Kimingala: celebrating Ian Kiming's mathematics
University of Copenhagen,
17 March 2023
This special session of the Number Theory seminar is dedicated to the celebration of Ian Kiming's mathematics.
The day will feature four research talks, connected to Galois representations and modular forms, which are two of the mostly featured topics in Ian Kiming's mathematical research.
|13:00 - 13:15||Welcoming words|
|13:15 - 14:15||Gabor Wiese (Université du Luxembourg)|
|14:15 - 14:30||Short break|
|14:30 - 15:30||Adel Betina (København Universiteit)
|15:30 - 16:00||Coffee and cake|
|16:00 - 17:00||Samuele Anni (Institut de mathématiques de Marseille)|
|17.00 - 17.15||Short break|
|17.15 - 18.15||Matthias Schütt (Leibniz Universität Hannover)|
Titles and abstracts
Title: Counting modular forms mod p satisfying constraints at p.
Abstract: The structure of the algebra of modular forms over finite fields has been widely studied, in part for its applications in establishing congruences. In this talk, after recalling classical geometric arguments of Ogg and Kenku, I will show how, for N coprime with p, one can count the number of classical modular forms of level Np and weight k with both a residual Galois representation and an Atkin-Lehner sign at fixed p, generalizing Martin's recent results, and dimension formulas given by Jochnowitz and by Bergdall-Pollack. Most of these results can be stated as equivariant isomorphisms for the Hecke operators between certain modules, thanks to a p-adic refinement of the Brauer-Nesbitt theorem. A theoretical framework for proving such isomorphisms is given, using the Eichler-Selberg trace formula. This method applies in the case where the level is divisible by the residual characteristic, contrary to the pre-existing approaches. This is work in progress with Alexandru Ghitza (University of Melbourne) and Anna Medvedovsky (Boston University).
Title: On the first derivative of cyclotomic Katz p-adic L-functions at exceptional zeros.
Abstract: This talk is based on a joint work with Ming-Lun Hsieh studying the exceptional zeros conjecture of Katz p-adic L-functions. We will present a formula relating the first derivative of the cyclotomic Katz p-adic L-function attached to a ring class character of a general CM field to the product of an L-invariant and the value of some improved Katz p-adic L-function at s=0. In particular, we show that these Katz p-adic L-functions have a simple trivial zero if and only if their cyclotomic L-invariants are non-zero. Our method uses congruences of Hilbert CM forms and the theory of deformations of reducible Galois representations. I will discuss at the end of this talk about how we can compute the first derivative beyond the case where the branch character is a ring class character using p-adic Eisenstein congruences for GU(2,1).
Title: Real multiplication and Hilbert modularity
Abstract: Real multiplication (RM) not only occurs on abelian varieties (in terms of endomorphisms), but also on K3 surfaces and other Calabi-Yau varieties via Hodge structures. Contrary to the CM case, RM remains rather mysterious, largely evading explicit examples. I will report on joint work in progress with Bert van Geemen which engineers new constructions of K3 surfaces with RM. I will also discuss applications to Hilbert modularity of Calabi-Yau threefolds.
Title: Splitting fields of X^n-X-1 and modular forms
Abstract: In his article "On a theorem of Jordan", Serre considered the family of polynomials f_n(X) = X^n-X-1 and the counting function of the number of roots of f_n over the finite field F_p, seen as function in p. He explicitly showed the `modularity' of this function for n=3,4. In this talk, I report on joint work with Alfio Fabio La Rosa and Chandrashekhar Khare, in which we treat the case n=5 in several different ways.