Conference "Arithmetic Days"

5 - 6 April, 2012

Moscow, Russia

Organizers: Ivan Fesenko (University of Notingham), Igor Zhukov (St. Petersburg State University), Alexey Zykin (HSE, IITP).

Towers of function fields over finite fields and their sequences of zeta-functions

Alexey Zaytsev (Kaliningrad)

Thursday 5 April, 17:00-18:00

Let $\Tau=(T_n)_{n \ge 1}$ be a tower of function fields over a finite field. Then there exits a sequence of zeta functions (or L-polynomials) (Z_n(t))_{n\ge 1} (or (L_n(t))_{n\ge 1} ) attached to it. It is well know that L-polynomial L_n(t) is a polynomial of degree $2g$, where $g$ is genus of $T_n$. It turns out that If a tower is recursive then we can attach certain graph to it. Using this graph we can express the first few coefficients of L-polynomials of function field $T_n$ as functions in $n$ explicitly. We also formulate some conjecture on coefficients of L-polynomial for particular example of towers. At the end of the talk we rephrase Beelen's result in term of asymptotic zeta functions in order to find asymptotic zeta functions of known examples of good tower and suggest a new definition of asymptotic zeta function of a tower of function fields. This is joint work with A. Zykin.