}(T)\otimes{\mathrm{id}}=\varepsilon^{\operatorname{{Iw}}}_{L}(\mathbf{D}^{\dagger}_{\mathrm{rig}}(T[1/p]))$, that is, the right hand side has an integral structure in the étale case.

The following is the main theorem of this paper, which can be regarded as an extension of the studies in [Nak14] and [Nak17a]. It roughly states that, for a general de Rham $(\varphi,\Gamma)$-module $M$ over $\mathcal{R}_{L}$ and the $p$-adic differential equation $\mathbf{N}_{{\operatorname{rig}}}(M)$ attached to $M$, the differences of $\varepsilon_{L}(M(\delta))$ and $\varepsilon_{L}(\mathbf{N}_{{\operatorname{rig}}}(M)(\delta))$ for the de Rham characters $\delta$ of $\Gamma$ are interpolated by the generalized Perrin-Riou map in [Nak14].

We remark that our theorem can be regarded as a refined interpolation formula for Bloch-Kato morphisms. The isomorphism $\mathrm{Exp}(M)$ is obtained by the generalized Perrin-Riou’s big exponential map

of [Nak14] for de Rham $(\varphi,\Gamma)$-module $M$, in conjunction with one of the main results, theorem $\delta(D)$. The big exponential maps are first introduced by Perrin-Riou [Per94] for crystalline representations and used essentially in her study of $p$-adic $L$-functions, and then generalized to de Rham representations [Col98] and to de Rham $(\varphi,\Gamma)$-modules [Nak14]. Their key feature is that they interpolate the Bloch-Kato’s morphisms of twists $\exp_{M(\chi^{k}\tilde{\delta})}$ and $\exp^{*}_{M(\chi^{k}\tilde{\delta})}$ for suitable $k\in\mathbf{Z}$. The theorem can be seen as a refinement of such interpolation formulae; our big exponential map $\mathrm{Exp}(M)$ interpolates, at any twists $\delta=\chi^{k}\tilde{\delta}$ for any $k\in\mathbf{Z}$ and $\tilde{\delta}$, not only the maps $\exp_{M(\delta)}$and $\exp^{*}_{M(\delta)}$ but also another exponential map $\exp_{f,M(\delta)}:\mathbf{D}_{{\operatorname{cris}}}(M(\delta))\to\mathrm{H}_{\varphi,\gamma}^{1}(M(\delta))$, which is closely related with the exceptional zeros for $p$-adic $L$-functions. We note that, even when $M$ comes from a crystalline $p$-adic representation, the map $\exp_{f,M(\delta)}$ is non-zero in general and we can obtain its information via our refined formula.

We also remark a relation of our theorem to the local $\varepsilon$-conjecture itself. The local $\varepsilon$-conjecture for the cyclotomic deformation of a general de Rham $(\varphi,\Gamma)$-module is not proved yet, and only the following special cases are proved.

• •

  The case of rank $1$ Galois representations (i.e. rank $1$ étale $(\varphi,\Gamma)$-modules) is proved by Kato in [Kat93b] (proofs taking account of signs is given in [FK06] briefly and in [Ven13] in detail.)

• •

  The case of crystalline representations is proved by Benois and Berger in [BB08], which is generalized by Loeffler, Venjakob, and Zerbes in [LVZ13], and by Bellovin and Venjakob in [BV19].

• •

  The case of trianguline $(\varphi,\Gamma)$-modules over relative Robba rings, including all semi-stable representations and also the representations associated to finite slope overconvergent modular forms, is proved by the second author in [Nak17a].

• •

  The case of rank $2$ Galois representations is proved by the second author in [Nak17b] in almost all cases and completed by Rodrigues Jacinto [RJ18], by showing its close relation to the $p$-adic local Langlands conjecture for $\mathrm{GL}_{2}({\mathbf{Q}_{p}})$.

