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．Seminars

	NCTS Seminar on Number Theory

13:30 - 14:30, July 4, 2018 (Wednesday)
Lecture Room B, 4th Floor, The 3rd General Building, NTHU
(清華大學綜合三館 4樓B演講室)
Endomorphism Rings of Reductions of Drinfeld Modules
Mihran Papikian (Pennsylvania State University)

Abstract:

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$ , and $F$ be the field of fractions of $A$ . Let $\phi$ be a Drinfeld $A$ -module of rank $r\geq 2$ over $F$ . For all but finitely many primes $\mathfrak{p}$ ⊲ $A$ , one can reduce $\phi$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$ -module $\phi\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$ . The endomorphism ring $\mathcal{E}_\mathfrak{p}=\text{End}_{\mathbb{F}_\mathfrak{p}}(\phi\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$ . Let $\mathcal{O}_\mathfrak{p}$ be the integral closure of $A$ in $K$ , and let $\pi_\mathfrak{p}\in \mathcal{E}_\mathfrak{p}$ be the Frobenius endomorphism of $\phi\otimes\mathbb{F}_\mathfrak{p}$ .

Then we have the inclusion of orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ in $K$ . We prove that if $\mathrm{End}_{F^\mathrm{alg}}(\phi)=A$ , then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$ there are infinitely many $\mathfrak{p}$

such that $\mathfrak{n}$ divides the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ and $\mathfrak{m}$ divides the index $\chi(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$ .

We show that the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ is related to a reciprocity law for the extensions of $F$ arising from the division points of $\phi$ . In the rank $r=2$ case we describe an algorithm for computing the orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ , and give some computational data. (This is a joint work with Sumita Garai.)