## Academic Year 2020/2021 - Number Theory Seminars

The Algebra and Number Theory group of the University of Luxembourg hosts three seminars.

• The Luxembourg Number Theory Seminar hosts invited speakers; the seminar will partly be held online.
• In the Research Seminar the group members study a topic together; during term time the seminar takes place weekly.
• In the Work in Progress Seminar the group discusses its work in progress; during term time the seminar alternates with the Number Theory Seminar.

#### Luxembourg Number Theory Seminar

This seminar typically meets on Tuesdays at 14:00. You will find below a collection of abstracts.

 Date (Room) Speaker Title 13/10/2020, 14:00 (MSA 3.220) Gautier Ponsinet (Max Planck Institute for Mathematics) Universal norms of p-adic Galois representations and the Fargues-Fontaine curve 27/10/2020, 15:00 (online) Paul Pollack (University of Georgia) Thoughts on the order of a mod p 03/11/2020, 14:00 (online) Nikita Karpenko (University of Alberta) An ultimate proof of Hoffmann-Totaro's conjecture 01/12/2020, 14:00 (online) Ariel Weiss (Einstein Institute of Mathematics) Lafforgue pseudocharacters and the construction of Galois representations 14/12/2020, all day (online) Luxembourg Number Theory Day 2020 04/02/2021, 14:00 (online) Carlo Pagano (Max Planck Institute for Mathematics) Distribution of fundamental units of varying real quadratic fields at a fixed place and Arakelov ray class groups 23/03/2021, 14:00 (online) Christian Johansson (Chalmers/GU) On the Calegari--Emerton conjectures for abelian type Shimura varieties 13/04/2021, 14:00 (online) Riccardo Pengo (ÉNS Lyon) Entanglement in the family of division fields of a CM elliptic curve 20/04/2021, 16:00 (online) Giovanni Rosso (Concordia university) Overconvergent Eichler--Shimura morphism for families of Siegel modular forms 27/04/2021, 14:00 (online) Ian Kiming (Københavns Universitet) Eisenstein series and overconvergence 04/05/2021, 14:00 (online) Dominik Bullach (King's College London) The equivariant Tamagawa Number Conjecture for abelian extensions of imaginary quadratic fields 11/05/2021, 14:00 (online) Peter Gräf (Universität Heidelberg) A residue map and a Poisson kernel for GL3 over a local field 18/05/2021, 14:00 (online) Matteo Verzobio (Università di Pisa) A recurrence relation for elliptic divisibility sequences 29/06/2021, 14:00 (online) Shaunak Deo (Indian Institute of Science) Density of modular points in pseudo-deformation rings

#### Work in Progress Seminar

This seminar alternates with the Luxembourg Number Theory Seminar. Each session lasts 60 to 90 minutes. You will find below a collection of abstracts.

 Date (Room) Speaker Title 29/09/2020, 14:00 (MSA 3.100) Alexandre Maksoud On Iwasawa theory and Bloch-Kato conjecture for Artin motives 06/10/2020, 13:30 (MSA 3.220) Daniel Berhanu Mamo Eisenstein series and newform theory 20/10/2020, 14:00 (MSA 3.220) Arturo Jaramillo/Xiaochuan Yang Selberg's theorem via Stein's method 27/10/2020, 13:45 (online) Sebastiano Tronto Kummer theory for algebraic groups 10/11/2020, 14:00 (online) Arturo Jaramillo A new probabilistic approach to Erdos-Kac theorem 17/11/2020, 14:00 (online) Andrea Conti Trianguline representations and functoriality 24/11/2020, 14:00 (online) Flavio Perissinotto Kummer theory for products of 1-dimensional tori 08/12/2020, 14:00 (online) Pietro Sgobba On the order of the reductions of algebraic numbers 23/02/2021, 14:00 (MNO 1.010) Alexandre Maksoud Generalized p-adic Stark conjectures and Iwasawa main conjectures for Artin motives 02/03/2021, 14:00 (MNO 1.010) Alexandre Maksoud On the Iwasawa-theoretic conjectures for Rubin-Stark elements 09/03/2021, 14:00 (MSA 2.220) Lassina Dembélé TBA (postponed) 16/03/2021, 14:00 (MSA 2.220) Lassina Dembélé Revisiting the modularity of the abelian surfaces of conductor 277 30/03/2021, 14:00 (online) Lassina Dembélé Revisiting the modularity of the abelian surfaces of conductor 277 - 2 15/06/2021, 14:00 (online) Luca Notarnicola Some questions related to Edwards curves

