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The Manin constant for 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.
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