Algebraicity modulo $p$ of G functions, hypergeometric series and strong Frobenius structure

Abstract: B. Dwork in his work about zeta function of a hypersurface over finite fields introduced the notion of strong Frobenius structure. In this talk we are going to take up this notion for the study of algebraicity modulo $p$ of Siegel G functions, where $p$ is a prime number. Firstly, we are going to see that if $f(t)$ is a power series (or Siegel G function) with coefficients in the ring of integers $\mathbb{Z}$ and if $f(t)$ is solution of a differential operator $L$ having strong Frobenius structure for $p$ of period $h$, then the reduction of $f$ modulo $p$ is algebraic over $\mathbb{F}_p(t)$ and its algebraicity degree is bounded by $p^{n^2h}$, where $n$ is the order of L and $\mathbb{F}_p$ is the field of $p$ elements. Secondly, we are going to show that, under reasonable hypotheses, rigid differential operators have a strong Frobenius structure for almost every prime number $p$. Finally, we are going to illustrate our results with several examples coming of hypergeometric series of type ${}_nF_n-1$.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar