Dr. Timo Keller

Contact information

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

Zimmer: Gebäude NW II, Raum 3.2.02.737
Telefon: +49-(0)921-55 3384
E-Mail: "Vorname""Punkt""Nachname" "at" uni"Minus"bayreuth"Punkt"de public GPG key
Arbeitsgruppe: Prof. Dr. Michael Stoll
Lehrstuhl: Mathematik II (Computeralgebra, Algorithmische Arithmetische Geometrie)
Sprechstunde: nach Vereinbarung


Research interests

Arithmetic Geometry, especially arithmetic of Abelian schemes, their L-functions and étale, crystalline and flat cohomology, Brauer groups, rational points on varieties over arithmetic fields, especially hyperelliptic curves (also with computational aspects)

Publications

Published

The published articles may differ from the preprints.
  1. On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields 79 pp., accepted for publication in Doc. Math. 24 (2019), 915–993
  2. On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields, manuscripta math. (2016) 150(1–2), 211–245. DOI: 10.1007/s00229-015-0803-1
  3. A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields, Abh. Math. Semin. Univ. Hambg. (2018) 88(2), 289–295. DOI: 10.1007/s12188-018-0196-7

Preprints

  1. Finiteness properties for flat cohomology of varieties over finite fields, 11 pp.
  2. K-theory mod ℓ, transfer of quadratic forms and Stiefel-Whitney invariants, 19 pp.

Theses

  1. PhD Thesis (advisor: Uwe Jannsen, Universität Regensburg): The conjecture of Birch and Swinnerton-Dyer for Jacobians of constant curves over higher dimensional bases over finite fields, see also Extended Edition: On the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields
  2. Diploma Thesis (advisor: Uwe Jannsen, Universität Regensburg): Kohomologieoperationen in Milnor-K-Theorie mod ℓ, Transfer quadratischer Formen und Stiefel-Whitney-Invarianten

Teaching

Wintersemester 2018/2019

Lecture on Brauer groups in arithmetic geometry with exercises, NW II room 3.2.02.740 Monday, Tuesday and Thursday 13:15 s.t.

Topics covered

  1. Crash course in Galois cohomology and Brauer groups of fields
  2. Crash course in étale cohomology and (cohomological) Brauer groups of schemes
  3. Brauer-Manin obstruction, theorems, examples, conjectures

Lectures

  1. Profinite groups, discrete G-modules and infinite Galois theory
  2. Cohomology of profinite groups
  3. Galois cohomology and Brauer groups of fields
  4. Examples of Brauer groups of fields
  5. Milnor K-theory and the theorem of Merkurjev-Suslin
  6. Motivation for étale cohomology
  7. Étale cohomology
  8. Brauer groups of schemes I: definition
  9. Brauer groups of schemes II: basic properties
  10. Brauer groups of schemes III: Brauer-Severi varieties and Hasse principle
  11. Brauer-Manin obstruction I: definition and basic properties (statements)
  12. Brauer-Manin obstruction II: definition and basic properties (proofs)
  13. Brauer-Manin obstruction III: functoriality and an example of a non-rational Châtelet surface
  14. Brauer-Manin obstruction IV: an example of a 2-torsion element of the Tate-Shafarevich group of an elliptic curve
  15. Brauer-Manin obstruction V: torsors under abelian varieties
References

Exercise sheets

  1. Profinite groups and discrete G-modules
  2. Group and Galois cohomology
  3. Brauer groups of fields
  4. Cohomology of schemes
  5. Zeta functions and the Weil conjectures
  6. Examples for Brauer groups of schemes
  7. Everywhere locally soluble varieties
  8. Preparations for the Brauer-Manin obstruction
  9. Basic properties of the Brauer-Manin obstruction
  10. The Brauer-Manin set of a del Pezzo surface of degree 4

Activities

Oberseminar Arithmetische Geometrie

Friday, 12:15–13:45 in S82

Kleine AG

Kleine Arbeitsgemeinschaft »Algebraische Geometrie und Zahlentheorie«
Next topic: Serre's Modularity Conjecture, Bonn, May 4, 2019

Bayerische Arbeitsgemeinschaft

Bayerische Arbeitsgemeinschaft »Algebraische Geometrie und Zahlentheorie«
Next topic: topological modular forms, autumn 2019

Rational Points 2019

Workshop, Franken-Akademie Schloss Schney, July 14–20, 2019

Last modified: July 6, 2019
Datenschutzerklärung