Calculating root numbers of elliptic curves over Q

LetE be an elliptic curve defined overQ. By making some results of Rohrlich computationaly explicit, the root numberW is calculated as a product of local root numbers except whenE is ‘nonabelian’ at 2 or 3. The derivation of these explicit formulas depends in an essential way on the classification, carried out by the author in an earlier paper, of thoseE that become everywhere good or semi-stable in an abelian extension ofQ.     

Finite descent obstruction for curves over function fields

We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of integral points on integral models of twists of modular curves over function fields.     

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     

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.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

Stark conjectures for CM elliptic curves over number fields

We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s=0. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.     

Lattices from elliptic curves over finite fields

In their well known book [6] Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.     

Modularity of CM elliptic curves over division fields

Let E be a CM elliptic curve defined over an algebraic number field F. In general E will not be modular over F. In this paper, we determine extensions of F, contained in suitable division fields of E, over which E is modular. Under some weak assumptions on E, we construct a minimal subfield of division fields over which E is modular.     

Efficient CM-constructions of elliptic curves over finite fields

We present an algorithm that, on input of an integer together with its prime factorization, constructs a finite field and an elliptic curve over for which has order . Although it is unproved that this can be done for all , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in , where is the number of distinct prime factors of . In the cryptographically relevant case where is prime, an expected run time can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order and nextprime.

     

An algorithm for DLP on anomalous elliptic curves over Fp

This paper improves the method of discrete logarithm on anomalous elliptic curves, and establishes an isomorphism from E(Fp) to Fp which can be more easily implemented. Fruthermore, we give an optimized algorithm for discrete logarithm on anomalous elliptic curves E(Fp).     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.