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 Winter Semester 2023/24 - Thursday, 4 p.m., Room E60

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

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 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