Oberseminar Algebra and Number Theory

The Oberseminar is a meeting that takes place during every week of the semester to discuss and disseminate new research results.

Planned talks for the Summer Semester 2024 - Thursday 2 p.m., Room 120

  • 13.06.2024: Tim Gehrunger, Reduction types of genus 2 curves in residue characteristic 2
  • 06.06.2024: Andreas Pieper, Reduction of plane quartics and Cayley Octads
  • 23.05.2024: Jeroen Sijsling, Modularity in genus 2
  • 16.05.2024: Xiaodong Zhang, On q-Weil Numbers, Part II
  • 25.04.2024: Miriam Ni-Chobhthaigh, Selmer groups and their structure over dihedral extensions
  • 19.04.2024 (Friday, Room E.04 in HeHo22): Yufei Qian, The moduli space of principally polarized abelian varieties and its toroidal compactification

Planned talks for the Winter Semester 2023/24 - Thursday, 4 p.m., Room E60

  • 29.02.2024, Xiaodong Zhang, On q-Weil numbers, Part I
  • 28.02.2024, Raum E20: Irene Bouw, Abelsche Varietäten über endlichen Körpern
  • 22.02.2024: Tim Evink, An equivalence of 2^n-descent and 2^(n-1)-descent using visualisation of Sh[2]
  • 15.02.2024: Irene Bouw, Abelian varieties over finite fields
  • 25.01.2024: Robert Nowak, Classifying reduction types of genus 2 hyperelliptic curves to characteristic 2 with equidistant branch locus
  • 21.12.2023: Sabrina Kunzweiler, Modular polynomials via Kani's theorem
  • 01.12.2023: Stefan Wewers, Models of hypersurfaces and BT buildings
  • 23.11.2023: Stefan Wewers, Instabilities of hypersurfaces
  • 16.11.2023: Valentijn Karemaker, When is a polarised abelian variety determined by its p-divisible group?
  • 09.11.2023: Leolin Nkuete, Maximal curves of genus 5
  • 02.11.2023: Jeroen Sijsling, Fundamentalgruppen und Überlagerungen
  • 28.09.2023: Andreas Pieper, The equation of a genus 4 curve from its theta constants

External Auditors (in case of virtual talks)

If you want to join an online talk, please ask us to send you the Zoom invitation. You can also register here for our mailing-list reine-announce@mawi.lists@uni-ulm.de to receive the Zoom invitation automatically. You will only have to click "Abonnieren" and then enter your email-address and name.

Abstracts Summer Semester 2024

A fundamental tool in studying the arithmetic of an algebraic variety over a local field is the theory of stable reduction. In this talk, we will focus on hyperelliptic curves over a field of mixed characteristic (0,2). We will start by outlying how the rigidification of the situation using Weierstrass points can simplify the construction of the stable model. Afterwards, we will apply this method to give a classification of the reduction types of genus 2 curves. This is joint work with Richard Pink.

 

Van Bommel et al. recently gave a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. In the talk we will introduce Cayley octads and discuss the main ideas of the conjecture.

Given an elliptic curve E over the rational numbers, the modularity theorem shows that its L-function L (E, s) can equally well be realized as an L-function L (f, s) associated to a modular form of weight 2 for GL_{2, QQ} with rational coefficients. This association gives rise to a bijection between the set of isogeny classes of elliptic curves E over QQ of a given conductor N and the set of normalized newforms f in S_2 (Gamma_0 (N), QQ).

This talk discusses how these results generalize (at least conjecturally) to curves of genus 2. Given such a genus 2 curve X over a number field F, we start by exploring the information contained in the L-function L (X, s). Things get considerably funkier, and in particular the above bijection breaks down because of the presence of false elliptic curves, so that we have to state carefully where to look for a modular form f for GL_{2, F} associated to X, if indeed any such form exists at all.

If the endomorphism ring of the Jacobian J of X is a quadratic number field M (so that X is of so-called GL_2-type), then a conjecture by Ribet claims the existence of a modular form f for GL_{2, F} with coefficient field M such that L (X, s) = L (f, s). We show what consequences this result has for curves X over QQ. It turns out that there always exists a quadratic number field F such that X is of GL_2-type over F. A consideration involving restrictions of scalars then shows that we have L (X, s) = L (f, s), where f is a modular form for GL_{2, F} associated to the base extension of X to F; moreover, the property that X_F comes from the curve X over QQ is reflected in a peculiar property of the modular form f that we call Galois alignment.

