Chirag Singhal : The field of definition of a Drinfeld module (I)

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

March 10, 2023, noon

In

612 SEO

Abstract

Let $A = \mathbb{F}_q[T]$ be the ring of polynomials in the indeterminate $T$ and with coefficients in the finite field $\mathbb{F}_q$ with $q$ elements, and let $K$ be an $A$-field. Let $\phi$ be a Drinfeld $A$-module over $K$, of rank $r \geq 2$. Denote by $j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi)$ the family of $j$-invariants associated to $\phi$ by I.Y. Potemine. Motivated by an argument of G. Shimura used in the context of abelian varieties, we show that $\mathbb{F}_q(T, j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi))$ is a field of definition for $\phi$.