#### Working Group: Spectral halo (Summer semester 2021)

The seminar will take place on Tuesdays from 10:30 to 12:30 in room MNO 1.010. You can find here a tentative schedule.

 Date (Room) Speaker Title 16/02/2021, 10:30 (MNO 1.010) Andrea Conti Overview 23/02/2021, 10:30 (MNO 1.010) Bryan Advocaat Hida theory 16/03/2021, 10:30 (MNO 1.010) Emiliano Torti Hida theory 2 23/03/2021, 10:30 (online) Alfio Fabio La Rosa Reminders on adic spaces 30/03/2021, 10:30 (online) Alfio Fabio La Rosa Reminders on adic spaces 2 13/04/2021, 10:30 (MNO 1.010) Alexandre Maksoud The weight space 04/05/2021, 10:30 (MNO 1.010) Lassina Dembélé Formal modular curves and partial Igusa towers 25/05/2021, 10:30 (MNO 1.010) Andrea Conti Formal modular curves and partial Igusa towers 2 15/06/2021, 10:30 (MNO 1.010) Andrea Conti TBA 29/06/2021, 10:30 (MNO 1.010) Alexandre Maksoud TBA

#### Working Group: Adic spaces (Winter semester 2020)

You can find here a tentative schedule.

 Date (Room) Speaker Title 29/09/2020, 10:30 (MSA 3.100) Andrea Conti Introduction 06/10/2020, 10:30 (MSA 3.220) Bryan Advocaat Valuations 13/10/2020, 10:30 (MSA 3.220) Alexandre Maksoud Spectral and sober spaces 20/10/2020, 10:30 (MSA 3.220) Daniel Berhanu Mamo Valuation spectra 27/10/2020, 10:30 (online) Alexandre Maksoud Non-archimedean rings 03/11/2020, 10:30 (online) Emiliano Torti f-adic rings and Tate rings 10/11/2020, 10:30 (online) Andrea Conti Adic spectra of affinoid rings, I 17/11/2020, 10:30 (online) Bryan Advocaat Adic spectra of affinoid rings, II 24/11/2020, 10:30 (online) Alexandre Maksoud Adic spaces, I 01/12/2020, 10:30 (online) Andrea Conti Adic spaces, II

#### Collection of abstracts

Alexandre Maksoud (uni.lu) On Iwasawa theory and Bloch-Kato conjecture for Artin motives

Inspired by the works of Perrin-Riou and of Benois we formulate a new cyclotomic Iwasawa Main Conjecture (IMC) for Artin motives, as well as an Exceptional Zeros Conjecture in this context. When the Artin representation is monomial, we show that our conjectures follow from the higher rank cyclotomic IMC recently introduced by Burns, Kurihara and Sano, together with the Iwasawa-theoretic Mazur-Rubin-Sano conjecture. We highlight some potential applications to a better understanding of special values of Artin L-functions at $s=0$, and to a conjecture on iterated p-adic integrals of Darmon-Lauder-Rotger.

Daniel Berhanu Mamo (uni.lu) Eisenstein series and newform theory

We set up a variant of strong multiplicity one theorems for Katz modular forms which admit a reducible mod $p$ Galois representation. An example that illustrates the main theorem will be presented.