This is joint work with Andrew Booker, Andrew Sutherland, John Voight, and Dan Yasaki.

The terminology "q-Weil numbers of weight m" was first introduced by Deligne in his work on the Weil conjecture. However, some properties of these numbers had already been mentioned by Kronecker in his 1857 paper. I will talk about basic definitions and properties of abelian varieties and some applications. The main goal of this talk is to present the idea of a comprehensive proof of the Honda-Tate theorem, which establishes a bijection between the set of simple abelian varieties over F_q, up to isogeny, and the set of q-Weil numbers of weight 1, up to conjugacy.

In 1983 Faltings gave a proof to Tate's conjecture that an elliptic curve E is determined up to isomorphism over a number field K by the Galois action of G_K on E_tors. The p-Selmer groups can be defined purely in terms of the Galois module E[p^infty], the p-group of E_tors. A result of this insight led to the paper "Finding large Selmer rank via an arithmetic theory of local constants" in 2007 by Barry Mazur and Karl Rubin, where in they used the twisted parity conjecture and pro-p Selmer groups to find a lower bound for the ranks of Selmer groups over dihedral extensions of number fields. This talk explains the main concepts and results of the paper.

My talk is about the moduli space of principally polarized abelian varieties and toroidal compactification. Abelian varieties are high-dimensional generalizations of elliptic curves. I will provide a brief introduction to abelian varieties and construct the analytic moduli space of principally polarized abelian varieties, which is the Siegel upper half space modulo the symplectic group action. This space is not compact. A good way to compactify it is by using toroidal compactification. I will introduce the toric geometry we need, give the general steps of toroidal compactification, work out the dim 1 case in detail and provide an outline for dim 2.

Abstracts Winter Semester 2023/24

Title : An equivalence of 2^n-descent and 2^(n-1)-descent using visualisation of Sh[2].

Abstract: visualisation of Sh[2] can be used to improve on a 2-descent of an elliptic curve over a number field, which in general is weaker than a 4-descent, and involves working with 2-Selmer groups over a quadratic extension of the base field. In the talk I will explain what this means, and give a condition under which this improvement is actually equivalent to a 4-descent. The argument actually shows that 2^n-descent yields the same rankbounds as a 2^(n-1)-descent over some quadratic field extension. As an example we illustrate the method to a family of quadratic twists of X_0(15).

Semistable reduction of hyperelliptic curves to characteristic 2 is known for requiring different approaches than for other characteristic, and for admitting a great number of reduction types. If we require equidistant branch locus -- a condition which roughly translates to the graph of components not admitting any loops of positive genus -- then the possible reduction types are greatly reduced.

We will classify and analyze these types, describe the MCLF algorithm for computing semistable reduction in these cases, and give bounds on the automorphism groups of the curves for the purpose of calculating conductor exponents.

The classical modular polynomial phi_l parametrizes pairs of elliptic curves connected by an isogeny of degree l. They are an important tool in algorithmic number theory. For instance, modular polynomials are used in point counting algorithms or for the computation of endomorphism rings.
This talk will be about a new method for computing the modular polynomial. It has the same asymptotic time complexity as the currently best known algorithms, but does not rely on any heuristics. One of the main ingredients is the use of Kani's theorem to represent isogenies of elliptic curves as smooth-degree isogenies in higher dimensions. This is based on joint work with Damien Robert.

We will study the Siegel moduli space of abelian varieties in characteristic p and in particular its supersingular locus. We first determine precisely when this locus is geometrically irreducible. Since it was known that the number of components is a class number, this comes down to solving a “class number one problem” or “Gauss problem”.

Next, we will show when a polarised abelian variety is determined by its p-divisible group. This can be viewed as a Gauss problem for central leaves, which are the loci consisting of points whose associated p-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus 4.

List of previous talks

  • 09.02.2023: Sachi Hashimoto, Geometric methods for finding rational points on curves
  • 17.02.2023: Art Waeterschoot, Base change of nonarchimedean analytic curves and logarithmic differents
  • 11.05.2023: Stefan Wewers, Key polynominals and inductive valuations
  • 25.05.2023: Irene Bouw, Conductor and invariants of Ciani curves
  • 01.06.2023: Jeroen Sijsling, The Revenge on the SIDH (1)
  • 22.06.2023: Jeroen Sijsling, The Revenge on the SIDH (2)
  • 06.07.2023: Tim Evink, Imaginary quadratic fields over which XO(15) has rank 0

