Jakob Top

On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms

Let E be an elliptic curve over Q. By the rank of E we shall mean the rank of the group of rational points of E. Mestre [31], improving on the work of Neron [34] (cf. [13], [39] and [46]), has shown that there is an infinite family of elliptic curves over Q with rank at least 12. However, computational work (see [3], [4], [7] and [47]) suggests that a typical elliptic curve will have much smaller rank, with curves of rank 0 or 1 being predominant. Indeed, Brumer [6] has proved, subject to the Birch, Swinnerton-Dyer conjecture, the Shimura, Taniyama, Weil conjecture and the generalized Riemann hypothesis, that the average rank of an elliptic curve, ordered according to its Faltings height, is at most 2.3. In this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over Q. That is, we shall restrict our attention to families of elliptic curves defined over Q which are isomorphic over C. There are families of quadratic, cubic, quartic and sextic twists (see, for example, Proposition 5.4 of Chapter X of [42]). Let E be an elliptic curve over Q with Weierstrass equation y2 = X + ax + b and for any non-zero integer d let Ed 2 3 denote a quadratic twist of E given by the equation dy2 = X + ax + b. Let r(d) denote the rank of Ed . Note that if d1 and d2 are non-zero integers, then Ed is isomorphic to Ed over Q if and only if d1 /d2 is the square of a rational number. Subject to the conjectures of Birch and Swinnerton-Dyer and of Shimura, Taniyama, and Weil, Goldfeld [14] conjectured in 1979 that

Heeke Eigenforms in the Cohomology of Congruence Subgroups of SL (3, Z)

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of Sl(3; Z). To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra–Lenstra–Lovasz algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Green and by Top and van Geemen in previous work. Comparing our tables with results from computations of Galois representations, we find some new numerical evidence for the conjectured relation between modular forms and Galois representations.

Curves of genus 3 over small finite fields

Abstract: We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q

Twists of genus three curves over finite fields

In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the twists of the Dyck-Fermat and Klein quartics. Our methods show how in special cases non-Abelian cohomology can be explicitly computed. They also show how questions which appear difficult from a function field perspective can be resolved by using the theory of the Jacobian variety.

An isogeny of K3 surfaces

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

Maximal hyperelliptic curves of genus three

This note contains general remarks concerning finite fields over which a so-called maximal, hyperelliptic curve of genus 3 exists. Moreover, the geometry of some specific hyperelliptic curves of genus 3 arising as quotients of Fermat curves, is studied. In particular, this results in a description of the finite fields over which a curve as studied here, is maximal.

Algebraic subgroups of GL2(C)

Abstract In this note we classify, up to conjugation, all algebraic subgroups of GL 2 (ℂ).

Elliptic curves over the rationals with bad reduction at only one prime

A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.

On the Mordell-Weil rank of an abelian variety over a number field

Let K be a number field and A an abelian variety over K. The K-rational points of A are known to constitute a finitely generated abelian group (Mordell-Weil theorem). The problem studied in this paper is to find an explicit upper bound for the rank r of its free part in terms of other invariants of A/K. This is achieved by a close inspection of the classical proof of the so-called ‘weak Mordell-Weil theorem’.

Explicit Elliptic K3 Surfaces with Rank 15

This note presents a relatively straightforward proof of the fact that, under certain congruence conditions on a, b, c is an element of Q, the group of rational points over (Q) over bar (t) on the elliptic curve given by y(2) = x(3) + t(3)(t(2) + at + b)(2)(t + c)x + t(5)(t(2) + at + b)(3) is trivial. This is used to show that a related elliptic curve yields a free abelian group of rank 15 as its group of (Q) over bar (t)-rational points.