Gautier Ponsinet (Max Planck Institute for Mathematics) Universal norms of $p$-adic Galois representations and the Fargues-Fontaine curve

In 1996, Coates and Greenberg computed explicitly the module of universal norms for abelian varieties in perfectoid field extensions. The computation of this module is essential to Iwasawa theory, notably to prove "control theorems" for Selmer groups generalising Mazur's foundational work on the Iwasawa theory of abelian varieties over $\mathbb{Z}_p$-extensions. Coates and Greenberg then raised the natural question on possible generalisations of their result to general motives. In this talk, I will present a new approach to this question relying on the classification of vector bundles over the Fargues-Fontaine curve, which enables to answer Coates and Greenberg's question affirmatively in new cases.

Arturo Jaramillo and Xiaochuan Yang (uni.lu) Selberg's theorem via Stein's method

Sebastiano Tronto (uni.lu) Kummer theory for algebraic groups

If G is a commutative and connected algebraic group over a number field K and A is a finitely generated and torsion-free subgroup of G(K), we can consider the set n^{-1}A={P in G(Kbar) | nP in A} of n-division points of A in an algebraic closure Kbar of K, for any positive integer n. The field extension K(n^{-1}A) of K generated by these points is Galois and it contains the n-torsion field K(G[n]) of G; we are interested in studying, among other things, its degree over this torsion field. This kind of extensions have been studied for example by Ribet [1], but the methods found in the literature are usually non-effective. In this talk I will present recent effective results of myself and Lombardo in the case of non-CM elliptic curves [2]. I will also outline a general framework, developped in [3], to reduce the study of these extensions to that of some arithmetic properties of A and certain properties of the Galois representations attached to G.

[1] Kennet A. Ribet, Kummer theory on extensions of abelian varieties by tori, Duke Mathematical Journal, 1979.
[2] Davide Lombardo and S. T., Explicit Kummer theory for elliptic curves, ArXiv preprint, 2019.
[3] S. T., Radical entanglement for elliptic curves, ArXiv preprint, 2020.

Paul Pollack (University of Georgia) Thoughts on the order of a mod p

I will discuss some old problems concerning the distribution of the orders of a modulo p, for a fixed integer a and varying primes p. The emphasis will be on connections between these problems and recent work of the speaker with K. Agrawal (UGA) --- on finite sets of integers at least one of which usually has a large order mod p --- and Z. Engberg (Wasatch Academy, UT) --- on arithmetic properties of Mersenne numbers.

Nikita Karpenko (University of Alberta) An ultimate proof of Hoffmann-Totaro's conjecture

We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic not 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the speaker in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.

Andrea Conti (uni.lu) Trianguline representations and functoriality

I will introduce the notion of triangulinity for a p-adic representation of the absolute Galois group of a p-adic field, explaining how having this property at p is linked to modularity for a global Galois representation. As an application of this theory, I will present some known results and work in progress on how to prove that an n-dimensional global Galois representation is modular if and only if it is modular after composition with an algebraic representation of GL_n of dimension at least 2.

Flavio Perissinotto (uni.lu) Kummer theory for products of 1-dimensional tori

Classical Kummer theory concerns the study of cyclotomic-Kummer extensions, which can be interpreted as torsion and division fields for the multiplicative group. It is natural to address the same problems for general 1-dimensional tori and for products of 1-dimensional tori. In this talk I will explain how to compute the degree of division fields for products of 1-dimensional tori defined over a number field. The strategy is reducing first to a single 1-dimensional torus and then to the case of the multiplicative group, which amounts to understanding whether the splitting field of the torus is contained or not in the given division field.

Ariel Weiss (Einstein Institute of Mathematics) Lafforgue pseudocharacters and the construction of Galois representations

A key goal of the Langlands program is to attach Galois representations to automorphic representations. In general, there are two methods to construct these representations. The first, and the most effective, is to extract the Galois representation from the étale cohomology of a suitable Shimura variety. However, most Galois representations cannot be constructed in this way. The second, more general method is to construct the Galois representation, via its corresponding pseudocharacter, as a p-adic limit of Galois representations constructed using the first method.

