Curves and L-functions

ICTP Trieste, Italy, 28 August - 8 September 2017
Sponsored by ICTP and EPSRC


Workshop photo (click to enlarge)

Week 1: PhD Summer school "Curves, L-functions, and Galois Representations"
Giambiagi Lecture Room, Adriatico Guest House

Lecture courses (4 lectures each)
Galois representations by Tim and Vladimir Dokchitser TD/VD
Lecture 1+2, Exc 1, Exc 2, Lecture 3, Exc 3, Lecture 4, Exc 4.
L-functions and BSD by Adam Morgan AM
Selmer groups and descent by Michael Stoll MS
Modularity by Jack Thorne JT
Monday Aug 28
09:00-11:10registration
11:10-11:20official welcome (Fernando Rodriguez Villegas)
11:20-12:20AM 1 Review of arithmetic of elliptic curves
lunch break
14:00-15:00VD 1 Local l-adic representations and elliptic curves
15:00-18:00work on exercises and projects, coffee break 16:15-16:45
18:00-19:00exercise presentations
19:30welcome reception
Tuesday Aug 29
09:00-10:00AM 2 Zeta functions, L-functions and BSD
10:10-11:10VD 2 l-adic and Weil/Weil-Deligne representations
11:20-12:20MS 1 Selmer sets
lunch break
14:00-15:00JT 1 Modular parametrisation of elliptic curves
15:00-18:00work on exercises and projects, coffee break 16:15-16:45
18:00-19:00exercise presentations
Wednesday Aug 30
09:00-10:00JT 2 Modularity over number fields
10:05-11:05MS 2 Selmer groups
11:25-12:25AM 3 BSD over functions fields
lunch break
14:00-15:00TD 3 Compatible systems of l-adic representations
15:00-18:00work on exercises and projects, coffee break 16:15-16:45
18:00-19:00exercise presentations
ThursdayAug 31
09:00-10:00MS 3 Selmer groups of elliptic curves
10:10-11:10JT 3 Potential modularity
lunch break
free afternoon
Friday Sep 1
09:00-10:00TD 4 H1 of curves
10:05-11:05AM 4 The BSD story in higher dimensions
11:25-12:25MS 4 Selmer sets of hyperelliptic curves
lunch break
14:00-15:00JT 4 Automorphic representations and abelian varieties
15:00-18:00work on exercises and projects, coffee break 16:15-16:45
18:00-19:00exercise presentations

Week 2: Workshop "Arithmetic of Hyperelliptic Curves"
Kastler Lecture Room, Adriatico Guest House

Monday Sep 4
09:00-09:50Jennifer Balakrishnan - Explicit Coleman integration for curves
10:00-registration
+ project presentations from the summer school
(workshop participants are welcome if they are there)
10:00-10:20project presentation: Elliptic curves with rational Z2xZ8-torsion
10:40-11:00project presentation: Generic rank of a family of elliptic curves
11:10-11:30project presentation: Szpiro's constants, the hyperelliptic case
11:40-12:00project presentation: Detecting good and bad reduction of curves
lunch break
14:50official welcome (Fernando Rodriguez Villegas)
15:00-15:50Qing Liu - New points on algebraic curves
coffee break
16:30-17:20Jennifer Park - A heuristic for boundedness of ranks of elliptic curves
Tuesday Sep 5
09:00-09:20Vladimir Dokchitser - Arithmetic of hyperelliptic curves over local fields
09:30-09:50Celine Maistret - Parity of ranks of abelian surfaces
10:00-10:20Alex Betts - Computing Tamagawa numbers of hyperelliptic curves
coffee break
11:00-11:50David Holmes - Arakelov geometry of hyperelliptic curves
lunch break
14:30-15:20Wei Ho - Odd degree number fields with odd class number
15:30-15:50Matthew Bisatt - Root Numbers of Abelian Varieties
coffee break
16:30-16:50Samuele Anni - The inverse Galois problem for symplectic groups
17:00-17:20Tim Dokchitser - Models of curves
19:00reception
Wednesday Sep 6
09:00-09:20Julie Desjardins - Density of rational points on rational elliptic surfaces
09:30-09:50Emmanuel Lecouturier - Higher Eisenstein elements in weight 2
 and prime level
10:00-10:20Gonzalo Tornaria - Explicit modularity for typical genus 2 curves
coffee break
11:00-11:50John Voight - Explicit modularity for atypical genus 2 curves
lunch break
free afternoon
Thursday Sep 7
09:00-09:20Adam Morgan - Parity of Selmer ranks in quadratic twist families
09:30-09:50Tom Fisher - Visualizing elements of order 7 in
 the Tate-Shafarevich group of an elliptic curve
10:00-10:20Raymond van Bommel - Numerical verification of
  the Birch and Swinnerton-Dyer conjecture
coffee break
11:00-11:50Steffen Muller - Canonical heights on Jacobians of curves of genus two
lunch break
14:30-15:20David Roberts - An inverse Hodge problem and some solutions
 from hypergeometric motives
15:30-15:50Bartosz Naskrecki - Hypergeometric motives of low degrees
 and their connection to elliptic and hyperelliptic curves
coffee break
16:30-16:50Davide Lombardo - Computing twists of hyperelliptic curves
17:00-17:20Fernando Rodriguez Villegas - Hypergeometric hyperelliptic curves
Friday Sep 8
09:00-09:20Netan Dogra - Rational points on hyperelliptic curves
 via the Chabauty-Kim method
09:30-09:50Michael Stoll - Arithmetic of hyperelliptic curves in Magma
10:00-10:20Jeroen Sijsling - Databases of curves
coffee break
11:00-11:50Kiran Kedlaya - L-functions via deformations: from hyperelliptic
 curves to hypergeometric motives + Jupyter notebook
lunch break
14:00-14:50Andrew Sutherland - Computing L-series of hyperelliptic curves

Participants

List of participants booklet by Celine Maistret


If you have questions on the scientific side, please contact one of the organisers
Tim Dokchitser, Vladimir Dokchitser, Fernando Rodriguez Villegas.
For organisational questions, please contact smr3175@ictp.it (School+Workshop) or smr3144@ictp.it (Workshop only).