  • 09.06.2022: Stefan Wewers, Berkovich curves
  • 02.06.2022: Stefan Wewers, Computing models of curve over local fields, II: Models and valuations
  • 19.05.2022: Stefan Wewers, Computing models of curves over local fields, I
  • 12.05.2022: Irene I. Bouw, Construction and reduction of Ciani curves
  • 21.04.2022: Magnus Heimpel, Uniformization with branched coverings and non-free group actions
  • 18.03.2021: Andreas Pieper, Constructing all genus 2 curves with supersingular Jacobian
  • 25.03.2021: Tim Evink, Two-descent on some genus two curves
  • 01.04.2021: Robert Slob, Primitive divisors of sequences associated to elliptic curves over function fields
  • 22.04.2021: Bogdan Dina, Isogenous (non-)hyperelliptic CM Jacobians: Constructions, results, and CM types
  • 29.04.2021: Jeroen Sijsling, Isogenous (non-)hyperelliptic CM Jacobians: Shimura class groups, algorithms, and equations
  • 06.05.2021: Stefan Wewers, Computing semistable reduction of curves via nonarchimedian analytic geometry
  • 20.05.2021: Stefan Wewers, Computing semistable reduction of curves via nonarchimedian analytic geometry
  • 27.05.2021: Ole Ossen, Semistable reduction of quartics at p=3
  • 15.07.2021: Jeroen Sijsling, Isomorphisms between hyperelliptic and quartic curves
  • 28.10.2021: Tim Evink, A remark on congruent numbers
  • 04.11.2021: Jan Sijsling, Kubische Erweiterungen mit vorgeschriebener Verzweigung (I)
  • 11.11.2021: Jan Sijsling, Kubische Erweiterungen mit vorgeschriebener Verzweigung (II)
  • 18.11.2021: Andreas Pieper, Introduction to Mumford´s theory of theta groups and algebraic theta nullvalues (I)
  • 25.11.2021:  Andreas Pieper, Introduction to Mumford´s theory of theta groups and algebraic theta nullvalues (II)
  • 02.12.2021: Stefan Wewers, Special divisors - an introduction (I)
  • 09.12.2021: Stefan Wewers, Introduction to special divisors, II: The Existence and Connectedness Theorem
  • 16.12.2021: Stefan Wewers, Inroduction to special divisors, III: Kempf's singularity theorem
  • 20.01.2022: David Hokken (Utrecht), Galois Groups of Littlewood Polynomials (VT)
  • 27.01.2022: Barinder Banwait (Heidelberg), Rational p-isogenies of elliptic curves (VT)
  • 03.02.2022: Andreas Pieper, Quotients of superficial abelian varieties by connected subgroups
  • 03.03.2022: Andreas Pieper, The Schottky problem for genus 4 curves
  • 20.10.2022: Stefan Wewers, The arithmetic of algebraic curves: a (biased) overview
  • 27.10.2022: Irene Bouw, Good reduction of hypersurfaces
  • 03.11.2022: Prof. Dr. em. Werner Lütkebohmert, Extension von rigid-analytischen Objekten
  • 10.11.2022: Catherine Ray, Modeling Formal Group Actions using Galois Theory
  • 17.11.2022:  Stefan Wewers: Introduction to GIT, and stability of hypersurfaces
  • 24.11.2022: Ole Ossen: Computing semistable reduction of quartics, I
  •  1.12.2022: Ole Ossen: Computing semistable reduction of quartics, II
  •  8.12.2022: Tim Evink, Rational points on families of hyperelliptic curves
  • 15.12.2022: Pip Goodman, Restrictions on endomorphism algebras

