Mazur module for elliptic curves

Under certain assumptions parameters of the Mazur module for an elliptic curve E over a extension K/K₀ are computed. This makes it possible, in particular, to prove in certain cases that the group E(K) is finitely generated without assuming that the groups E(K₀) and III(K₀/K₀, E) are finite.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 319–328, February, 1975.The author thanks Yu. I. Manin for his useful discussions and attention to my work.     

Spectral curves for Cauchy-Riemann operators on punctured elliptic curves

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve under appropriate boundary conditions. The algebraic curves thus obtained arise, for example, as irreducible components of the spectral curves of minimal tori with planar ends in ?³. It turns out that these curves coincide with the spectral curves of certain elliptic KP solitons studied by Krichever.     

Pesenti-Szpiro inequality for optimal elliptic curves

We study Pesenti-Szpiro inequality in the case of elliptic curves over F_q(t) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.     

On universal elliptic curves over Igusa curves

Summary The purpose of this note is to introduce the arithmetic, study of the universal elliptic curve over Igusa curves. Specifically, its Hasse-WeilL-function is computed in terms of modular forms and is shown to have interesting zeros. Explicit examples are presented for which the Birch and Swinnerton-Dyer conjecture is verified.This paper summarizes part of the author's Ph.D. thesis. He wishes to thank the Sloan Foundation for financial support in the form of a Doctoral Dissertation Fellowship and his advisor, Dick Gross, for mathematical guidance and inspirational enthusiasm.To my parents in their 50th year     

Hasse invariants for the Clausen elliptic curves

Gauss’s hypergeometric function gives periods of elliptic curves in Legendre normal form. Certain truncations of this hypergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we prove that certain truncations of and in $\mathbb {F}_{p}[x]$ are related to the characteristic p Hasse invariants.     

Explicit identities for invariants of elliptic curves

New explicit formulas are given for the supersingular polynomial ssp(t) and the Hasse invariant of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves En in Tate normal form with distinguished points of order n. This yields a proof that and are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ1/4 of supersingular parameters of the Legendre normal form Y²=X(X−1)(X−λ) in characteristic p has octahedral symmetry. For general n?4, the field of definition of a supersingular En is determined, along with the field of definition of the points of order n on En.     

Quaternion algebras over elliptic curves

All composition algebras of rank 2 and 4 over elliptic curves are enumerated and partly classified, and examples of octonion algebras are constructed using the generalized Cayley-Dickson doubling process. The underlying field is assumed to be perfect, and of characteristic not two. Some applications are given.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Icosahedral representations and elliptic curves

In this paper we show a connection between icosahedral Artin representations of the rationals and elliptic curves. More specifically, we prove for each suitable elliptic curve defined over there is an associated icosahedral Artin representation defined over the rationals.     

Bounded torsion of elliptic curves

An explicit bound is obtained for the torsion of elliptic curves over the field of rational numbers. Let be an elliptic curve over the field of rational numbers R, and Q_m a primitive R-point of order m on it; here m is a prime or a double prime. Hence if m=2p, then p 509, whereas if m=p, then p < 6144.Translated from Matematicheskie Zametki, No. 1, pp. 53–58, July, 1972.     

Three-Selmer groups for elliptic curves with 3-torsion

We consider a specific family of elliptic curves with rational 3-torsion subgroup. We arithmetically define 3-Selmer groups through isogeny and 3-descent maps, then associate the image of the 3-descent maps to solutions of homogeneous cubic polynomials affiliated with the elliptic curve E and an isogenous curve E′. Thanks to the work of Cohen and Pazuki, we have solubility conditions for the homogeneous polynomials. Using these conditions, we give a graphical approach to computing the size of 3-Selmer groups. Finally, we translate the conditions on graphs into a question concerning ranks of matrices and give an upper bound for the rank of the elliptic curve E by calculating the size of the Selmer groups.     

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier–Mukai transforms for coherentsystems consisting of a vector bundle over an elliptic curveand a subspace of its global sections, showing that these transformsare indexed by positive integers. We prove that the naturalstability condition for coherent systems, which depends on aparameter, is preserved by these transforms for small and largevalues of the parameter. By means of the Fourier–Mukaitransforms we prove that certain moduli spaces of coherent systemscorresponding to small and large values of the parameter areisomorphic. Using these results we draw some conclusions aboutthe possible birational type of the moduli spaces. We provethat for a given degree d of the vector bundle and a given dimensionof the subspace of its global sections there are at most d differentpossible birational types for the moduli spaces.     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.