Abstracts

Kenichi Bannai: Syntomic realization of the polylogarithm for the algebraic torus associated to a totally real field

The Beilinson conjecture for algebraic varieties and more general motives defined over algebraic number fields is a conjecture relating the non-critical values of Hasse-Weil $𝐿$-functions to regulator of certain elements in motivic cohomology. For the case of the Dirichlet L-function for the rational number field $\mathbb{Q}$, the special value is related to the regulator of the cyclotomic element -- which may be constructed as the specialization to torsion points of the motivic polylogarithm in this case. In order to prospect $p$-adic analogues of such results in the case of totally real fields, we consider the syntomic realization of the polylogarithm for the algebraic torus associated to a totally real field, and its relation to special values of $p$-adic $L$-functions associated to Hecke characters of the totally real field. This is a work in process, and includes results of work with H. Bekki, K. Hagihara, T. Oshita, K. Yamada, and S. Yamamoto.

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Laurent Berger: Integer-valued polynomials and p-adic Fourier theory

Let K be a finite extension of Qp and let o_K be its ring of integers. The theory of Lubin-Tate formal groups allows us to construct many integer-valued polynomials on o_K. To what extent do these polynomials generate all integer-valued polynomials on o_K? I will explain some results of de Shalit and Iceland, as well as of Ardakov, Sprang and myself on this question. I will also explain the motivation (p-adic Fourier theory) for considering this problem.

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Massimo Bertolini: p-adic L-functions and rational points

This talk will present partly conjectural formulas relating rational points to certain triple product $p$-adic $L$-functions. It is based on work in progress with F. Andreatta, M. Seveso and R. Venerucci.

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Pierre Colmez: Completed cohomology and group cohomology of arithmetic groups

Using Shapiro's lemma one can express Iwasawa cohomology groups as Galois cohomology of the base field with values in a big module with a lot of structures; these extra structures makes it possible to define a multitude of operators on these Iwasawa modules. In this vein, I will explain that Emerton's completed Cohomology has a natural definition as the group cohomology of arithmetic groups with values in spaces of functions on adelic groups.

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Christopher Deninger: Aspects of zeta functions

We will give an introduction to arithmetic zeta functions and discuss some of the deep conjectures about them that have been fascinating mathematicians for more than a century. We also discuss the behaviour of Dedekind zeta functions at $s=1/2$ which is outside the standard conjectural framework. Cohomological considerations and a conjecture by Serre lead to a prediction that can be proved unconditionally.

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Annette Huber: Species and dimensions of periods spaces

Prima facie, the period conjecture makes a qualitative prediction about relations between period numbers of motives over number fields: all relations are induced from a small list of obvious relations. However, the structure theory of the category of motives allows us to turn this into (conjectural) formulas for the dimension of the space of periods of a single motive in terms of the species of the category generated by the object. The results are unconditional and complete in the case of 1-motives. They also shed new light on the known upper bounds for spaces of multiple zeta values. This is joint work with Martin Kalck.

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Shinichi Kobayashi: A local sign decomposition for symplectic self-dual Galois representations

We present a new structure on the first Galois cohomology of families of symplectic self-dual p-adic representations of $G_{\mathbb{Q}_p}$ of rank two. This is a functorial decomposition into free rank one Lagrangian submodules encoding Bloch-Kato subgroups and epsilon factors, mirroring an underlying symplectic structure. This local sign decomposition has local as well as global applications, including compatibility of the Mazur-Rubin arithmetic local constants and epsilon factors, and new cases of the parity conjecture. It also leads to a formulation and proof of an analogue of Rubin's conjecture over ramified quadratic extensions of $\mathbb{Q}_p$, which initiates an integral Iwasawa theory for CM elliptic curves at primes of additive reduction. (Joint with A. Burungale, K. Nakamura, and K. Ota.)

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David Loeffler: Ultraprimes, Kolyvagin systems and local conditions

(Joint work with Sarah Livia Zerbes.) The machinery of Euler and Kolyvagin systems gives a very powerful tool for studying Selmer groups of p-adic Galois representations. Any non-trivial Euler system will give a bound for the "strict" Selmer group (with zero local condition at p). However, for applications one would like to bound Selmer groups with more general local conditions, assuming suitable local restrictions on the Euler system classes. I will describe a simple criterion for which local conditions are compatible with the Kolyvagin-system machinery, using an insight due to Sweeting that one can interpret Kolyvagin's arguments using ultrafilters. I will then explain some cases in which this criterion in satisfied, and applications to Pottharst's "trianguline" local conditions.

