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Pure and Applied Mathematics Quarterly Volume 2, Number 2

Summary: Pure and Applied Mathematics Quarterly
Volume 2, Number 2
(Special Issue: In honor of
John H. Coates, Part 2 of 2)
617--636, 2006
The Manin Constant
Amod Agashe, Kenneth Ribet and William A. Stein
Abstract: The Manin constant of an elliptic curve is an invariant that
arises in connection with the conjecture of Birch and Swinnerton-Dyer. One
conjectures that this constant is 1; it is known to be an integer. After
surveying what is known about the Manin constant, we establish a new
sufficient condition that ensures that the Manin constant is an odd integer.
Next, we generalize the notion of the Manin constant to certain abelian
variety quotients of the Jacobians of modular curves; these quotients are
attached to ideals of Hecke algebras. We also generalize many of the results
for elliptic curves to quotients of the new part of J0(N), and conjecture
that the generalized Manin constant is 1 for newform quotients. Finally an
appendix by John Cremona discusses computation of the Manin constant
for all elliptic curves of conductor up to 130000.
1. Introduction


Source: Agashe, Amod - Department of Mathematics, Florida State University
Stein, William - Department of Mathematics, University of Washington at Seattle


Collections: Mathematics