In the finite-temperature Yang-Mills theory we calculate the functional determinant for fermions in the fundamental representation of SU(N) gauge group in the background of an instanton with non-trivial holonomy at spatial infinity. This object, called the Kraan–van Baal – Lee–Lu caloron, can be viewed as composed of N Bogomolny–Prasad–Sommerfeld monopoles (or dyons). We compute analytically tw...

Abstract. The two centre problem is studied both in the phase plane and in spacetime, assuming first a trajectory collinear, and, as a second case, a circular one through the newtonian attractors, finding a saddle equilibrium. For the latter problem a probably new differential equation is met and solved. Time is then obtained in both cases through elliptic integrals of all kinds and Jacobian fu...

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.

Let f1, . . . , fr ∈ K[x], K a field, be homogeneous polynomials and put F = ∑r i=1 yifi ∈ K[x, y]. The quotient J = K[x, y]/I, where I is the ideal generated by the ∂F/∂xi and ∂F/∂yj , is the Jacobian ring of F . We describe J by computing the cohomology of a certain complex whose top cohomology group is J .

Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (Quebec). Some results in section 3 of this work are taken from Jha's Ph.D. thesis [Jha 1992]. After a brief review of partial results regarding Case I of Fermat’s Last Theorem, we discuss the relationship between the number of points on Fermat’...

In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...

A genus one curve defined over Q which has points over Qp for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of Q.

Let K be a finite field of characteristic not equal to 2, and L a quadratic extension of K. For a large class of elliptic curves E defined over L we construct hyperelliptic curves over K of genus 2 whose jacobian is isogenous to the Weil restriction ResK(E).

Contents 1. Introduction 2 2. Prym varieties for covers of curves 3 3. Galois covers 8 4. Degree two covers 10 5. Covers of degree three 13 6. Covers of degree four 15 6.1. The cyclic case 15 6.2. The Klein case 17 7. The dihedral case 22 7.1. The bigonal construction 34 8. The alternating case 37 8.1. The trigonal construction for the case A 4 43 9. The symmetric case 44 9.1. The classical cas...

According to [LO], the Prym variety of any non-cyclic étale triple cover f : Y → X of a smooth curve X of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map Pr of a moduli space S3M̃2 of admissible S3-covers of genus 7 to the moduli space A2 of principally...