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Ben Moonen: Bézout's theorem for abelian varieties

Abstract: I'll report on joint work with Olivier Debarre. The main result is that if $A$ is an absolutely simple abelian variety over some field and $X_1,..., X_t$ are subvarieties of $A$, then the dimension of their sum $X_1 + ... + X_t$ equals the minimum of $\dim(A)$ and $\sum \dim(X_i)$. In characteristic $0$, there is a simple geometric proof for this, but that argument breaks down in characteristic $p$. Instead, we prove this result as a consequence of a theorem on perverse sheaves, building upon work of Krämer and Weissauer.

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Wiesława Nizioł: p-adic pro-etale cohomology of rigid analytic varieties

The last ten years have seen a a major progress in understanding p-adic pro-etale cohomology of rigid analytic varieties. I will review briefly key theorems, computations, and examples with a bias towards applications in p-adic Langlands Program.

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Sujatha Ramdorai: Iwasawa theory of elliptic curves in a family of Z_p extensions

Suppose that $p$ is an odd prime. Let $K$ be a number field and $K_\infty$ the composite of all linearly disjoint $\mathbb{Z}_p$-extensions of $K. $ Suppose $E$ is an elliptic curve defined over $K.$ This talk will focus on the invariance of certain properties of the dual Selmer group of elliptic curves in a family of $Z_p$ extensions. We also study the Galois cohomology of the Selmer groups over the anticyclotomic extension, vastly simplifying results in existing literature. The results are in two categories, one part is joint work with Ahmed Matar and Sören Kleine, the other is joint with Nguyen Tam.

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Damian Rössler: On a possible generalisation of a formula of Colmez

We shall discuss a possible generalisation of the formula of Colmez, which compute the Faltings height of abelian varieties with complex multiplication. We will also show how classical results of Siegel give a proof of this generalisation in the case of CM elliptic curves, where the formula of Colmez is equivalent to the Chowla-Selberg formula. Along the way, we obtain a new class invariant, whose minimal polynomial seems to have smaller coefficients than many other class invariants.

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Otmar Venjakob: On Lubin-Tate regulator maps and Kato's explicit reciprocity law

Kato proved his explicit reciprocity law for Lubin-Tate formal groups by means of syntomic cohomology. After the development of Lubin-Tate $(\varphi,\Gamma)$-modules we reproved a special case of it in joint work with Peter Schneider. We then continued to define Lubin-Tate regulator maps for certain crystalline representations and proved a reciprocity formula for them. In this talk we report on joint work with Takamichi Sano in which we firstly extend the interpolation property of the Lubin-Tate regulator map to Artin characters and secondly show a reciprocity law in the sense of Cherbonnier-Colmez. This allows us to provide a new proof of Kato's explicit reciprocity law for Lubin-Tate formal groups in general.

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Shanwen Wang: The $p$-adic Eichler-Shimura map

Using the $C_{\rm dR}$ conjecture for modular forms, we give an explicit description of the $p$-adic Eichler-Shimura map. This talk is based on joint work with Pierre Colmez.

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Sarah Zerbes: Euler systems and Pottharst–Iwasawa main conjectures

(Joint work with David Loeffler.) To formulate a version of the Iwasawa main conjecture for some Galois representation V, one assumes that V satisfies some kind of "ordinarity" condition at p. An influential 2013 paper of Pottharst showed how to define a variant of the Main Conjecture under a much weaker assumption, namely that the Robba-ring (phi, Gamma)-module associated to V have a submodule of some specific rank (not necessarily arising from a subrepresentation). However, while the method of Euler systems has been very successfully applied to prove one inclusion in the Main Conjecture in many "ordinary" settings, this is much more difficult for Pottharst's Selmer groups, since Pottharst's spaces have no natural integral structure. In this talk (and in David's sequel talk), we will explain how the method of Euler systems can be modified to circumvent this difficulty. As an application, I will outline a proof of one inclusion in the Pottharst–Iwasawa main conjecture for Rankin–Selberg convolutions of non-ordinary modular forms.

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