  • 17.01.19: Jeroen Sijsling, Picard-Kurven: Gleichungen und Invarianten
  • 24.01.19: Irene Bouw, Spezielle Picard-Kurven: Twists und Führerexponenten
  • 31.01.19:  Duc Khoi Do, Rekonstruktion von Frobenius-halbeinfachen Weildarstellungen mithilfe ihrer lokalen Polynome
  • 07.02.19: Paula Truöl,  Massey products and linking numbers
  • 02.05.19: Irene Bouw, Der Parshin-Trick
  • 09.05.19: Jeroen Hanselman, Semi-abelsche Varietäten und ihre Néron-Modelle
  • 23.05.19: Duc Khoi Do, Galoisdarstellungen und Endlichkeitssätze
  • 06.06.19: Sabrina Kunzweiler, Die Faltingshöhe, I
  • 13.06.19: Stefan Wewers, Die Faltingshöhe, II
  • 27.06.19:  Jeroen Sijsling, Die Tate-Vermutung
  • 27.06.19: Angel Villanueva, Joint Distribution of Hecke and Casimir Eigenvalues for Automorphic Forms
  • 04.07.19: Irene Bouw, p-dividierbare Gruppen und die Hodge-Tate-Zerlegung
  • 27.06.19: Jeroen Sijsling, Beweis der Mordell-Vermutung
  • 17.10.219: Dr. Sophie Schmieg (Google, USA), Cryptography at Google
  • 24.10.19: Felix Göbler (Frankfurt), Das Zahlkörpersieb
  • 31.10.19: Jeroen Sijsling, Kubische Erweiterung mit vorgeschriebener Verzweigung
  • 14.11.19: Andreas Pieper, Die Leopoldt-Vermutung und die Geometrie der Zahlen
  • 21.11.19: Andreas Pieper, Die Leopoldt-Vermutung und die Geometrie der Zahlen II: Gitterpunkte in Kreuzpolytopen und Kugeln
  • 28.11.19: Sabrina Kunzweiler, Reduktionstypen von ebenen Quartiken
  • 05.12.19: Stefan Wewers, Torische Flächensingularitäten
  • 12.12.19: Bogdan Dina, Hyperelliptic curves with complex multiplication via Shimura reciprocity
  • 23.01.20: Jeroen Hanselman, Algorithmen für das Verkleben von Jacobischen
  • 24.01.20: Tim Evink (Universiteit Groningen), 2-Descent on hyperelliptic curves of genus 2
  • 30.01.20: Duc Do, Galois-Darstellungen elliptischer Kurven mit potentiell guter Reduktion
  • 06.02.20: Robert Slob (Universiteit Utrecht), Divisibility sequences of elliptic curves of characteristic 0
  • 07.02.20: Mike Daas (Universiteit Amsterdam), The sympletic method
  • 28.05.20: Sabrina Kunzweiler, Superelliptische Kurven und ganze Differentialformen
  • 15.10.20: Jeroen Sijsling, Abstieg algebraischer Kurven
  • 22.10.20: Jeroen Sijsling, Abstieg algebraischer Kurven returns
  • 29.10.20: Stefan Wewers, The Tate curve: Illustrating the connection between reduction and non-archimedean uniformization

  • 12.01.17: Stefan Wewers, Picard-Kurven mit kleinem Führer
  • 26.01.17: Jeroen Sijsling, Die Rekonstruktion quartischer Kurven aus ihren Invarianten
  • 02.02.17: Jeroen Hanselman, Non-hyperelliptic genus 3 covers of curves
  • 09.02.17: Mohamed Barakat, Category theory is a programming language
  • 23.02.17: Stefan Wewers, Rigid analytic spaces a la Berkovich and adic spaces: a short introduction
  • 09.03.17: Irene Bouw, Dynamical Belyi maps
  • 20.04.17 Tudor Micu, Etale morphisms
  • 26.05.17 Pinar Kilicer (Oldenburg), On primes dividing the invariants of Picard curves. 
  • 01.06.17 Angelos Koutsianas, The Chabauty-Coleman method and rational points on curves.
  • 20.06.17 Roman Kohls, Spurformel und Gauss-Summen
  • 29.06.17 Irene Bouw, Reduction of Picard curves, I
  • 06.07.17  Jeroen Sijsling, Canonical models of arithmetic (1,∞)-curves
  • 13.07.17 Stefan Wewers, Reduction of Picard curves, II
  • 20.07.17 Jeroen Hanselman, Pairings and line bundles on abelian varieties
  • 26.10.17: Stefan Wewers, "MCLF: ein Werkzeugkasten zur Berechnung von Modellen von Kurven über lokalen Körpern"
  • 02.11.17: Christian Steck, Auflösen von zahmen zyklischen Quotientensingularitäten auf gefaserten Flächen
  • 16.11.17: Jeroen Sijsling, Endomorphismen algebraischer Kurven
  • 23.11.17: Stefan Wewers, Duality and canonical sheaf
  • 30.11.17: Martin Djukanovic, Some remarks on split Jacobians
  • 07.12.17: Martin Djukanovic, Split Jacobians--continued
  • 25.01.18: Jeroen Hanselman, Gluing Curves along their 2-torsion
  • 08.02.18: Sabrina Kunzweiler, Ogg's formula
  • 15.02.18: Tudor Micu, Models, valuations, and Berkovich trees
  • 22.02.18: Stefan Wewers, What is Intersection theory?
  • 19.04.18: Jeroen Sijsling, Split Jacobians in genus 3: an inventory
  • 03.05.18: Jeroen Sijsling, Endomorphisms of kind of special Picard curves and Richelot isogenies
  • 24.05.18: Jeroen Sijsling, Endomorphisms and Divisors
  • 07.06.18: Martin Djukanovic, Families of (3, 3)-split Jacobians
  • 14.06.18: Jeroen Sijsling, Endomorphisms and Divisors II
  • 21.06.18: Sabrina Kunzweiler, Discriminants of hyperelliptic curves
  • 28.06.18: Jeroen Hanselman, Prym Varieties
  • 12.07.18: Stefan Wewers, Resolution of wild arithmetic quotient singularities
  • 25.10.18: Jeroen Sijsling, Eine Datenbank von Belyi-Morphismen
  • 08.11.18: Andreas Pieper, Abelsche Varietäten mit komplexer Multiplikation
  • 15.11.18: Sabrina Kunzweiler, Differentialformen auf hyperelliptischen Kurven mit semistabiler Reduktion
  • 22.11.18: Roman Kohls, Eine obere Schranke für den Führerexponenten einer hyperelliptischen Kurve
  • 29.11.18: Stefan Wewers, Rationale Singularitäten und die kanonische Garbe
  • 06.12.18: Jeroen Hanselman, Ein Algorithmus zum Verkleben einer Kurve vom Geschlecht 1 und einer Kurve vom Geschlecht 2 entlang ihrer 2-Torsion
  • 13.12.18: Matthew Bisatt, Root numbers of abelian varieties