In this talk, I will demonstrate how the second construction can be refined by using V. Lafforgue's G-pseudocharacters in place of classical pseudocharacters. As an application, I will prove that the Galois representations attached to certain irregular automorphic representations of U(a,b) are odd, generalising a result of Bellaïche-Chenevier in the regular case. This work is joint with Tobias Berger.

Pietro Sgobba (uni.lu) On the order of the reductions of algebraic numbers

Let K be a number field, and let G be a torsion-free and finitely generated subgroup of K*. We consider the order of the reduction (G mod p) for all but finitely many primes p of K. Assuming that K is normal over Q and without relying on GRH, we prove an asymptotic formula for the number of primes p for which the order of (G mod p) is divisible by a fixed integer. This result is a generalization of Pappalardi's work concerning the case K=Q.

Carlo Pagano (Max Planck Institute for Mathematics) Distribution of fundamental units of varying real quadratic fields at a fixed place and Arakelov ray class groups

The motivating question of this talk is the behavior of the fundamental units of varying real quadratic fields when reduced modulo a fixed integer. I will explain how this question is linked to the formulation of a Cohen-Lenstra heuristic for ray class groups of real quadratic fields. To this aim I will discuss first the case of imaginary quadratic fields and then, by introducing the notion of Arakelov ray class groups, the case of real quadratic fields. Along the way I will explain the new developments on Cohen-Lenstra at p=2, and some first results on the motivating question. The works discussed are joint with Alex Bartel, Peter Koymans and Efthymios Sofos.

Alexandre Maksoud (uni.lu) Generalized p-adic Stark conjectures and Iwasawa main conjectures for Artin motives

We will state a precise conjecture describing a generator of the Selmer group attached to Artin motives that are unramified at an odd prime p. This conjecture generalizes various classical conjectures in Iwasawa theory. We will then see how Coleman's theory applies to the study of Selmer groups. If time permits we will discuss potential applications to the Tamagawa Number Conjecture (aka. the Bloch-Kato Conjecture).

Alexandre Maksoud (uni.lu) On the Iwasawa-theoretic conjectures for Rubin-Stark elements

Burns, Kurihara and Sano recently proposed a strategy to tackle the equivariant Tamagawa Number Conjecture via the proof of Iwasawa-theoretic conjectures on Rubin-Stark elements. The aim of this talk is to introduce their work and to show the equivalence between their conjectures and our conjecture in the case where the Artin motive is induced from a character.

Lassina Dembélé (uni.lu) Revisiting the modularity of the abelian surfaces of conductor 277

The modularity (or paramodularity) of the abelian surfaces of conductor 277 was proved by a team of six people: Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaria, John Voight and David Yuen. They did so by using the so called Faltings-Serre method. This was the first known case of the paramodularity conjecture. In this work in progress, I will discuss how to (re-)prove the modularity of these surfaces by directly applying deformation theory. This could be seen an explicit approach to deformation theory.

Christian Johansson (Chalmers/GU) On the Calegari--Emerton conjectures for abelian type Shimura varieties

Emerton's completed cohomology gives, at present, the most general notion of a space of p-adic automorphic forms. Important properties of completed cohomology, such as its 'size', is predicted by a conjecture of Calegari and Emerton, which may be viewed as a non-abelian generalization of the Leopoldt conjecture. I will discuss the proof many new cases of this conjecture, using a mixture of techniques from p-adic and real geometry. This is joint work with David Hansen.

Riccardo Pengo (ÉNS Lyon) Entanglement in the family of division fields of a CM elliptic curve

Division fields associated to an algebraic group defined over a number field, which are the extensions generated by its torsion points, have been the subject of a great amount of research, at least since the times of Kronecker and Weber. For elliptic curves without complex multiplication, Serre's open image theorem shows that the division fields associated to torsion points whose order is a prime power are "as big as possible" and pairwise linearly disjoint, if one removes a finite set of primes. Explicit analogues of this result have recently been featured in the work of Campagna-Stevenhagen and Lombardo-Tronto. In this talk, based on joint work with Francesco Campagna (arXiv:2006.00883), I will present an analogue of these results for elliptic curves with complex multiplication. Moreover, I will present a necessary condition to have entanglement in the family of division fields, which is always satisfied for elliptic curves defined over the rationals. In this last case, I will describe in detail the entanglement in the family of division fields.

