HUJI-BGU Workshop in Arithmetic

The HUJI-BGU Workshop in Arithmetic

The HUJI-BGU Workshop in Arithmetic meets twice a semester, alternating locations, starting from the year 5780. All are welcome.
The workshop is currently organized by Ari Shnidman (HUJI) and Daniel Disegni (BGU).

1st meeting: Monday December 16th, 2019 - HUJI

The day will be dedicated to the memory of John Tate, who passed away on October 16th, 2019. Tate's ideas "have dominated the development of arithmetic algebraic geometry" (from the 2003 Wolf Prize citation).

Schedule

10.30 - 11.00 (Manchester, Faculty lounge) Welcome
11.00 - 11:45 (Ross 63) Shay Ben Moshe (HUJI), Tate elliptic curves and p-adic uniformization
12.00 - 13.00 (Ross 63) Ariel Weiss (HUJI), l-adic Tate modules and good reduction
13.00 - 14.00 (Manchester, Faculty lounge) Lunch
14.00 - 15:30 (Ross 70) David Jarossay (BGU), p-adic Tate modules and good reduction

Abstracts of talks

Shay Ben Moshe (11:00-11:45)
We start by considering elliptic curves over the complex numbers. They admit analytic uniformization, namely they are given as a quotient of the complex plane by a lattice. This allows us to draw many conclusions about them, for example understand their torsion. We then try to mimic this in the p-adic situation. We describe a family of p-adic elliptic curves, the Tate elliptic curves, and sketch a proof that they admit p-adic uniformization. We apply this to give some structural results about them, such as describing the Galois action on their Tate module explicitly.

Ariel Weiss (12:00-13:00)
If E is an elliptic curve over a number field K, then, by a deep conjecture of Tate (now a theorem of Faltings), E is determined up to isogeny by its l-adic Tate module. In particular, any isogeny invariant of E should be encoded in its Tate module. In this expository talk, we study the criterion of N\'eron--Ogg--Shafarevich, which states that E has good reduction at a prime of K that is coprime to l if and only if its l-adic Tate module is unramified. We will begin by reviewing the arithmetic theory of elliptic curves and their Tate modules, before stating the N\'eron--Ogg--Shafarevich criterion, sketching its proof and giving several examples and applications.

David Jarossay (14.00-15.30)

This talk will be about a paper by Coleman and Iovita, which shows two descriptions of a filtered Frobenius monodromy module attached to the H^1_dR of elliptic curves (more generally Abelian varieties) with split semistable reduction over Q_p. The equivalence between the two descriptions is induced by Colmez's p-adic integration pairing. A corollary is that, for A an Abelian variety over a local field, the p-adic Tate module of A is crystalline if and only if A has good reduction.

Dates to be saved

2nd meeting: January 13th, 2020 - BGU

Locations

HUJI: Einstein Institute of Mathematics, Hebrew University Giv'at Ram Campus, Jerusalem.
BGU: Deichmann Building for Mathematics, Ben-Gurion University of the Negev, Be'er Sheva.

Directions: you may use Google Maps or Moovit. (Please note that Google is sometimes optimistic about the travel time of local buses.)