Josep González

Luminescence and impurity states in CuInSe2

The photoluminescence spectrum of melt‐grown n‐CuInSe2 single crystal at 7 K, has been determined near the fundamental absorption edge. Three peaks, at approximately 0.980, 0.990, and 1.013 eV, have been observed. Considering the transition probabilities of recombination processes, donor‐to‐acceptor pair and the corresponding free‐to‐bound transitions have been identified. From the data, an absorption edge at (1.023±0.003) eV, a donor state at (10±2) meV, and an acceptor state at (33±2) meV have been determined.

Equations of Shimura Curves of Genus Two

LetBD be the indefinite quaternion algebra overQ of reduced discriminantD=p1· · · · ·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown by Shimura [23], XD is the coarse moduli space of abelian surfaces with quaternionic multiplication by BD. Let W = {ωm : m | D} ⊆ Aut Q(XD) be the group of Atkin-Lehner involutions. For any m | D, we will denote X D = XD/〈ωm〉, the quotient of the Shimura curve XD, by ωm. The importance of the curves X D is enhanced by their moduli interpretation as curves embedded in Hilbert-Blumenthal surfaces and Igusa’s threefold A2 (cf. [21, 22]). The classical modular case arises when D = 1. In this case, automorphic forms of these curves admit Fourier expansions around the cusp of infinity and we know explicit generators of the field of functions of such curves. Also, explicit methods are known to determine bases of the space of their regular differentials, which are used to compute equations for quotients of modular curves. When D = 1, the absence of cusps has been an obstacle for explicit approaches to Shimura curves. Explicit methods to handle functions and regular differential forms on these curves are less accessible and we refer the reader to [3] for progress in this regard. For this reason, at present, few equations of Shimura curves are known, all of them of genus 0 or 1 (cf. [6, 11, 13]). In addition, in a later work, Kurihara conjectured equations for all Shimura curves of genus two and for several curves of genera three and five, though he was not able to give a proof for his guesses (cf. [14]).

Modular curves of genus 2

We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π : X1 (N) → C defined over Q and the jacobian of C is Q-isogenous to the abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding newforms and present equations for all these curves.

On the Hasse–Witt Invariants of Modular Curves

We briefly discuss the relationship between several characterizations of the Hasse–Witt invariant of curves in characteristic p with the goal of computing its value in concrete instances. We study its asymptotic behaviour when dealing with the geometric fibres of curves of genus ≥ 2 defined over the rationals. Numerical evidence gathered for several modular curves supports certain conjectural distribution laws.

On finiteness conjectures for endomorphism algebras of abelian surfaces

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

Computations on Modular Jacobian Surfaces

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties Af attached by Shimura to normalized newforms f ? S2(?0(N)). We present all the curves corresponding to principally polarized surfaces Af for N ? 500.

The automorphism group of the non-split Cartan modular curve of level 11

Abstract We derive equations for the modular curve X ns ( 11 ) associated to a non-split Cartan subgroup of GL 2 ( F 11 ) . This allows us to compute the automorphism group of the curve and show that it is isomorphic to Kleins four group.

The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms

The Sato–Tate conjecture has been recently settled in great generality. One natural question now concerns the rate of convergence of the distribution of the Fourier coefficients of modular newforms to the Sato–Tate distribution. In this paper, we address this issue, imposing congruence conditions on the primes and on the Fourier coefficients as well. Assuming a proper error term in the convergence to a conjectural limiting distribution, supported by experimental data, we prove the Lang–Trotter conjecture, and in the direction of Lehmers conjecture, we prove that τ(p)=0 has at most finitely many solutions. In fact, we propose a conjecture, much more general than Lehmers, about the vanishing of Fourier coefficients of any modular newform.

Frobenius distribution for quotients of Fermat curves of prime exponent

Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of (l,k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the l-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Plane Quartic Twists of X(5,3)

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the representation. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N=5.