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Spring Semester 2022

Date / Time Speaker Title Location
25 February 2022
14:15-15:15 		Prof. Dr. Yingkun Li
TU Darmstadt
Details

Number Theory Seminar

Title Algebraicity of higher Green functions at CM points
Speaker, Affiliation Prof. Dr. Yingkun Li, TU Darmstadt
Date, Time 25 February 2022, 14:15-15:15
Location Zoom
Abstract By the classical theory of complex multiplication, the modulari-function takes algebraic values at CM points. It is an interesting question to ask about the algebraic nature of other types of automorphic functions at CM points. For the automorphic Green function at integral parameters, Gross and Zagier conjectured in the 1980s that their values at CM points are essentially logarithms of algebraic numbers. In this talk,we will discuss recent progress toward this conjecture and its generalization to the setting of orthogonal Shimura varieties. This is partly joint with Jan Bruinier and Tonghai Yang.
Algebraicity of higher Green functions at CM pointsread_more
Zoom
4 March 2022
14:15-15:15 		Dr. Matthew Welsh
University of Bristol
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Number Theory Seminar

Title Fine-scale distribution of roots of quadratic congruences
Speaker, Affiliation Dr. Matthew Welsh, University of Bristol
Date, Time 4 March 2022, 14:15-15:15
Location HG G 43
Abstract In addition to being interesting objects of study in their own right, statistical information on the roots of quadratic congruences has been crucial input for results in analytic number theory. In this talk we will discuss limit laws for fine-scale statistics of these roots, in particular, an explicit expression for the pair correlation density, that are established by translating the problem to convergence of certain random geodesic-line processes in the hyperbolic plane. This is based on joint work with Jens Marklof and Zonglin Li.
Fine-scale distribution of roots of quadratic congruencesread_more
HG G 43
11 March 2022
14:15-15:15 		Prof. Dr. Henri Darmon
McGill University
Details

Number Theory Seminar

Title Singular moduli for real quadratic fields and modular generating series
Speaker, Affiliation Prof. Dr. Henri Darmon, McGill University
Date, Time 11 March 2022, 14:15-15:15
Location HG G 43
Abstract I will discuss an approach to studying the algebraicity of the special values of rigid meromorphic cocycles at real multiplication points, which rests on transposing an approach of Gross and Zagier to a p-adic setting. This is a preliminary report on ongoing work with Jan Vonk.
Singular moduli for real quadratic fields and modular generating seriesread_more
HG G 43
18 March 2022
14:15-15:15 		Prof. Dr. Olivier Fouquet
Université de Franche-Comté
Details

Number Theory Seminar

Title The Iwasawa Main Conjecture for modular forms with residually irreducible Galois representations
Speaker, Affiliation Prof. Dr. Olivier Fouquet, Université de Franche-Comté
Date, Time 18 March 2022, 14:15-15:15
Location HG G 43
Abstract The Iwasawa Main Conjecture for modular forms is a conjecture describing the p-adic variation of special values of L-functions of eigencuspforms in terms of arithmetic objects. In this talk, I will outline a joint work with X.Wan in which we prove this conjecture provided the residual Galois representation attached to the eigencuspform is irreducible (and a couple of mild technical assumptions). This shows in particular that for most rational elliptic curves, L(E,1)≠0 if and only if the Selmer group of E is finite.
The Iwasawa Main Conjecture for modular forms with residually irreducible Galois representationsread_more
HG G 43
25 March 2022
14:15-15:15 		Dr. Michael Mertens
Universität Köln
Details

Number Theory Seminar

Title Explicit formulas for 1-point functions in certain classes of vertex operator algebras
Speaker, Affiliation Dr. Michael Mertens, Universität Köln
Date, Time 25 March 2022, 14:15-15:15
Location Zoom
Abstract The study of vertex operator algebras in Mathematics originates in the famous Monstrous Moonshine Conjecture by McKay-Thompson and Conway-Norton, famously proved by Borcherds, building on important previous work by Fränkel-Lepowsky-Meurman. Since then, a general theory has been developed and one important result due to Huang, Zhu, Dong-Li-Mason and others in this context gives a very close connection between VOAs and modular forms: For a sufficiently nice VOA, its character (i.e. the generating function of the dimensions of its graded components) and more generally the so-called 1-point functions of states in the VOA define modular forms for some congruence subgroup. It is however still rather mysterious which congruence subgroup this might be in general or how to determine the modular form in question. In my talk I will give a short introduction to the theory of VOAs and the problem just described and then present joint work with Geoffrey Mason in which we provide a solution to this problem for certain special VOAs in that we give explicit formulas for these 1-point functions on a basis of the respective VOA. The main ingredient is a new variant of so-called Zhu recursion.
Explicit formulas for 1-point functions in certain classes of vertex operator algebrasread_more
Zoom
1 April 2022
14:15-14:15 		Dr. George Boxer
Université Paris-Saclay
Details

Number Theory Seminar

Title On the ordinary part of coherent cohomology of Hilbert modular varieties
Speaker, Affiliation Dr. George Boxer, Université Paris-Saclay
Date, Time 1 April 2022, 14:15-14:15
Location HG G 43
Abstract In this talk, I will explain that the integral structure on the coherent cohomology of Shimura varieties is not so well behaved in general. Then I will explain that at least in the case of Hilbert modular varieties, the situation improves when we pass to the ordinary part in the sense of Hida. In particular we have vanishing theorems and Hida style control theorems. This is joint work in progress with Vincent Pilloni.
On the ordinary part of coherent cohomology of Hilbert modular varietiesread_more
HG G 43
8 April 2022
14:15-15:15 		Prof. Dr. Philipp Habegger
Universität Basel
Details

