The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q\ge 5$ elements, $A:={\mathbb{F}}_q[T]$ and $F:={\mathbb{F}}_q(T)$. Assume that $q$ is odd and take $|\cdot |$ to be the absolute value at $\infty$ that is normalized by $|T|=q$. Given a pair $w=(g_1,g_2)\in A^2$ with $g_2\ne 0$, consider the associated Drinfeld module ${\varphi }^w:A\to A\{\tau \}$ of rank $2$ defined by ${\varphi }_T^w=T+g_1\tau +g_2{\tau }^2$. Fix integers $c_1,c_2\ge 1$ and define $|w|:=max\{|g_1{|}^{\frac{1}{c_1}},|g_2{|}^{\frac{1}{c_2}}\}$. I show that when ordered by height, there is a positive density of pairs $w=(g_1,g_2)$, such that the $T$-adic Galois representation attached to ${\varphi }^w$ is surjective.

References in corpus (1)

• Surjectivity of Galois Representations in Rational Families of Abelian Varieties