Program
| Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|
| 9:00–9:25 Registration 9:25–9:30 Welcome |
9:00–10:00 Burmester |
9:00–10:00 Mini-course Bullach |
9:00–10:00 Mini-course Bullach |
9:00–10:00 Mini-course Lang |
| 9:30–10:30 Mini-course Bullach |
10:00–10:30 Coffee break |
10:00–10:30 Coffee break |
10:00–10:30 Coffee break |
10:00–10:30 Coffee break |
| 10:30–11:00 Coffee break |
10:30–11:00 Tosi |
10:30–11:00 Kaur |
10:30–11:00 Stern & Zindulka |
10:30–11:30 Mini-course Bullach |
| 11:00–12:00 Mini-course Lang |
11:15–12:15 Mini-course Lang |
11:15–11:45 Kızıldere Mutlu |
11:15–12:15 Mini-course Lang |
11:45–12:15 Pajaziti |
| 12:00–13:30 Lunch break |
12:15–13:45 Lunch break |
12:00–12:30 Thøgersen |
12:15–13:45 Lunch break |
|
| 13:30–14:30 Ponsinet |
13:45–14:15 Kufner |
Free afternoon | 13:45–14:15 Qi |
|
| 14:45–15:15 Marannino |
14:30–15:00 Gamarra Segovia |
14:30–15:00 Honnor |
||
| 15:15–15:45 Coffee break |
15:00–16:00 Poster session + coffee |
15:00–15:30 Coffee break |
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| 15:45–16:45 Rosu |
15:30–16:00 Angurel Andres |
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| 17:00–17:30 Zerman |
16:00–17:00 Rivero |
16:00–16:30 Angdinata |
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| 17:15–17:45 Steingart |
17:00–17:30 Sheth |
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| 18:30 Dinner |
The schedule is subject to change.
Mini-courses
Dominik Bullach (University College London): On leading terms of $L$-series and Euler systems
The relationship between leading terms of $L$-series and arithmetic is a central theme in modern algebraic number theory, with much current research undertaken on this topic. Ever since their introduction, Euler systems have played an important role in the story and, more importantly, in progress towards our understanding of it. In this series of talks, I will introduce the core principles in the general theory of Euler systems and highlight some recent progress in the field. To do this, I will focus on the accessible example of Dirichlet $L$-series attached to real abelian fields and the Euler system of cyclotomic units.
Jaclyn Lang (Temple University): An Introduction to Eisenstein Congruences and Arithmetic Applications
Eisenstein series are one of the first and simplest examples of modular forms that one encounters. Nevertheless, understanding the congruences admitted by Eisenstein series has led to remarkable insights about arithmetic in the last 50+ years. In this course we will give a tour of some classic examples where Eisenstein congruences have been used to deduce arithmetic information with an emphasis on key techniques that are useful in many settings. It is impossible to cover all of the significant examples in four lectures, so we shall restrict our attention to congruences between forms of a fixed weight and level. This still leaves a vast landscape that encompasses Mazur’s Eisenstein ideal paper and Ribet’s converse to Herbrand’s theorem.
In Lecture 1 we will give an overview of this landscape including the Eisenstein series and congruences that are relevant to Mazur's and Ribet’s work, a summary of their arithmetic applications, and other theorems inspired by their work.
Lecture 2 will be devoted to describing techniques for establishing the existence of the type of Eisenstein congruences needed for Mazur, Ribet, and other applications.
Lecture 3 contains a brief survey of Ribet’s method of proof for the converse to Herbrand’s theorem, followed by an application of those techniques to class groups of certain number fields due to myself and Wake.
In Lecture 4 we address the question of counting congruences: how many cusp forms of a given weight and level are congruent to a particular Eisenstein series? This question was asked in Mazur’s Eisenstein ideal paper, and in that context it is difficult to say anything (though progress has been made, which we will mention). However, in the context of my work with Wake from Lecture 3, we can answer this question completely in a surprisingly uniform way. Talking about our proof will give me an excuse to demonstrate the power of deformation theoretic techniques and a simple case of the use of Wiles’ numerical criterion.
The lectures will start at a level that should be accessible to anyone having some familiarity with modular forms, say from a first graduate course. For material relating to Ribet’s method, class field theory (and algebraic number theory more generally) will be assumed. The last lecture will have a somewhat higher bar since it will use deformation theory and pseudodeformations. However, the setting will be fairly concrete, and the goal is for everyone to be able to follow the chain of reasoning even if they are not comfortable with these topics.
