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Spring Semester 2025

Date / Time Speaker Title Location
21 February 2025
14:15-15:15 		Francesco Naccarato
ETH Zurich, Switzerland
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Number Theory Seminar

Title On the arithmetic of critical values of polynomials
Speaker, Affiliation Francesco Naccarato, ETH Zurich, Switzerland
Date, Time 21 February 2025, 14:15-15:15
Location HG G 43
Abstract Let d>1 be an integer and let k be an algebraically closed field with char(k)=0 or char(k)>d. Given a polynomial f in k[x], its critical values C_f are the branching points of the induced map f: P^1 —> P^1. When suitably normalized, polynomials of degree d with given critical values can be parametrized in terms of their critical points, the roots of the derivative, yielding a covering T_d: A^{d-1} —> A^{d-1}. We briefly discuss the geometry of these coverings, giving a generalization of a classical result of Arnold. Motivated by this and by some applications, we consider two questions of arithmetical flavour: given a subfield L of k, for which subsets V of k is there a degree d polynomial f in L[x] with C_f=V, and are two such polynomials always related by a linear change of variable? We answer these questions in the case of quartics over any number field and discuss a link to the arithmetic of elliptic curves, as well as (time permitting) the case of higher degrees.
On the arithmetic of critical values of polynomialsread_more
HG G 43
28 February 2025
14:15-15:15 		Dr. Ananyo Kazi
UniDistance Suisse
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Number Theory Seminar

Title Twisted triple product p-adic L-function for finite slope families and a p-adic Gross–Zagier formula
Speaker, Affiliation Dr. Ananyo Kazi, UniDistance Suisse
Date, Time 28 February 2025, 14:15-15:15
Location HG G 43
Abstract In the theory of p-adic L-functions a p-adic Gross–Zagier formula gives interpretation to special values of p-adic L-functions outside the region of interpolation using p-adic integration. Seen as a first step towards "explicit reciprocity laws" they have important applications towards proving various instances of the Bloch-Kato conjecture. We construct a p-adic twisted triple product L-function associated to finite slope families of Hilbert modular forms, assuming p unramified in the totally real fields. In joint work with Ting-Han Huang, we prove a p-adic Gross–Zagier formula for this L-function for a pair of an elliptic modular form and a quadratic Hilbert modular form. This generalises work of Blanco-Chacon and Fornea for the case of Hida families, and we overcome a technical assumption in their work of p being split in the quadratic field.
Twisted triple product p-adic L-function for finite slope families and a p-adic Gross–Zagier formularead_more
HG G 43
7 March 2025
14:15-15:15 		Dr. Seoyoung Kim
UBasel
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Number Theory Seminar

Title Certain families of K3 surfaces and their modularity
Speaker, Affiliation Dr. Seoyoung Kim, UBasel
Date, Time 7 March 2025, 14:15-15:15
Location HG G 43
Abstract We start with a double sextic family of K3 surfaces with four parameters with Picard number $16$. Then by geometric reduction (specializing at fiber) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number $17, 18$ and $19$ respectively. All these families turn out to be of hypergeometric type in the sense that their Picard-Fuchs differential equations are given by hypergeometric or Heun functions. We will study the geometry of 2-parameter families and related finite hypergeometric series in detail. We will then discuss some relevant modularity questions. This is a joint work with A. Clingher, A. Malmendier, and N. Yui.
Certain families of K3 surfaces and their modularityread_more
HG G 43
14 March 2025
14:15-15:15 		Prof. Dr. Mattia Cavicchi
Institut de Mathématiques de Bourgogne
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Number Theory Seminar

Title Motivic Hecke algebras
Speaker, Affiliation Prof. Dr. Mattia Cavicchi, Institut de Mathématiques de Bourgogne
Date, Time 14 March 2025, 14:15-15:15
Location HG G 43
Abstract The action of Hecke operators on cuspidal modular forms of weight 2 can be realized as the action of an algebra of algebraic cycles on the square of a smooth projective curve, with composition being defined in terms of intersection theory. Having in mind applications to the arithmetic of the associated Galois representations, one would like to generalize this description to the case of higher weight. The goal of the talk is to introduce this circle of ideas and to explain the construction of a so-called motivic Hecke algebra, providing such a generalization. This actually works in the much broader context of automorphic forms living in the cohomology of any PEL Shimura variety.
Motivic Hecke algebrasread_more
HG G 43
21 March 2025
14:15-15:15 		Dr. Luca Marannino
CNRS
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Number Theory Seminar

