Conference "Zeta Functions"

Order of 0 of a L-function of twisted Carlitz modules

Dimitry Logachev (Caracas, Venezuela)

Monday, November 19, 16.30 - 17:30

Video: [mp4]

Abstract

The $L$-function under consideration is the simplest type of the $L$-function $L(\phi, U)$ of a Drinfeld module $\phi$ of rank 1 over $F_q[\theta]$. These Drinfeld modules are the twisted Carlitz modules. Its analytic rank is the order of 0 of $L(\phi, U)$ at $U=1$. A version of the Lefschetz trace formula gives us an explicit expression of $L(\phi, U)$. We get immediately that there exists a coset of index $(q-1)^2$ in $\Hom(\Gal(F_q(\theta)), Z/(q-1))$ --- the set of all twists --- such that the analytic rank of corresponding Drinfeld modules is $\ge 1$. We present results of computer calculations of rank for the case $q=3$ which show that there are examples of twists whose rank is $\le 3$. It is unknown whether there exist examples of higher rank, and what is the distribution of twists of rank 2, 3.