  • 12.05.15: Michel Börner, L-Reihen hyperelliptischer Kurven
  • 26.05.15: Michael Eskin, Semistabile Reduktion von 3-Punkt-Überlagerungen
  • 02.06.15: Stefan Wewers, Semistabile Reduktion einer gewissen Kurve vom Geschlecht 4 über Q_2
  • 09.06.15: Irene Bouw, Konstruktion von Kurven mit schlechter Reduktion an vorgegebenen Stellen
  • 16.06.15: Stefan Wewers, Was ist etale Kohomologie?
  • 23.06.15: Christian Steck, Etale Morphismen
  • 14.07.15: Roman Kohls, Lokale Konstanten und elliptische Kurven
  • 03.08.15: Tudor Micu, A weak version of Beilinson's conjecture for Rankin-Selberg products of modular forms
  • 22.10.15: Stefan Wewers, Reguläre und semistabile Modelle elliptischer Kurven: ein Miniprojekt
  • 05.11.15 Jeroen Hanselman, Bounding the field extension necessary to find a semistable model of finite covers of curves
  • 12.11.15: Stefan Wewers, Arithmetische Flächen
  • 19.11.15: Christian Steck, Reguläre Modelle und Quotientensingularitäten
  • 26.11.15: Christian Steck, ______, Teil 2
  • 03.12.15: Tudor Micu, The Kodaira classification of  the special fiber of the minimal proper regular model of an elliptic curve
  • 10.12.15: Roman Kohls, Classification of automorphism groups of elliptic curves
  • 21.01.16: Christina Höhn, Faktorisieren mit elliptischen Kurven
  • 28.01.16: Tudor Micu, MacLane valuations
  • 04.02.16: Michel Börner, Picard curves and L-functions
  • 11.02.16: Angelos Koutsianas, Computing elliptic curves with good reduction outside S
  • 18.02.16: Christian Steck, Blow-ups
  • 25.02.16: Roman Kohls, The sign of the functional equation of the L-function of an elliptic curve
  • 27.10.16: Angelos Koutsianas, Lebesgue-Nagell equation and the modular Approach
  • 10.11.16: Dimitris Xatzakos (Bristol), Hyperbolic lattice point counting in conjugacy classes 
  • 17.11.16 November: Gunther Cornelissen (Utrecht), A combinatorial Li-Yau equality and rational points on curves
  • 24.11.16 November: Tudor Micu: Sheaves, cohomology and local systems
  • 01.12.16: Roman Kohls, Semistable reduction of curves and Weil-Deligne representations, I
  • 08.12.16 Dezember: Roman Kohls, Semistable reduction and Weil-Deligne representations, II
  • 12.12.16 Dezember: Davide Lombardo (Orsay), On division fields of CM abelian varieties