Oberseminar Algebra & Number Theory, Ulm

Expected talks - Scheduled to take place: Thursdays, 4 p.m. - 6 p.m. (virtual seminar)

  • March 18, 2021 / MSc Andreas Pieper (Ulm)
  • March 25, 2021 / MSc Tim Evink (Ulm)
  • April 1, 2021 / MSc Robert Slob (Ulm)
  • April 22, 2021 / MSc Bogdan Dina (Ulm)
  • April 29, 2021 / Jun.-Prof. Dr. Jeroen Sijsling (Ulm)
  • May 6, 2021 / Prof. Dr. Stefan Wewers (Ulm)
  • May 20, 2021 / Prof. Dr. Stefan Wewers (Ulm)
  • May 27, 2021 / MSc Ole Ossen (Ulm)
  • July 15, 2021 / Jun.-Prof. Dr. Jeroen Sijsling (Ulm)

External Auditors

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.


F. Oort showed that the moduli space of principally polarized supersingular abelian surfaces is a union of rational curves. This is proven by showing that every principally polarized supersingular abelian surface is the jacobian of a fibre of one of the families of genus 2 curves $\pi: \mathcal{C}\rightarrow \mathbb{P}^1$ constructed by L. Moret-Bailly. We present an algorithm that makes this construction effective: Given a point $x\in \mathbb{P}^1$ we compute a hyperelliptic model of the fibre $\pi^{-1}(x)$. The algorithm uses Mumford's theory of theta groups to compute
quotients by the group scheme $\alpha_p$.


The talk is based on a paper recently submitted for publication by G.J. vd Heiden, J. Top and myself, which in turn is based on my masters' thesis (which in turn is based on vd Heidens masters' thesis...).
We consider for a prime p>3 the hyperelliptic curve C defined over the rationals by the equation y²=x(x²-p²)(x²-4p²). The talk can be divided into three parts:

- Rédei symbols, which satisfy a reciprocity law based on the product formula for quadratic Hilbert symbols in quadratic number fields. 
- Rankbounds on J = Jac(C) over Q obtained by a 2-Selmergroup computation, which for certain classes of primes p of positive Dirichlet density can be improved by the use of Rédei symbols. 
- Determination of the rational points of C for some primes p by use of 'baby-Elliptic Curve Chabauty' (baby because in our case, which is working over Q, we can completely avoid the general machinery of the method). For example, I will prove that for p = 5 mod 24 (even though J(Q) has rank 1 if Sha(J/Q) is finite), the rational points C(Q) consist of the Weierstrass points, unless p=5.

In the first part of the talk, I will give a gentle introduction into the subject of divisibility sequences over the
rational numbers and discuss the notion of a primitive divisor/Zsigmondy bound. I then explain how these notions can be extended to number fields and function fields, and how to obtain a divisibility sequence from a non-torsion point on an elliptic curve over any of these fields. I will also provide some nice examples. 

In the second part of the talk, we discuss the typical methods that are used to prove the existence of a Zsigmondy bound for a divisibility sequence obtained from a non-torsion point on an elliptic curve E over a function field K (but the same approach works for number fields). Let P be this non-torsion point in E(K), and suppose Q is a torsion point in E(K). We can also associate a sequence of divisors {D_{nP+Q}} on K to the sequence of points {nP+Q}.

In my preprint, I have proved the existence of a Zsigmondy bound for this sequence {D_{nP+Q}} (under some minor conditions), extending the analogous result of Verzobio over number fields. I end my talk by providing the crucial ideas I used to be able to apply the existing methods to my case {D_{nP+Q}}.

Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous.

Joint work with Sorina Ionica considers this question in genus 3 by usingthe catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions and results. Furthermore, it gives a classification of CM types and their reflex fields for this particular genus.

Resuming the theme of last week, we consider the Galois action on the set of Jacobians with given CM, and use the Shimura class group to get some grip on the size of the corresponding orbits. This leads to algorithms that given a maximal CM order ZZ_K determine a small set of period matrices that represent all these Galois orbits. Using them and evaluating theta values by work of Labrande gives the results mentioned in the first talk. We also discuss how (heuristically) to determine defining equations for the corresponding CM curves.

I will report on my long term efford to make the computation of the semistable reduction of curves over p-adic fields effective and practical. I will focus on some particular cases, and on the use of methods from nonarchimedian analytic geometry to achieve this goal.

In this talk I will try to explain how the difficulty of certain search problems
on "isogeny graphs" can be used for cryptographic purposes.

I will focus on the isogeny graphs of "supersingular elliptic curves", avoiding all technicalities from number theory and algebraic geometry to make
the talk understandable and enjoyable for mathematicians from any area. Members of the Institute of Optimzation and Operations Research are particularly welcome. 

In his talk from May 6, Stefan Wewers explained how to use the language of nonarchimedean analytic geometry for explicitly determining the semistable reduction of curves. I will expand on this and explain a method which allows us to determine the semistable reduction of many plane quartic curves (at p=3), with emphasis on intuition and concrete examples.

Former talks

  • 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