Kenneth Ribet

William A. Stein

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
, and conjecture that the
generalized Manin constant is
for newform quotients.
Finally an appendix by John Cremona discusses computation
of the Manin constant for all elliptic
curves of conductor up to
.

Amod Agashe
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Kenneth A. Ribet
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William A. Stein
Department of Mathematics
Harvard University
Cambridge, MA 02138
`[email protected]`

- Introduction
- Optimal Elliptic Curve Quotients
- Quotients of arbitrary dimension
- Generalization to quotients of arbitrary dimension
- Motivation: connection with the conjecture of Birch and Swinnerton-Dyer
- Results and a conjecture
- Examples of nontrivial Manin constants

- Proofs of some of the Theorems

- Appendix by J. Cremona: Verifying that
- Bibliography
- About this document ...

William Stein 2006-06-25