The Manin constant The Manin constant Amod Agashe
Kenneth Ribet
William A. Stein1 Agashe, Ribet, Stein {\caps 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 $ J_0(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$ .


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





William Stein 2006-06-25