Title Anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classes
Speaker, Affiliation Dr. Luca Marannino, CNRS
Date, Time 21 March 2025, 14:15-15:15
Location HG G 43
Abstract In this talk, we outline an approach to the study of anticyclotomic Iwasawa theory of modular forms when the fixed prime p is inert in the relevant quadratic imaginary field. Following ideas of Castella-Do for the “p split" case, one can envisage a construction of an anticyclotomic Euler system arising from a suitable manipulation of certain Galois cohomology classes known as diagonal classes. We will report on this work in progress, trying to underline the main difficulties arising in the “p inert” setting.
Anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classesread_more
HG G 43
28 March 2025
14:15-15:15 		Prof. Dr. Gergely Harcos
Alfréd Rényi Institute of Mathematics
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Number Theory Seminar

Title A new zero-free region for Rankin–Selberg L-functions
Speaker, Affiliation Prof. Dr. Gergely Harcos, Alfréd Rényi Institute of Mathematics
Date, Time 28 March 2025, 14:15-15:15
Location HG G 43
Abstract I will present a new zero-free region for all GL(1)-twists of GL(m)×GL(n) Rankin–Selberg L-functions. The proof is inspired by Siegel’s celebrated lower bound for Dirichlet L-functions at s=1. I will also discuss briefly some applications and a very recent development. Joint work with Jesse Thorner.
A new zero-free region for Rankin–Selberg L-functionsread_more
HG G 43
11 April 2025
14:15-15:15 		Prof. Dr. William Duke
UCLA
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Number Theory Seminar

Title A Diophantine divisor function
Speaker, Affiliation Prof. Dr. William Duke, UCLA
Date, Time 11 April 2025, 14:15-15:15
Location HG G 43
Abstract I will describe recent work joint with Olga Balkanova and Dmitry Frolenkov on a restricted divisor function and its associated divisor problem. This problem shares properties with the usual Dirichlet divisor problem and the Hardy-Littlewood problem to count lattice points in a right triangle. The latter depends deeply on the arithmentic nature of the slope of the triangle the restricted divisor problem has a similar property. For the H-L problem, Hecke showed how to go much further when the slope comes from a real quadratic field. We apply Hecke’s idea to our problem, which introduces a number of interesting new difficulties,
A Diophantine divisor functionread_more
HG G 43
17 April 2025
14:15-15:15 		Dr. Arthur Forey
Université de Lille
Details

Number Theory Seminar

Title Title tba: Arthur Forey
Speaker, Affiliation Dr. Arthur Forey, Université de Lille
Date, Time 17 April 2025, 14:15-15:15
Location HG G 43
Title tba: Arthur Forey
HG G 43
2 May 2025
14:15-15:15 		Dr. Ju-Feng Wu
University College Dublin
Details

Number Theory Seminar

Title Title tba: Ju-Feng Wu
Speaker, Affiliation Dr. Ju-Feng Wu, University College Dublin
Date, Time 2 May 2025, 14:15-15:15
Location HG G 43
Title tba: Ju-Feng Wu
HG G 43
9 May 2025
14:15-15:15 		Dr. Gabriel Ribeiro
ETH Zurich
Details

Number Theory Seminar

Title Title tba: Gabriel Ribeiro
Speaker, Affiliation Dr. Gabriel Ribeiro, ETH Zurich
Date, Time 9 May 2025, 14:15-15:15
Location HG G 43
Title tba: Gabriel Ribeiro
HG G 43
23 May 2025
14:15-15:15 		Prof. Dr. Mariusz Mirek
Rutgers University
Details

Number Theory Seminar

Title title tba: Mariusz Mirek
Speaker, Affiliation Prof. Dr. Mariusz Mirek, Rutgers University
Date, Time 23 May 2025, 14:15-15:15
Location
title tba: Mariusz Mirek

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