By the last assertion of the theorem, we can reduce the local $\varepsilon$-conjecture for the cyclotomic deformation of arbitrary de Rham $(\varphi,\Gamma)$-module $M$ to that of $\mathbf{N}_{{\operatorname{rig}}}(M)$. This reduction seems a useful approach, since $\mathbf{N}_{{\operatorname{rig}}}(M)$ is relatively simple (all of its Hodge-Tate weights are zero) and also has an additional structure of a $p$-adic differential equation with a Frobenius structure so that we can utilize the theory of $p$-adic differential equations. We note that such a reduction is implicitly used to prove the trianguline case, and this theorem is stated as a conjecture [Nak17a, Remark 4.15]; see also Remark 4.2.2.

The structure of the paper is as follows. In section 2, we recall definitions about $(\varphi,\Gamma)$-modules over Robba rings and prove the key lemma Lemma 2.2.5 on a relation of Bloch-Kato’s morphisms and distributions. In section 3, we recall (a special case of) the local $\varepsilon$-conjecture for $(\varphi,\Gamma)$-modules studied in [Nak17a], introduce the $p$-adic differential equation $\mathbf{N}_{{\operatorname{rig}}}(M)$ for a de Rham $(\varphi,\Gamma)$-module $M$, and construct our big exponential map $\mathrm{Exp}(M):\Delta^{\operatorname{{Iw}}}_{L}(\mathbf{N}_{{\operatorname{rig}}}(M))\xrightarrow{\sim}\Delta^{\operatorname{{Iw}}}_{L}(M)$; it is induced by distribution, and the construction depends heavily on [Nak14]. In section 4, we state our main theorem and prove it, by introducing the notion of genericity, deducing the proof of the general case to the case of generic, and proving the generic case by applying the key lemma.

Notation. Let $p$ be a prime number. We fix the algebraic closure $\overline{\mathbf{Q}}_{p}$ of the $p$-adic number field $\mathbf{Q}_{p}$. Let $L$ be a finite extension of $\mathbf{Q}_{p}$. Let $\mu_{p^{\infty}}$ denote the group of $p$-power roots of unity in $\overline{\mathbf{Q}}_{p}$. We fix primitive $p^{n}$-th roots of unity $\zeta_{p^{n}}\in\mu_{p^{\infty}}$ such that $\zeta^{p}_{p^{n+1}}=\zeta_{p^{n}}$ for any $n\geqslant\mathbf{Z}_{\geqslant 1}.$ The set $\Gamma=\mathrm{Gal}(\mathbf{Q}_{p}(\mu_{p^{\infty}})/{\mathbf{Q}_{p}})$. Let $\Delta\subseteq\Gamma$ be the $p$-torsion subgroup of $\Gamma$ and put $p_{\Delta}=\frac{1}{|\Delta|}\sum_{\sigma\in\Delta}\sigma$. We fix an element $\gamma\in\Gamma$ whose image in $\Gamma/\Delta$ is a topological generator. The cyclotomic character on $\Gamma$ is denoted by $\chi:\Gamma\xrightarrow{\sim}\mathbf{Z}_{p}^{\times}$, which is characterized by $\gamma(\zeta)=\zeta^{\chi(\gamma)}$ for all $\zeta\in\mu_{p^{\infty}}$ and $\gamma\in\Gamma$. For a ring $R$, the objects of the category of graded invertible $R$-modules are written as the pairs $(\mathcal{L},r)$ of an invertible $R$-module $\mathcal{L}$ and a continuous function $r:\mathrm{Spec}(R)\to\mathbf{Z}$, and the product $\boxtimes$ is defined by $(\mathcal{L}_{1},r_{1})\boxtimes(\mathcal{L}_{2},r_{2}):=(\mathcal{L}_{1}\otimes_{R}\mathcal{L}_{2},r_{1}+r_{2})$. We put $1_{R}\coloneqq(R,0)$.

In this section, we first recall the definition of $(\varphi,\Gamma)$-modules over Robba rings, their cohomologies, and some notions of $p$-adic Hodge theory. Then, we study several kinds of morphisms defined by a distribution. Theorem 2.2.5 is the key result, which describes a relation between such morphisms and Bloch-Kato’s morphisms.