Number Theory Seminar

Title Some cases of the Schinzel-Zassenhaus Conjecture in Arithmetic Dynamics
Speaker, Affiliation Prof. Dr. Philipp Habegger, Universität Basel
Date, Time 8 April 2022, 14:15-15:15
Location HG G 43
Abstract The Schinzel-Zassenhaus Conjecture states that a non-zero algebraic integer of degree d that is not a root of unity has at least one conjugate with absolute value greater than 1+c/d where c>0 is an absolute constant. This conjecture was proved in a recent breakthrough by Vesselin Dimitrov. In joint work with Harry Schmidt we prove a variant of the Schinzel-Zassenhaus Conjecture for a class of polynomials in the setting of arithmetic dynamics. The class contains T^2-1. For this polynomial we conclude a lower bound for the Call-Silverman height of a wandering point that decays like the inverse of the square of the field degree.
Some cases of the Schinzel-Zassenhaus Conjecture in Arithmetic Dynamicsread_more
HG G 43
15 April 2022
Details

Number Theory Seminar

Title Easter holidays - No seminar
Speaker, Affiliation
Date, Time 15 April 2022,
Location
Easter holidays - No seminar
22 April 2022
Details

Number Theory Seminar

Title Easter holidays - No seminar
Speaker, Affiliation
Date, Time 22 April 2022,
Location
Easter holidays - No seminar
29 April 2022
14:15-15:15 		Dr. James Newton
University of Oxford
Details

Number Theory Seminar

Title Modularity over CM fields
Speaker, Affiliation Dr. James Newton, University of Oxford
Date, Time 29 April 2022, 14:15-15:15
Location HG G 43
Abstract Since the seminal works of Wiles and Taylor-Wiles, robust methods were developed to prove the modularity of 'polarised' Galois representations. These include, for example, those coming from elliptic curves defined over totally real number fields. Over the last 10 years, new developments in the Taylor-Wiles method (Calegari, Geraghty) and the geometry of Shimura varieties (Caraiani, Scholze) have broadened the scope of these methods. One application is the recent work of Allen, Khare and Thorne, who prove modularity of a positive proportion of elliptic curves defined over a fixed imaginary quadratic field. I'll review some of these developments and work in progress with Caraiani which has further applications to modularity of elliptic curves over imaginary quadratic fields.
Modularity over CM fieldsread_more
HG G 43
13 May 2022
14:15-15:15 		Prof. Dr. Emmanuel Kowalski
ETH Zurich, Switzerland
Details

Number Theory Seminar

Title Sidon sets in arithmetic and algebraic geometry
Speaker, Affiliation Prof. Dr. Emmanuel Kowalski, ETH Zurich, Switzerland
Date, Time 13 May 2022, 14:15-15:15
Location HG G 43
Abstract (Joint work with A. Forey and J. Fresán) Sidon sets are subsets of abelian groups distinguished by the property that the restriction of the addition map is injective, up to switching the summands. This notion arose in the study of Fourier series, and turns out to have surprising applications in other areas of harmonic analysis. In particular, it can be extremely useful for the determination of Galois groups or monodromy groups in various circumstances. The talk will present these relations, focusing on new examples of Sidon sets arising from classical algebraic geometry (involving jacobians -- classical, generalized, intermediate), and will explain in particular their applications to equidistribution questions.
Sidon sets in arithmetic and algebraic geometryread_more
HG G 43
27 May 2022
14:15-15:15 		Dr. Anna von Pippich
University of Konstanz
Details

Number Theory Seminar

Title Title T.B.A.
Speaker, Affiliation Dr. Anna von Pippich, University of Konstanz
Date, Time 27 May 2022, 14:15-15:15
Location
Abstract Talk postponed to next semester.
Title T.B.A.read_more (CANCELLED)
3 June 2022
16:15-17:15 		Prof. Dr. Kevin Buzzard
Imperial College London
Details

Number Theory Seminar

Title FIM Lecture: Can Computers Prove Theorems?
Speaker, Affiliation Prof. Dr. Kevin Buzzard, Imperial College London
Date, Time 3 June 2022, 16:15-17:15
Location HG E 1.2
Abstract *This is not a talk in the Number Theory Seminar, but an advertisement for the FIM Lectures. Please see the link below for more information* Abstract: We taught computers the rules of chess, and now they can beat all humans at chess. Now we’ve taught computers the rules of mathematics. What will happen next? I think it is much too optimistic to suggest that computers will soon be proving theorems at Fields Medal level. However, just like modern chess computers help humans to play chess better, I think that computer proof systems might help us become better mathematicians. This software has the potential to help us to prove theorems, to check technically difficult parts of our work, to search mathematical databases for proofs and counterexamples, and to help us to teach and communicate mathematics. I’ll give an overview of the area as it stands today, including an update on the project being led by Johan Commelin to formally verify the Clausen-​Scholze result that the reals are a liquid vector space, using the Lean theorem prover.
FIM Lecture: Can Computers Prove Theorems?read_more
HG E 1.2

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