**Workshop "Towers of Function Fields", February 23-24, Sabanci University**

Place: Minerva Palace, Karakoy (How to reach Minerva Palace)

**Program**

February 23, Thursday

09.30 - 10.30 Michael Tsfasman (Institute for Information Transmission Problems and Independent University of Moscow)

* Sphere packings in large dimensions*

10.45 - 11.45 Henning Stichtenoth (Sabanci University)

*Explicit towers over non-prime finite fields I*

14.00 - 15.00 Arnaldo Garcia (IMPA, Rio de Janeiro)

*Explicit towers over non-prime finite fields II*

15.30 - 16.30 Stephane Ballet (Institut de Mathematiques de Luminy, Marseille)

*Class numbers in algebraic function fields defined over finite fields, and related problems*

16.45 - 17.45 Ignacio Cascudo (CWI, Amsterdam)

* Arithmetic codices*

19.00 Workshop dinner (details will be announced at the workshop)

February 24, Friday

09.30 - 10.30 Marc Perret (Institut de mathematique de Toulouse)

*Some remarks on the loci of decomposed and of singular points in recursive towers*

10.45 - 11.45 Kit-Ho Mak (University of Illinois at Urbana-Champaign)

* On lower bounds for the Ihara constants A(2) and A(3)*

14.00 - 15.00 Peter Beelen (DTU, Copenhagen)

* Explicit towers over non-prime finite fields III*

15.30 - 16.30 Alp Bassa (Sabanci University)

* Explicit towers over non-prime finite fields IV*

**Abstracts:**

**Ballet: Class numbers in algebraic function fields defined over finite fields, and related problems**

We give lower bounds for the class number in algebraic function fields defined over finite fields, which improve the Lachaud - Martin-Deschamps bounds. We give examples of towers of algebraic function fields having a large class number and we present related problems.

**Bassa, Beelen, Garcia, Stichtenoth: Explicit towers over non-prime finite fields I-IV**

A new family of towers over any non-prime finite field is presented. These towers are recursively defined, and they have a surprisingly large limit. For quadratic base fields, the Drinfeld-Vladut bound is attained; for cubic base fields, the Zink bound is attained. For all other finite base fields, the limit of the tower is much larger than the limits of previously known families of function fields. Hence we obtain a significant improvement of lower bounds for Ihara's quantity A(q), for all q=p^n with odd n > 3.

**Cascudo: Arithmetic codices**

In this talk I will present the notion of arithmetic codices, recently introduced in joint work with Ronald Cramer (CWI Amsterdam) and Chaoping Xing (NTU Singapore). Arithmetic codices have important applications in cryptography (in secure multiparty and two party computation) and in the complexity of multiplications in extension fields. Good towers of function fields provide the only known way of constructing 'asymptotically good families' of codices. In addition, I will motivate our introduction and study of what we have called the 'torsion limit' of a tower.

**Mak: On lower bounds for the Ihara constants A(2) and A(3)**

Let F_q be the finite field with q elements. The Ihara constant A(q) is an asymptotic measure of how many points a curve over F_q can have compared to its genus when the genus goes to infinity. In this talk, we will describe how to improve the best known lower bounds for A(2) and A(3). This is joint work with I. Duursma.

**Perret: Some remarks on the loci of decomposed and of singular points in recursive towers**

With the help of a graph closely related to Beelen's one and to intersection theory on surfaces, we will give some insights into the sets of decomposed points and of singular points in good recursive towers.

This workshop is supported by Sabanci University. The visit of A. Garcia to Sabanci University is supported by Tübitak.