Results 1  10
of
13
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 ..."
Abstract

Cited by 24 (16 self)
 Add to MetaCart
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.
Torsion points on elliptic curves defined over quadratic fields
 Nagoya Mathematical Journal
, 1988
"... Let He a quadratic field and E an elliptic curve defined over k. The authors [8,12, 13] [23] discussed the ^rational points on E of prime power order. For a prime number p, let n = n{k,p) be the least non negative integer such that E p~(k) = U ker (p»: E> E)(k) c E pn for all elliptic curves E def ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
Let He a quadratic field and E an elliptic curve defined over k. The authors [8,12, 13] [23] discussed the ^rational points on E of prime power order. For a prime number p, let n = n{k,p) be the least non negative integer such that E p~(k) = U ker (p»: E> E)(k) c E pn for all elliptic curves E defined over a quadratic field k ([15]). For prime
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 ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
. 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...
Heegner divisors, Lfunctions and harmonic weak Maass forms, preprint
"... Abstract. Recent works, mostly related to Ramanujan’s mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as “generating functions ” for central ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Abstract. Recent works, mostly related to Ramanujan’s mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as “generating functions ” for central values and derivatives of quadratic twists of weight 2 modular Lfunctions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of Lfunctions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier. 1.
Algebraic hypergeometric transformations of modular origin
, 2006
"... Abstract. It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 � N � 7 the Nfold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transfor ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 � N � 7 the Nfold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic–geometric mean iterations, but the N = 5, 6,7 transformations are new. In the final two the change of variables is not parametrized by rational functions, since X0(6), X0(7) are of genus 1. Since their quotients X + 0 (6), X+ 0 (7) under the Fricke involution (an Atkin–Lehner involution) are of genus 0, the parametrization is by twovalued algebraic functions. The resulting hypergeometric transformations are closely related to the twovalued modular equations of Fricke and H. Cohn. 1.
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, which are parametrized by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equation, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algebraic transformation formulas for special functions. The ones for N = 4,3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3,4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
Complete classification of torsion of elliptic curves over quadratic cyclotomic fields, preprint
"... In a previous paper ([10]), the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and Q ( √ −3). Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if t ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
In a previous paper ([10]), the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and Q ( √ −3). Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and Q ( √ −3), we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and Q ( √ −3). 1
ON THE ARITHMETIC OF CERTAIN MODULAR CURVES
"... Abstract. In this work, we estimate the genus of the intermediate ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. In this work, we estimate the genus of the intermediate
Rational Points on ... and Quadratic QCurves
"... . The rational points on X0 (N)=WN in the case where N is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases N = 91 and N = 125 and the jinvariants of the corresponding quadr ..."
Abstract
 Add to MetaCart
. The rational points on X0 (N)=WN in the case where N is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases N = 91 and N = 125 and the jinvariants of the corresponding quadratic Qcurves are exhibited. 1. Introduction Let N be an integer greater than one and consider the modular curve X 0 (N) whose noncusp points correspond to isomorphism classes of isogenies between elliptic curves : E ! E 0 of degree N with cyclic kernel. The rational points of X 0 (N) have been studied by many authors. Results of Mazur [22], Kenku [20] and others have provided a classication of them. The conclusion is that rational points usually arise from cusps or elliptic curves with complex multiplication. There are a nite number of values of N for which other rational points arise, and we call such rational points `exceptional'. For X 0 (N) the largest N for which there are exc...