Giovanni Rosso (Concordia University) Overconvergent Eichler--Shimura morphism for families of Siegel modular forms

Classical results of Eichler and Shimura decompose the cohomology of certain local systems on the modular curve in terms of holomorphic and anti-holomorphic modular forms. A similar result has been proved by Faltings' for the etale cohomology of the modular curve and Falting's result has been partly generalised to Coleman families by Andreatta--Iovita--Stevens. In this talk, based on joint work with Hansheng Diao and Ju-Feng Wu, I will explain how one constructs a morphism from the overconvergent cohomology of GSp_2g to the space of families of Siegel modular forms. This can be seen as a first step in an Eichler--Shimura decomposition for overconvergent cohomology and involves a new definition of the sheaf of overconvergent Siegel modular forms using the Hodge--Tate map at infinite level.

Ian Kiming (Københavns Universitet) Eisenstein series and overconvergence

I will talk about recent work with Nadim Rustom on the rate of overconvergence of certain modular functions that arise from classical Eisenstein series. Knowledge of this kind can be put to use to understand the finer structure of the Coleman-Mazur eigencurve, in concrete terms, this knowledge has implications for the study of Coleman families passing through classical forms. Among other things we prove a theorem that is a direct generalization of a theorem of Coleman-Wan that was instrumental in Wan's celebrated work regarding the Coleman-Mazur conjecture. Other of our theorems are general versions for p\ge 5 of statements that were proved for p=2 and p=3 by Buzzard-Kilford and Roe. I will start the talk very lightly, talking about overconvergent modular forms and "Katz expansions" before moving on to explain the problems and our results.

Dominik Bullach (King's College London) The equivariant Tamagawa Number Conjecture for abelian extensions of imaginary quadratic fields

The equivariant Tamagawa Number Conjecture is a far-reaching equivariant refinement of the analytic class number formula. I will report on joint work with Martin Hofer that establishes new cases of this conjecture for abelian extensions of imaginary quadratic fields.

Peter Gräf (Universität Heidelberg) A residue map and a Poisson kernel for GL3 over a local field

I will discuss the relationship between certain holomorphic discrete series representations on the Drinfeld period domain and spaces of harmonic cocycles on the Bruhat-Tits building for the group GL3 over a local field of any characteristic. The main novelty is that we allow non-trivial coefficients in a situation beyond the well-known theory for GL2, which extends work of Schneider and Teitelbaum. I will explain how to construct a residue map and a Poisson kernel in this situation. Moreover, I will explain how the existence of the relevant boundary distributions follows from a conjectural non-criticality statement for certain (generalized) automorphic forms. If time permits, I will discuss an application in function field arithmetic, namely to Drinfeld modular forms.

Matteo Verzobio (Università di Pisa) A recurrence relation for elliptic divisibility sequences

Let $N$ be an integer, $p$ be an odd prime and $\bar\rho_0$ be a continuous, odd and reducible $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of characteristic $p$. We will prove that the maximal reduced quotient of the universal deformation ring of the pseudo-representation corresponding to $\bar\rho_0$ (pseudo-deformation ring) is isomorphic to the local component of the big $p$-adic Hecke algebra of level N corresponding to $\bar\rho_0$, if a certain global Galois cohomology group has dimension $1$. This partially extends the results of B\"{o}ckle to the case of residually reducible representations. As an application of our methods and results, we will prove a level-raising result for newforms lifting $\bar\rho_0$ in the spirit of Diamond--Taylor which gives a partial answer to a conjecture of Billerey--Menares.