Paul van Wamelen

Examples of genus two CM curves defined over the rationals

We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example y 2 = x 5 - 1 we find 19 non-isomorphic such curves. We believe that these are the only such curves.

Proving that a genus 2 curve has complex multiplication

Recently examples of genus 2 curves defined over the rationals were found which, conjecturally, should have complex multiplication. We prove this conjecture. This involves computing an explicit representation of a rational map defining complex multiplication.

Integral bases for TQFT modules and unimodular representations of mapping class groups

Abstract We construct integral bases for the

Equations for the Jacobian of a hyperelliptic curve

SO(3)

Stark’s conjecture over complex cubic number fields

-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus three at a fifth root of unity, we still give an explicit basis.

Poonen's question concerning isogenies between Smart's genus 2 curves

We give an explicit embedding of the Jacobian of a hyperelliptic curve, y2 = f(x), into projective space such that the image is isomorphic to the Jacobian over the splitting field of f . The embedding is a modification of the usual embedding by theta functions with half integer characteristics.

On linear congruence relations for Kubota–Leopoldt 2-adic L-functions

jjSystematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Starks conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Starks conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

Divisibility properties of generalized Vandermonde determinants

We describe a method for proving that two explicitly given genus two curves have isogenous jacobians. We apply the method to the list of genus 2 curves with good reduction away from 2 given by Smart. This answers a question of Poonen.

Computing with the analytic Jacobian of a genus 2 curve

Abstract Our purpose in the paper is to find the most general linear congruence relation of the Hardy–Williams type for linear combinations of special values of Kubota–Leopoldt 2-adic L-functions L2(k,χω1−k) with k running over any finite subset of Z not necessarily consisting of consecutive integers (see Acta Arith. 47 (1986) 263; Publ. Math. Fac. Sci. Besancon, Theorie des Nombres, 1995/1996; Publ. Math. Debrecen 56 (2000) 677 and cf. Mathematics and Its Applications, Vol. 511, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000). If k runs over finite subsets of Z consisting of consecutive integers see Compositio Math. 111 (1998) 289; Publ. Math. Debrecen 56 (2000) 677; Hardy and Williams, 1986; Compositio Math. 75 (1990) 271; Acta Arith. 71 (1995) 273; 52 (1989) 147; J. Number Theory 34 (1990) 362. In order to obtain the most general congruences of this type we make use of divisibility properties of the generalized Vandermonde determinants obtained in Spiez et al. (Divisibility properties of generalized Vandermonde and Cauchy determinants, Preprint 627, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2002). This allows us to simplify our main Theorem 2 and obtain Theorem 3 where the most general form of the linear congruence relation is given.

Jacobi sums over finite fields

Given n ≥ 2 let a denote an increasing n-tuple of non-negative integers ai and let x denote an n-tuple of indeterminates xi. Denote by Va(x) the generalized Vandermonde determinant, the polynomial obtained by computing the determinant of the matrix with (i, j) entry equal to x aj i . Let s be the standard n-tuple of consecutive integers from the interval [0, n−1] and given c ≥ 1 assume that x is an n-tuple of distinct 2-integral odd rational numbers xi such that xi ≡ xj ( mod 2 ). Several years ago one of the authors, investigating some properties of KubotaLeopoldt 2-adic L-functions, asked whether for any n-tuples a and x with c = 1 the identity ord2Va(x) = ord2Vs(x) + ord2Vs(a)− ord2Vs(s) (1.1)