103 research outputs found

### The modularity of the Barth-Nieto quintic and its relatives

The moduli space of (1,3)-polarized abelian surfaces with full level-2
structure is birational to a double cover of the Barth-Nieto quintic. Barth and
Nieto have shown that these varieties have Calabi-Yau models Z and Y,
respectively. In this paper we apply the Weil conjectures to show that Y and Z
are rigid and we prove that the L-function of their common third \'etale
cohomology group is modular, as predicted by a conjecture of Fontaine and
Mazur. The corresponding modular form is the unique normalized cusp form of
weight 4 for the group \Gamma_1(6). By Tate's conjecture, this should imply
that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's
rigid Calabi-Yau Z_{A_3}, which all have the same L-function, are in
correspondence over Q. We show that this is indeed the case by giving explicit
maps.Comment: 30 pages, Latex2

### Genus three curves and 56 nodal sextic surfaces

Catanese and Tonoli showed that the maximal cardinality for an even set of nodes on a sextic surface is 56 and they constructed such nodal surfaces. In this paper we give an alternative, rather simple, construction for such surfaces starting from non-hyperelliptic genus three curves. We illustrate our method by giving explicitly the equation of such a sextic surface starting from the Klein curve

### Modular symbols and Hecke operators

We survey techniques to compute the action of the Hecke operators on the
cohomology of arithmetic groups. These techniques can be seen as
generalizations in different directions of the classical modular symbol
algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in
papers of the author and the author with Mark McConnell. Some results are
unpublished work of Mark McConnell and Robert MacPherson.Comment: 11 pp, 2 figures, uses psfrag.st

### Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We characterize genus g canonical curves by the vanishing of combinatorial
products of g+1 determinants of Brill-Noether matrices. This also implies the
characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities.
A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of
Szego kernels, reduces such identities to a simple rank condition for matrices
whose entries are logarithmic derivatives of theta functions. Such a basis,
together with the Fay trisecant identity, also leads to the solution of the
question of expressing the determinant of Brill-Noether matrices in terms of
theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added.
Accepted for publication in Math. An

### Symmetry breaking from Scherk-Schwarz compactification

We analyze the classical stable configurations of an extra-dimensional gauge
theory, in which the extra dimensions are compactified on a torus. Depending on
the particular choice of gauge group and the number of extra dimensions, the
classical vacua compatible with four-dimensional Poincar\'e invariance and zero
instanton number may have zero energy. For SU(N) on a two-dimensional torus, we
find and catalogue all possible degenerate zero-energy stable configurations in
terms of continuous or discrete parameters, for the case of trivial or
non-trivial 't Hooft non-abelian flux, respectively. We then describe the
residual symmetries of each vacua.Comment: 24 pages, 1 figure, Section 4 modifie

### A new geometric description for Igusa's modular form $(azy)_5$

The modular form $(azy)_5$ notably appears in one of Igusa's classic
structure theorems as a generator of the ring of full modular forms in genus 2,
being exhibited by means of a complicated algebraic expression. In this work a
different description for this modular form is provided by resorting to a
peculiar geometrical approach.Comment: 10 page

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