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Autumn Semester 2023

Date / Time Speaker Title Location
22 September 2023
14:15-15:15 		Dr. Peter Hubrecht Koymans
ETH Zurich, Switzerland
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Number Theory Seminar

Title The negative Pell equation
Speaker, Affiliation Dr. Peter Hubrecht Koymans, ETH Zurich, Switzerland
Date, Time 22 September 2023, 14:15-15:15
Location HG G 43
Abstract In this talk we will study the negative Pell equation, which is the conic C_D : x2 - D y2 = -1 to be solved in integers x, y in Z. We shall be concerned with the following question: as we vary over squarefree integers D, how often is C_D soluble? Stevenhagen conjectured an asymptotic formula for such D. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and, time permitting, potential applications of the new proof techniques. This is joint work with Carlo Pagano.
The negative Pell equationread_more
HG G 43
29 September 2023
14:15-15:15 		Dr. Vivian Kuperberg
ETH Zurich, Switzerland
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Number Theory Seminar

Title Consecutive sums of two squares in arithmetic progressions
Speaker, Affiliation Dr. Vivian Kuperberg, ETH Zurich, Switzerland
Date, Time 29 September 2023, 14:15-15:15
Location HG G 43
Abstract In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.
Consecutive sums of two squares in arithmetic progressionsread_more
HG G 43
13 October 2023
14:15-15:15 		Prof. Dr. William Duke
UCLA
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Number Theory Seminar

Title Integral and rational representations of forms
Speaker, Affiliation Prof. Dr. William Duke, UCLA
Date, Time 13 October 2023, 14:15-15:15
Location HG G 43
Integral and rational representations of forms
HG G 43
20 October 2023
14:15-15:15 		Dr. Brandon Williams
RWTH Aachen
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Number Theory Seminar

Title Computing Fourier coefficients of paramodular eigenforms
Speaker, Affiliation Dr. Brandon Williams, RWTH Aachen
Date, Time 20 October 2023, 14:15-15:15
Location HG G 43
Abstract I will talk about an ongoing project of computing the Fourier expansions of cuspidal paramodular eigenforms, particularly in low weight. This is joint work with Eran Assaf.
Computing Fourier coefficients of paramodular eigenformsread_more
HG G 43
27 October 2023
14:15-15:15 		Javier Fresán
Sorbonne Université
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Number Theory Seminar

Title A construction of the polylogarithm motive
Speaker, Affiliation Javier Fresán, Sorbonne Université
Date, Time 27 October 2023, 14:15-15:15
Location HG G 43
Abstract Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the projective line minus three points, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives, whose existence is proved by computing the corresponding spaces of extensions both in the Hodge and the motivic settings. I will present a joint work with Clément Dupont, in which we construct the polylogarithm motive as an explicit, easy to remember, relative cohomology motive.
A construction of the polylogarithm motiveread_more
HG G 43
3 November 2023
14:15-15:15 		Prof. Dr. Sarah Zerbes
ETH Zurich, Switzerland
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Number Theory Seminar

Title Iwasawa theory for the symmetric square of an elliptic curve
Speaker, Affiliation Prof. Dr. Sarah Zerbes, ETH Zurich, Switzerland
Date, Time 3 November 2023, 14:15-15:15
Location HG G 43
Abstract The arithmetic of the adjoint, or symmetric square, of an elliptic curve over Q (or, more generally, of a modular form) is a particularly interesting case from the viewpoint of Iwasawa theory, not least because of its close connection with modularity-lifting problems and hence with Fermat's last theorem. In this talk I will describe ongoing work with David Loeffler in which we prove the cyclotomic Iwasawa main conjecture in this setting, using Euler systems for Hilbert modular surfaces.
Iwasawa theory for the symmetric square of an elliptic curveread_more
HG G 43
10 November 2023
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Number Theory Seminar

Title Swiss Number Theory Days
Speaker, Affiliation
Date, Time 10 November 2023,
Location
Swiss Number Theory Days
17 November 2023
14:15-15:15
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Number Theory Seminar

Title Arithmetica Transalpina
Speaker, Affiliation
Date, Time 17 November 2023, 14:15-15:15
Location
More information https://people.math.ethz.ch/~zerbess/ArithmeticaTransalpina.html
Arithmetica Transalpinaread_more
24 November 2023
14:15-15:15 		Prof. Dr. Claudia Alfes-Neumann
Universität Bielefeld
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Number Theory Seminar

Title On harmonic weak Maass forms associated to even integer weight newforms
Speaker, Affiliation Prof. Dr. Claudia Alfes-Neumann, Universität Bielefeld
Date, Time 24 November 2023, 14:15-15:15
Location HG G 43
Abstract In this talk we review results on several types of harmonic weak Maass forms that are related to integral even weight newforms. We start with a brief introduction to the theory of harmonic weak Maass forms. These can be related to classical modular forms via a certain differential operator, the so-called \xi-operator. Starting with an integral weight newform, we will review different constructions of integral weight harmonic weak Maass forms via (generalized) Weierstrass zeta functions that map to the newform under the \xi-operator. A second construction via theta liftings gives a half-integral weight harmonic weak Maass form whose coefficients are given by periods of certain meromorphic modular forms with algebraic coefficients and periods of the integer even weight newform. This is joint work with Jens Funke, Michael Mertens, and Eugenia Rosu resp. Jan Bruinier and Markus Schwagenscheidt.
On harmonic weak Maass forms associated to even integer weight newformsread_more
HG G 43
8 December 2023
14:15-15:15 		Dr. Paul Kiefer
Bielefeld University
Details

Number Theory Seminar

Title Injectivity of the Kudla-Millson-Lift in genus two
Speaker, Affiliation Dr. Paul Kiefer, Bielefeld University
Date, Time 8 December 2023, 14:15-15:15
Location HG G 43
Abstract In the eighties, Kudla and Millson constructed a linear map between certain spaces of vector-valued Siegel modular cusp forms to the space of closed differential forms on some orthogonal Shimura variety. The injectivity of this map in genus 1 has been of great interest and has many applications, including the surjectivity of Borcherds' lift. The aim of this talk is to introduce orthogonal Shimura varieties and indicate why they might be of interest. We then proceed to explain the Kudla-Millson lift and its injectivity in genus 2. We end the talk with a cohomological application. This is joint work with Riccardo Zuffetti.
Injectivity of the Kudla-Millson-Lift in genus tworead_more
HG G 43

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