It is well known that every closed Riemann surface S of genus g amp;gt;= 2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of Q. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over Q. This is the situation if G is either abelian or isomorphic to A x Z(2), where A is an abelian group. On the other hand, as shown by Streit and Wolfart, if G congruent to Z(p) x Z(q), where p, q amp;gt; 3 are prime integers, then S is not necessarily definable over Q. In this paper, we observe that if G congruent to Z(2)(2) x Z(m) with m amp;gt;= 3, then S can be defined over Q. Moreover, we describe explicit models for S, the corresponding groups of automorphisms, and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.

Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.

Jiménez, Leslie

Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Univ Chile, Chile.

Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We consider the set of equisymmetric Riemann surfaces M(2n -1, D-2n, theta) for all n amp;gt;= 2. We study the group algebra decomposition of the Jacobian JX of every curve X is an element of M (2n - 1, D-2n, theta) for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We describe some of the factors of JX as Jacobian (or Prym) varieties of intermediate coverings. Finally, we compute the dimension of the corresponding Shimura domains.

This paper uses the model of Mathematical Working Spaces to analyse the area of geometry in the Swedish upper secondary mathematics curriculum. By applying this framework, we describe how one can understand the mathematical work advocated in the curriculum in terms three geometrical paradigms as well as in terms of different emphasis on a set of three ways of working connecting epistemological and cognitive aspects of geometrical work.