Research talks
Monday
Gautier Ponsinet (IHES): On a characterisation of perfectoid fields by Iwasawa theory
Monday
Gautier Ponsinet (IHES): On a characterisation of perfectoid fields by Iwasawa theory
With a $p$-adic representation of the Galois group of a $p$-adic field are associated the Bloch-Kato groups defined via $p$-adic Hodge theory. Iwasawa theory motivates the study of these Bloch-Kato groups over infinite algebraic extensions of the field of $p$-adic numbers.
Over perfectoid fields, several results (by Coates-Greenberg, Perrin-Riou, Berger, P. …) state that the Bloch-Kato groups admit a simple description.
In this talk, we will present a reciprocal statement: the structure of the Bloch-Kato groups associated with certain crystalline representations characterises the algebraic extensions of the field of $p$-adic numbers whose completion are perfectoid fields. In particular, we will recover, via a different method, results by Coates and Greenberg for abelian varieties, and by Bondarko for $p$-divisible groups.
Luca Marannino (Universität Duisburg-Essen): Triple product $p$-adic $L$-functions in universal deformation families
Darmon and Rotger attached to a triple of Hida families $(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ a $p$-adic $L$-function which interpolates the central values of triple product $L$-functions corresponding to suitable triples of specializations of $(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$. In the last years, this construction has been improved and generalized in various directions and has led to several arithmetic applications. Following ideas of Loeffler, in this talk we discuss what happens when $\boldsymbol{g}$ and $\boldsymbol{h}$ are allowed to be universal deformation families (without any ordinarity or finite slope condition at the prime $p$). Time permitting, we will also sketch some applications.
Eugenia Rosu (Leiden University): Special cycles on compactifications of Shimura varieties
Special cycles have been at the heart of the Kudla program in arithmetic geometry, their intersections being related to L-functions and Eisenstein series. In joint work with Bruinier and Zemel, we construct special cycles of dimension 0 on toroidal compactifications of Shimura varieties. We show the modularity of the generating series that have these special cycles as coefficients, generalizing the open case.
Francesco Zerman (FernUni Schweiz): Big Heegner points in towers of Shimura curves
Let $p>3$ be a prime and $N^+$, $N^-$ be coprime positive integers. Let also $K$ be an imaginary quadratic field such that all the primes dividing $N^+$ (resp. $N^-$) are split (resp. inert) in $K$. If $N^-$ is squarefree, using the theory of Eichler orders of level $N^+p^n$ in the quaternion algebra of discriminant $N^-$ over $\mathbb{Q}$, Longo and Vigni were able to build an anticyclotomic Euler system of Heegner classes for the Galois representation attached to a Hida family of modular forms of tame level $N=N^+N^-$. In this talk, we show how one can adapt their method to deal with the case when $N^-$ is not squarefree, studying the arithmetic of Pizer orders and their associated Shimura curves.
Tuesday
Annika Burmester (Bielefeld University): Towards a Hopf algebra structure for multiple q-zeta values
Multiple q-zeta values are specific q-series, which degenerate to multiple zeta values under the limit q to 1 and have a deep connection to quasi-modular forms and multiple Eisenstein series. We present a model for multiple q-zeta values, which satisfies particularly simple and conjecturally graded relations. This gives the starting point for revealing a Hopf algebra structure on multiple q-zeta values, inspired by Racinet’s work on multiple zeta values. We outline the steps needed to establish this Hopf algebra structure and discuss the current progress.
Riccardo Tosi (Universität Duisburg-Essen): A geometric approach to irrationality proofs for zeta values
The values of the Riemann zeta function at odd positive integers greater than 1 are conjectured to be transcendental, yet even their irrationality remains a mostly open question. Recently, Brown has drawn some connections with period integrals over the moduli space of smooth projective curves of genus zero with marked points, which have given new inputs to irrationality proofs for zeta values. In this talk, I will discuss Brown’s result and sketch the main ideas of the theory. I will then present some work in progress regarding extensions of these methods to more general varieties and thus to irrationality proofs for a broader range of numbers, mainly concerning multiple polylogarithmic values.
Han-Ung Kufner (Universität Regensburg): Deligne's conjecture on the critical values of Hecke $L$-functions
Deligne's conjecture expresses the critical values of an $L$-function of a motive $M$ in terms of a certain period $c^+(M)$. We discuss a proof of this conjecture in the case of $L$-functions attached to arbitrary Hecke characters $\chi$ generalizing a result of Blasius for Hecke characters of CM-fields. A key insight of Blasius' proof is an alternative expression of the $c^+$-period in terms of the so-called reflex motive attached to $\chi$. In our approach, we make use of the recently constructed Eisenstein-Kronecker classes of Kings-Sprang, which allow for a cohomological interpretation of the value $L(\chi,0)$, even when $\chi$ is defined over an arbitrary totally imaginary number field. By means of an alternative construction, we are able to naturally regard these classes as de Rham classes attached to the reflex motive. Investigating Blasius' approach further from this new context finally yields the desired relationship between $L$-value and $c^+$-period.
Guillermo Gamarra Segovia (University of Duisburg-Essen): Construction of Eisenstein - Kronecker classes for families of abelian schemes
Eisenstein-Kronecker classes were constructed by Kings and Sprang as special classes of equivariant cohomology of an abelian scheme with coefficients in the completion of its Poincaré bundle. Via this method, one can obtain a $p$-adic measure interpolating the $p$-adic $L$-values of certain Hecke characters. In this talk, I aim to discuss a generalization of this construction for families of abelian schemes, by instead considering the cohomology of the moduli space of abelian schemes with real multiplication and some level structure. We show that they have an explicit description relating them to classical Eisenstein series, and which extends again results from Kings and Sprang. In the end, this should lead to a new construction of Katz's $p$-adic measure.
Oscar Rivero (Universidade de Santiago de Compostela): Critical Eisenstein series and Euler systems
Euler systems have become one of the most prominent techniques for studying arithmetic problems, such as the Bloch--Kato conjecture and the Iwasawa main conjecture. Additionally, their variation in Coleman families has been a highly relevant area of study. However, this presents certain challenges, particularly around the so-called "critical points".
In this talk, we focus on what occurs at the points corresponding to the critical p-stabilization of an Eisenstein series, illustrating how this leads to a fascinating interaction between seemingly distinct Euler systems. Time permitting, we will also discuss specific instances where these phenomena are more subtle, and where the arithmetic of the adjoint p-adic L-function plays a crucial role.
The contents of the talk are based on joint work with David Loeffler and on an ongoing project with Javier Polo.
Rustam Steingart (Universität Heidelberg): $\varepsilon$-isomorphisms for rank one $(\varphi,\Gamma)$-modules in the Lubin-Tate case
Inspired by the work of Nakamura on $\varepsilon$-isomorphisms for $(\varphi,\Gamma)$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for $L$-analytic Lubin-Tate $(\varphi_L,\Gamma_L)$-modules over (relative) Robba rings for any finite extension $L$ of $\mathbb Q_p$. In contrast to Kato's and Nakamura's setting, our conjecture involves $L$-analytic cohomology instead of continuous cohomology within the generalized Herr complex. This work is joint with Milan Malcic, Otmar Venjakob and Max Witzelsperger.
Wednesday
Sumandeep Kaur (Panjab University): On certain sextic number fields
Computation of discriminant as well as integral basis of an algebraic number field has been one of the most important problems in algebraic number theory. This has attracted the attention of several mathematicians who determined the discriminant and integral basis of various classes of number fields which are defined over the field $\mathbb Q$ of rational numbers by certain types of irreducible polynomials. In this talk, we discuss this problem for the fields $K = \mathbb Q(\theta)$ with $\theta$ a root of an irreducible trinomial $f (x) = x^6 + ax + b$ belonging to $\mathbb Z[x]$. For each prime number $p$, we compute the highest power of $p$ dividing the discriminant of $K$. An explicit $p$- integral basis of $K$ will also be given for each prime $p$. A simple method will be described to obtain an integral basis of $K$ from these $p$-integral bases. In the end, we give some sufficient conditions on $a$, $b$ for which $K$ will be non-monogenic.
Elif Kızıldere Mutlu (Bursa Uludag University): On the solutions of some generalized Lebesgue-Ramanujan-Nagell type equations
Let $d$ and $\delta$ be fixed positive integers. Consider the Diophantine equation $x^2+d^s=\delta y^n$ where $x, y, n$ and $s$ are nonnegative integer unknowns. This equation is usually called the generalized Lebesgue-Ramanujan-Nagell equation. It has a long history and rich content. Recently, a survey paper on the generalized Lebesgue-Ramanujan-Nagell equation has been written by M.-H. Le and G. Soydan, [LS].
Denote by $h=h(-p)$ the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ with $p$ prime. It is well known that $h=1$ for $p\in\{3,7,11,19,43,67,163\}$. Recently, all the solutions of the Diophantine equation $x^2+p^s=4y^n$ with $h=1$ were given by Chakraborty et al. in [Cetal]. In this talk, we consider the Diophantine equation $x^2+p^s=2^ry^n$ in unknown integers $(x,y,s,r,n)$ where $s\ge 0$, $r\geq 3$, $n \geq 3$, $h\in\{1,2,3\}$ and $\gcd(x,y)=1$. Our main tools include the known results from the modularity of Galois representations associated with Frey-Hellegoaurch elliptic curves (i.e. modular approach) [BS], the symplectic method [FK], a Thue-Mahler solver which was improved by Gherga and Siksek [GheSi] and elementary methods of classical algebraic number theory. This work was supported by the Research Fund of Bursa Uludağ University under Project No: FGA-2023-1545. This is a joint work with Gökhan Soydan, [MS].
[BS] M. A. Bennett and C. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad.~J.~Math. 56 (2004) 23–54.
[Cetal] K. Chakraborty, A. Hoque and R. Sharma, Complete Solutions of Certain Lebesgue-Ramanujan-Nagell Type equations, Publ. Math. Debrecen, 97(3-4) (2020), 339–352.
[FK] N. Freitas, A. Kraus, On the symplectic type of isomorphism of the p-torsion of elliptic curves, Mem. Amer. Math. Soc. 277 (2022) 1–104.
[GheSi] A. Gherga and S. Siksek, Efficient resolution of Thue-Mahler equations, arXiv preprint arXiv:2207.14492 2022.
[LS] M. H. Le and G. Soydan, A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation, Surv. Math. Appl., 15 (2020), 473–523.
[MS] E.K. Mutlu and G. Soydan, On the solution of some generalized Lebesgue-Ramanujan-Nagell type equations, Int. Journal of Number Theory (2024), to appear, DOI 10.1142/S1793042124500593
Frederick E. Thøgersen (University of Nottingham): $p$-adic $L$-functions and parabolic methods
Glenn Steven's ideas on constructions of finite slope $p$-adic $L$-functions for modular forms has served as the seminal inspiration behind numerous finite slope $p$-adic $L$-functions attached to more general RACARs $\pi$ of reductive groups $\mathbf{G}/F$ with $F$ a number field. When considering the $p$-adic interpolation, the iwahoric groups $I_{\mathfrak{p}}:=\{g\in\mathbf{G}(\mathcal{O}_{F_{\mathfrak{p}}}): g\text{ }(\text{mod }\mathfrak{p})\in \mathbf{B}(\mathcal{O}_{F_{\mathfrak{p}}}/\mathfrak{p})\}$ where $\mathbf{B}$ a Borel subgroup naturally arise. In work of M. Dimitrov, F. Januszewski, and A. Raghuram, it was shown it is more natural to consider the parahoric groups $J_{\mathfrak{p}}$ where $\mathbf{B}$ is replaced with a certain parabolic $\mathbf{P}$. We intend to explore the impact of this substitution on the constructions of the $p$-adic $L$-functions attached to $\mathbf{GL}_{2n}$ as in the work of D. Barrera Salazar, M. Dimitrov, A. Graham, A. Jorza and C. Williams.
Thursday
David Stern & Mikulas Zindulka (Charles University): Partitions of algebraic numbers
Integer partitions are at the intersection of additive number theory and combinatorics. They attracted the attention of great mathematicians such as Euler, Hardy, and Ramanujan, who discovered many beautiful partition identities. The notion of partition can be extended to a real (quadratic) field $K$, where positive integers are replaced by totally positive integral elements. The properties of the associated partition function $p_K(n)$ are little understood. In the first part of the talk, we will discuss a recurrence formula for $p_K(n)$, its parity, and the following problem: given an integer $ r\in\mathbb{Z}_{\geq 1}$, characterize the real quadratic fields $K$ which contain an element with exactly $r$ partitions. Besides integer partitions, one may consider partitions whose parts belong to a fixed set $S \subset \mathbb{Z}_{\geq 1}$. When $S$ is the set of powers of a fixed integer $m \in \mathbb{Z}_{\geq 2}$, these are the so-called $m$-ary partitions. What happens when we replace $m$ by an algebraic number $\beta$? In the second part of the talk, we will show how to characterize the $\beta$'s such that every $\alpha\in\mathbb{C}$ has finitely many partitions into powers of $\beta$. The first part is based on a joint work of the speakers and the second part on a paper by M. Zindulka and V. Kala.
Peikai Qi (Michigan State University): Iwasawa $\lambda$ invariant and Massey products
How does the class group of the number field change in field extensions? This question is too wild to have a uniform answer, but there are some situations where partial answers are known. I will compare two such situations. First, in Iwasawa theory, instead of considering a single field extension, one considers a tower of fields and estimates the size of the class groups in the tower in terms of some invariants called $\lambda$ and $\mu$. Second, in a paper by Lam-Liu-Sharifi-Wake-Wang, they relate the relative size of Iwasawa modules to values of a "generalized Bockstein map", and further relate these values to Massey products in Galois cohomology in some situations. I will compare these two approaches to give a description of the cyclotomic Iwasawa $\lambda$-invariant of some imaginary quadratic fields and cyclotomic fields in terms of Massey products.
Matthew Honnor (Imperial College London): On the refined 'Birch—Swinnerton-Dyer type' conjectures of Mazur and Tate
In the 1980's Mazur—Tate refined the Birch—Swinnerton-Dyer Conjecture, to give an equivariant statement for a certain group ring element which relates to the twisted Hasse—Weil $L$-series of elliptic curves. In this talk I will explain the conjectures of Mazur—Tate and report on work in progress, joint with Dominik Bullach, in which we prove new results towards these conjectures.
Alberto Angurel Andres (University of Nottingham): Kolyvagin systems and Selmer structures of rank 0
Selmer structures provide a powerful framework for studying a wide range of arithmetic objects, including class groups, elliptic curves and more. The theory of Kolyvagin systems is an important tool in understanding the structure of the associated Selmer groups. However, when the rank of a Selmer structure is 0, Kolyvagin systems do not naturally exist. In this talk, I will describe how to modify a rank 0 structure to construct a Kolyvagin system that describes the structure of its associated Selmer group. Additionally, I will show how this theory relates the arithmetic information of an elliptic curve with its Kurihara numbers, which are analytic quantities defined in terms of the modular symbols of the curve.
David Kurniadi Angdinata (London School of Geometry and Number Theory): Twisted elliptic L-values over global fields
The $L$-function of an abelian variety base changed over an extension of global fields decomposes into a product of twisted $L$-functions. Their central algebraic values encode crucial arithmetic information in the context of refined Birch–Swinnerton-Dyer conjectures, but modern techniques in Iwasawa theory only explore the ideals generated by these values. I will explain a trick that determines their actual values in the simplest case of elliptic curves twisted by cubic Dirichlet characters. If time permits, I will describe what has been known and what can be done in positive characteristic.
Arshay Sheth (University of Warwick): Control Theorems for Hilbert Modular Varieties
The classical results of Iwasawa on class groups of cyclotomic fields, and of Hida on p-adic families of modular forms are prototypical examples of control theorems in number theory. We will first briefly recall these results and then explain our recent work on establishing control theorems for certain Hilbert modular varieties. At the end of the talk, we will briefly explain applications of this result to Euler systems and the Bloch—Kato conjecture in the setting of Hilbert modular forms.
Friday
Antigona Pajaziti (University of Luxembourg & Leiden University): On congruence classes of orders of reductions of elliptic curves
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $E_{\text{ep}} (\mathbb{F}_p )$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. Given an integer $m \geq 2$ and any $a$ modulo $m$, we consider how often the congruence $|E_{\text{ep}} (\mathbb{F}_p )| \equiv a \mod m$ holds. We then exhibit elliptic curves over $\mathbb{Q}(t)$ with trivial torsion for which the orders of reductions of every smooth fiber modulo primes of positive density at least $1/2$ are divisible by a fixed small integer. We show that the greatest common divisor of the integers $|E_{\text{ep}} (\mathbb{F}_p )|$ over all rational primes $p$ cannot exceed $4$. We also show that if the torsion of $E$ grows over a quadratic field $K$, then one may explicitly compute $|E_{\text{ep}} (\mathbb{F}_p )|$ modulo $|E(K)_{\text{tor}}|$. More precisely, we show that there exists an integer $N \geq 2$ such that $|E_{\text{ep}} (\mathbb{F}_p )|$ is determined modulo $|E(K)_{\text{tor}}|$ according to the arithmetic progression modulo $N$ in which $p$ lies. It follows that given any $a$ modulo $|E(K)_{\text{tor}}|$, we can estimate the density of primes $p$ such that the congruence $|E_{\text{ep}} (\mathbb{F}_p )| \equiv a \mod |E(K)_{\text{tor}}|$ occurs. This is joint work with Assoc. Prof. Mohammad Sadek.