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

be the polynomial ring over

, and

be the field of fractions of

. Let

be a Drinfeld

-module of rank

over

. For all but finitely many primes

⊲

, one can reduce

modulo

to obtain a Drinfeld

-module

of rank

over

. The endomorphism ring

is an order in an imaginary field extension

of

of degree

. Let

be the integral closure of

in

, and let

be the Frobenius endomorphism of

.

Then we have the inclusion of orders

in

. We prove that if

, then for arbitrary non-zero ideals

of

there are infinitely many

such that

divides the index

and

divides the index

.

We show that the index

is related to a reciprocity law for the extensions of

arising from the division points of

. In the rank

case we describe an algorithm for computing the orders

, and give some computational data. (This is a joint work with Sumita Garai.)