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.
Abstract. This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A f that are optimal quotients of J 0 (N ) attached to newforms. We prove theorems about the ratio L(A f , 1)/Ω A f , develop tools for computing with A f , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties A f of level ≤ 2333. Over half of these A f have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of (A f ). We find that there are at least 168 such A f for which the Birch and Swinnerton-Dyer conjecture implies that (A f ) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of (A f ) really divides # (A f ) by constructing nontrivial elements of (A f ) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.
Abstract.The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.
Elliptic curves with a known number of points over a given prime field Fn are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H D (X) for a fundamental discriminant D. The usual way is to compute H D (X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute H D (X) modulo n directly from the knowledge of H D (X) modulo enough small primes. Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods.
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