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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)


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.)


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