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.     

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.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

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.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.     

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.

     

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over Fq(t1/d) whose members obtain arbitrarily large rank as d→∞.     

A ruled residue theorem for function fields of elliptic curves

It is shown that a valuation of residue characteristic different from 2 and 3 on a field E has at most one extension to the function field of an elliptic curve over E, for which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.     

Statistics about elliptic curves over finite prime fields

We derive formulas for the probabilities of various properties (cyclicity, squarefreeness, generation by random points) of the point groups of randomly chosen elliptic curves over random prime fields.     

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) .     

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P _λ of infinite order in the Mordell-Weil group of the elliptic curve Y ² = X ³ − n ² X over ℚ(λ). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain).     

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.     

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a -tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal.