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Visible evidence in the Birch and SwinnertonDyer Conjecture for modular abelian varieties of analytic rank zero
, 2004
"... This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic ..."
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Cited by 31 (20 self)
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This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of �(Af). We find that there are at least 168 such Af for which the Birch and SwinnertonDyer conjecture implies that �(Af) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of �(Af) really divides # �(Af) by constructing nontrivial elements of �(Af) 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 ShafarevichTate groups of elliptic curves.
J1(p) Has Connected Fibers
 DOCUMENTA MATH.
, 2002
"... We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup H ⊆ (Z/pZ) × /{±1} the map XH(p) = X1(p)/H → X0(p) induces an injection Φ(JH(p)) → Φ(J0(p)) on mod p component groups, with image equal to that of H in Φ(J0(p)) when the l ..."
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Cited by 19 (2 self)
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We study resolution of tame cyclic quotient singularities on arithmetic surfaces, and use it to prove that for any subgroup H ⊆ (Z/pZ) × /{±1} the map XH(p) = X1(p)/H → X0(p) induces an injection Φ(JH(p)) → Φ(J0(p)) on mod p component groups, with image equal to that of H in Φ(J0(p)) when the latter is viewed as a quotient of the cyclic group (Z/pZ) × /{±1}. In particular, Φ(JH(p)) is always Eisenstein in the sense of Mazur and Ribet, and Φ(J1(p)) is trivial: that is, J1(p) has connected fibers. We also compute tables of
Component groups of purely toric quotients
"... Suppose : J! A is an optimal quotient of abelian varieties over a padic ¯eld, optimal in the sense that ker() is connected. Assume that J is equipped with a symmetric principal polarization µ (e.g., any Jacobian of a curve has such a polarization), that J has semistable reduction, and that A has ..."
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Cited by 15 (5 self)
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Suppose : J! A is an optimal quotient of abelian varieties over a padic ¯eld, optimal in the sense that ker() is connected. Assume that J is equipped with a symmetric principal polarization µ (e.g., any Jacobian of a curve has such a polarization), that J has semistable reduction, and that A has purely toric reduction. In this paper, we express the group of connected components of the N¶eron model of A in terms of the monodromy pairing on the character group of the torus associated to J. We apply our results in the case when A is an optimal quotient of the modular Jacobian J0(N). For each prime p that exactly divides N, we obtain an algorithm to compute the component group of A at p. 1
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
The Modular Degree, Congruence Primes, and Multiplicity One
"... 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) ..."
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Cited by 13 (9 self)
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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.
The Manin constant
"... The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and SwinnertonDyer. 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 t ..."
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Cited by 11 (6 self)
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The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and SwinnertonDyer. 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.
Euler systems and Jochnowitz congruences
 Amer. J. Math
, 1999
"... This article relates the GrossZagier formula with a simpler formula of Gross for special values of Lseries, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign −1 and 1 respectively, ..."
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Cited by 8 (4 self)
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This article relates the GrossZagier formula with a simpler formula of Gross for special values of Lseries, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign −1 and 1 respectively, and an imaginary quadratic field K satisfying certain auxiliary conditions, the main result gives a congruence between the algebraic part of L ′(f/K, 1) (expressed in terms of Heegner points) and the algebraic part of the special value L(g/K, 1). Congruences of this type were anticipated by Jochnowitz, and for this reason are referred to as “Jochnowitz congruences”. 1
A visible factor of the special Lvalue
, 2008
"... Let A be a quotient of J0(N) associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. We give a formula for the ratio of the special Lvalue to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this for ..."
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Cited by 7 (4 self)
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Let A be a quotient of J0(N) associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. We give a formula for the ratio of the special Lvalue to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is nontrivial in general and is related to certain congruences of f with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and SwinnertonDyer conjecture on rank), if an odd prime q divides this factor, then q divides either the order of the ShafarevichTate group or the order of a component group of A. Suppose p is an odd prime such that p2 does not divide N, p does not divide the order of the rational torsion subgroup of A, and f is congruent modulo a prime ideal over p to an eigenform whose associated abelian variety has positive MordellWeil rank. Then we show that p divides the factor mentioned above; in particular, p divides the numerator of the ratio of the special Lvalue to the real period of A. Both of these results are as implied by the second part of the Birch and SwinnertonDyer conjecture, and thus provide theoretical evidence towards the conjecture. 1