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Autumn Semester 2024

Date / Time Speaker Title Location
20 September 2024
14:15-15:15 		Prof. Dr. Irene Bouw
Universität Ulm
Details

Number Theory Seminar

Title Computing the Weil-Deligne representation of a curve via stable reduction
Speaker, Affiliation Prof. Dr. Irene Bouw, Universität Ulm
Date, Time 20 September 2024, 14:15-15:15
Location HG G 19.1
Abstract Let Y be a curve defined over a p-adic field. The local Galois representation of Y is determined by a Weil-Deligne representation. We explain how to compute this representation from the stable reduction of Y to characteristic p>0. This is joint works with D.K. Do, R. Nowak, S. Wewers, and X. Zhang.
Computing the Weil-Deligne representation of a curve via stable reductionread_more
HG G 19.1
4 October 2024
14:15-15:15 		Sebastian Herrero
University of Santiago de Chile
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Number Theory Seminar

Title Counting rational points on Hirzebruch-Kleinschmidt varieties.
Speaker, Affiliation Sebastian Herrero, University of Santiago de Chile
Date, Time 4 October 2024, 14:15-15:15
Location HG F 26.1
Abstract This talk is about asymptotic formulas for the number of rational points of bounded (or large) height on Hirzebruch-Kleinschmidt varieties over global fields. These varieties are realizations of split toric varieties with Picard rank 2, and their explicit models enable computations that go beyond the general expectation of the Manin-Peyre conjecture. This is joint work with Tobías Martínez (University of El Salvador) and Pedro Montero (UTFSM, Chile).
Counting rational points on Hirzebruch-Kleinschmidt varieties.read_more
HG F 26.1
11 October 2024
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Number Theory Seminar

Title Swiss Number Theory Days
Speaker, Affiliation
Date, Time 11 October 2024,
Location UniDistance Suisse, Schinerstrasse 18, 3900 Brig
More information https://unidistance.ch/en/mathematics-and-computer-science/event/swiss-number-theory-days-2024
Swiss Number Theory Daysread_more
UniDistance Suisse, Schinerstrasse 18, 3900 Brig
18 October 2024
14:15-15:15 		Christopher Birkbeck
University of East Anglia
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Number Theory Seminar

Title Formalising the Regular Case of Fermat’s Last Theorem in Lean
Speaker, Affiliation Christopher Birkbeck, University of East Anglia
Date, Time 18 October 2024, 14:15-15:15
Location HG G 43
Abstract I will discuss some recent joint work on formalising the regular case of Fermat’s Last Theorem in Lean. The regular case of FLT, where the exponent is a prime p that does not divide the class number of the p-th cyclotomic field, has long been known to be much simpler than the full proof, making it a good target for formalisation. In my talk, I will explain what Lean is, why one would want to formalise mathematics, and some of the challenges encountered in this process. No prior knowledge of Lean or formalisation is required.
Formalising the Regular Case of Fermat’s Last Theorem in Leanread_more
HG G 43
1 November 2024
14:15-15:15 		Tim Gehrunger
ETH Zurich, Switzerland
Details

Number Theory Seminar

Title Classification of genus 2 curves in residue characteristic 2
Speaker, Affiliation Tim Gehrunger, ETH Zurich, Switzerland
Date, Time 1 November 2024, 14:15-15:15
Location HG G 43
Classification of genus 2 curves in residue characteristic 2
HG G 43
8 November 2024
14:15-15:15 		Dr. Francesco Zerman
UniDistance Suisse
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Number Theory Seminar

Title Heegner points on Pizer curves
Speaker, Affiliation Dr. Francesco Zerman, UniDistance Suisse
Date, Time 8 November 2024, 14:15-15:15
Location HG G 43
Abstract Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $K$ be an imaginary quadratic field. Assume that $N=N^+ N^-$ with $N^+$ split in $K$ and $N^-$ squarefree and inert in $K$. Under this "generalised" Heegner hypothesis, in the last thirty years there have been many works building $K$-Heegner points on $E$ by studying the arithmetic of Eichler orders of level $N^+$ inside the quaternion algebra of discriminant $N^-$ over $\mathbb{Q}$. The existence of nontrivial systems of Heegner points has always deep consequences, leading to rank one results for $E(K)$ and to control on the arithmetic of $E$ over anticyclotomic $p$-extensions of $K$. Much less is known when $N^-$ is not squarefree. In this talk, I will explain how one can use the arithmetic of Pizer orders to build Heegner points in this setting, building on a recent work of Longo, Rotger and de Vera-Piquero. I will then show how their work could be generalized to study other Galois representations, mainly focusing on Hida families of modular forms. This is a joint work with Luca Dall'Ava.
Heegner points on Pizer curvesread_more
HG G 43
15 November 2024
Details

Number Theory Seminar

Title Arithmetica Transalpina
Speaker, Affiliation
Date, Time 15 November 2024,
Location
More information https://people.math.ethz.ch/~zerbess/ArithmeticaTransalpina.html
Arithmetica Transalpinaread_more
29 November 2024
14:15-15:15 		Dr. Manuel Hauke
Norwegian University of Science and Technology
Details

Number Theory Seminar

Title Metric Diophantine approximation: What comes after Duffin--Schaeffer?
Speaker, Affiliation Dr. Manuel Hauke, Norwegian University of Science and Technology
Date, Time 29 November 2024, 14:15-15:15
Location HG G 43
Abstract Given a function $\psi: \mathbb{N} \to [0,1/2]$, the theorems of Khintchine and Koukoulopoulos--Maynard (formerly known as Duffin--Schaeffer conjecture) provide a satisfying answer about the number of good rational approximations $\vert \alpha - \frac{p}{q} \vert < \frac{\psi(q)}{q}$ for Lebesgue almost every $\alpha \in \mathbb{R}$.\\ However, the picture is much less understood when we allow an inhomogeneous parameter $\gamma \in \mathbb{R}$ and ask for solutions to $\vert \alpha - \frac{p+ \gamma}{q} \vert < \frac{\psi(q)}{q}$, and even less if we allow the parameter $\gamma_q$ to change with $q$. In this talk, I will explain the genuine difficulties of these questions that are surprisingly deeply connected with analytic and combinatorial number theory, and discuss recent positive results both in dimension $1$ as well as and in higher-dimensional analogues. This is partially joint work with Victor Beresnevich and Sanju Velani, respectively with Felipe Ramírez.
Metric Diophantine approximation: What comes after Duffin--Schaeffer?read_more
HG G 43
5 December 2024
17:15-18:45 		Prof. Dr. David Loeffler
UniDistance Suisse
Details

Number Theory Seminar

Title Inaugural Lecture : The Legacy of Fermat's last Theorem
Speaker, Affiliation Prof. Dr. David Loeffler , UniDistance Suisse
Date, Time 5 December 2024, 17:15-18:45
Location UniDistance Suisse & online, Schinerstrasse 18, 3900 Brig
Abstract In the 17th century, Pierre de Fermat stated – and claimed to have proved – an elegant mathematical theorem, stating that a certain equation has no solutions in the whole numbers. Generations of mathematicians tried to find a proof of this theorem, but the problem resisted attack for more than 350 years, until it was solved in 1995 by Andrew Wiles and Richard Taylor. Professor David Loeffler will explain the problem, and some of the beautiful and intricate ideas that played a role in its solution; and he will explain some more recent mathematical developments arising from the same circle of ideas which are still the focus of intense research today.
More information https://fernuni.ch/mathematik-und-informatik/event/lecon-inaugurale-david-loeffler
Inaugural Lecture : The Legacy of Fermat's last Theoremread_more
UniDistance Suisse & online, Schinerstrasse 18, 3900 Brig
6 December 2024
14:15-15:15 		Prof. Dr. Philippe MICHEL
EPFL
Details

Number Theory Seminar

Title On mixed cubic moments for Dirichlet L-functions
Speaker, Affiliation Prof. Dr. Philippe MICHEL, EPFL
Date, Time 6 December 2024, 14:15-15:15
Location HG G 43
Abstract A few months ago, Etienne fouvry, Emmanuel Kowalski and myself initiated the study of "mixed" moments of $L$-functions of Dirichlet characters of prime modulus $q$. The mixed second moments are of the shape $$\sum_{\chi\mod q}L(\chi^a,1/2)L(\chi^b,1/2)$$ with $a$ and $b$ non-zero integers not necessarily equal (hence the term "mixed"; we could establish an asymptotic formula for these moments when $q$ is large, with a power saving error term in $q$. In this talk, I will explain what we (EF, EK, myself together with Will Sawin) can currently prove in the much more complicated case of cubic mixed moments: moments of the shape $$\sum_{\chi\mod q}L(\chi^a,1/2)L(\chi^b,1/2)L(\chi^c,1/2).$$ Our methods belong to the area of "applied $\ell$-adic cohomology" and combine a variety of techniques from analytic number theory with the theory of algebraic exponential sums, in particular the properties of Kloosterman and hypergeometric sheaves studied by Katz.
On mixed cubic moments for Dirichlet L-functionsread_more
HG G 43
13 December 2024
14:15-15:15 		Dr. Xie Jianfeng
ETH Zurich
Details

Number Theory Seminar

Title The growth of Tate-Shafarevich groups in cyclic extensions
Speaker, Affiliation Dr. Xie Jianfeng, ETH Zurich
Date, Time 13 December 2024, 14:15-15:15
Location HG G 43
Abstract Let K be a global field and p be a prime number with p\neq char K. A classical theorem in algebraic number theory asserts that when L varies in Z/pZ-extensions of K, the p-rank of the class group of L is unbounded. It is expected that similar unboundenss results also hold for other arithmetic objects. For example, K. Česnavičius proved that for fixed abelian variety A over K, the p-Selmer group of A over L is unbounded when L varies in Z/pZ-extensions of K. And he raised a further problem: in the same setting, does the Tate-Shafarevich group of A also grow unboundedly? Using a machinery developed by B. Mazur and K. Rubin, we give a positive answer to this problem. This is a joint work with Yi Ouyang.
The growth of Tate-Shafarevich groups in cyclic extensionsread_more
HG G 43

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