Dr. Timo Keller

Contact

Lehrstuhl für Computeralgebra
Mathematisches Institut
Universität Bayreuth
Universitätsstraße 30
95440 Bayreuth, Germany

Office: building NW II, room 3.2.02.737
Phone: +49 (0)921 55 3384
E-Mail: public GPG key
Workgroup: Prof. Dr. Michael Stoll
Chair: Mathematik II (computer algebra)
Office hours: by appointment

Research

Arithmetic geometry, especially arithmetic and computational aspects of Jacobian and abelian varieties over arithmetic fields, their rational points, L-functions, and cohomology

• over number fields:
  • Birch–Swinnerton-Dyer conjecture, explicit methods for modular abelian varieties, their Galois representations, Selmer and Shafarevich–Tate groups, Heegner points, Euler systems, p-adic L-functions, Iwasawa theory (funded by the DFG)
  • Chabauty method (both classical and non-abelian) and its applications to rational points on modular and Shimura curves
• in positive characteristic: theoretical aspects of abelian varieties and schemes over higher dimensional bases over finitely generated fields, their L-functions, Shafarevich–Tate groups, Brauer groups, étale, flat and p-adic cohomology theories

Publications

Published

• On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields ^DOI
  Doc. Math. 24 (2019), 915–993 abstract
  We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic p. We prove the prime-to-p part conditionally on the finiteness of the p-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-p part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-p part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of \Acal[p] or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.
• A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields ^{DOI PDF}
  Abh. Math. Semin. Univ. Hambg. (2018) 88(2), 289–295 abstract
  In this note, we prove a duality theorem for the Tate-Shafarevich group of a finite discrete Galois module over the function field $K$ of a curve over an algebraically closed field: There is a perfect duality of finite groups $Sha^1(K,F) \times Sha^1(K,F') \to Q/Z$ for $F$ a finite étale Galois module on $K$ of order invertible in $K$ and with $F' = \Hom(F,Q/Z(1))$. Furthermore, we prove that $H^1(K,G) = 0$ for $G$ a simply connected, quasisplit semisimple group over $K$ not of type $E_8$.
• On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields ^{DOI PDF}
  manuscripta math. (2016) 150(1–2), 211–245 abstract
  We study analogues for the Tate-Shafarevich group for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.

Accepted

• Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces, 2021,
  with Michael Stoll
  accepted for publication in Comptes Rendus de l'Académie des Sciences - Série Mathématique abstract
  Let $X$ be one of the $28$ Atkin-Lehner quotients of a curve $X_0(N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich-Tate group $\Sha(J/\Q)$ is trivial. This verifies the strong BSD~conjecture for $J$.
• On the p-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields ^arXiv, 2021,
  accepted for publication in Journal de Théorie des Nombres de Bordeaux abstract
  We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich and the Brauer group in this situation.

Preprints

• Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings, 2022,
  with Nikola Adžaga, Shiva Chidambaram, and Oana Padurariu
  submitted abstract
  We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q-rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the Q-rational points on all of their modular coverings.
• Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves, 2021,
  submitted abstract
  Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over finitely generated fields to the the specialization theorem for N\'eron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after an alteration of the base surface $S$, for almost all curves $C$ on $S$ the Mordell-Rank of $A$ over $S$ stays the same when restricting to $C$.
• Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6, 2021,
  with Nikola Adžaga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, and Boya Wen
  submitted abstract
  We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.

Non-refereed publications

• Exact verification of the strong Birch–Swinnerton-Dyer conjecture for some absolutely simple modular abelian surfaces, 2022,
  with Michael Stoll
  Computeralgebra-Rundbrief Nr. 70

Activities (seminars, conferences, and talks)

Conference »Quadratic Chabauty for Atkin-Lehner quotients of modular curves and Shimura curves«

WiN-UBT Conference Grant 2021, Universität Bayreuth
Homepage
August 21 – 27, 2022

See also: Modular Curves Workshop at MIT

Oberseminar Arithmetische Geometrie

Thursday, 10:15–12:00 (Homepage)

Kleine AG/Bayerische Arbeitsgemeinschaft

Kleine/Bayerische Arbeitsgemeinschaft »Algebraische Geometrie und Zahlentheorie«

Recent talks

• Rational Points 2022, March 28, 2022 slides (the slides from the previous talk contain additional explanation!)
  Verification of the strong BSD conjecture for abelian surfaces over Q
• Boston University, number theory seminar, March 14, 2022 slides
  Exact verification of the strong BSD conjecture for some absolutely simple modular abelian surfaces
• Tagung der Fachgruppe Computeralgebra 2022, March 9, 2022
  
  Exakte Verifizierung der starken BSD-Vermutung für einige absolut einfache abelsche Flächen
  ausgezeichnet mit dem »Preis für den besten Vortrag einer Nachwuchswissenschaftler|in«
  (awarded with the prize for the best talk of a junior researcher)

Teaching

Wintersemester 2021/2022

Einführung in die Theorie der Modulformen und Modulkurven
Skript und Übungsaufgaben

Wintersemester 2018/2019

Brauer groups in arithmetic geometry