Modular curves of genus 2

Mathematical Research Letters, 2007

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2004

Rational torsion of J 0 (N ) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5, 7 or 10

Journal of Number Theory, 2010

A curve C defined over Q is modular of level N if there exists a nonconstant morphism X1(N) −→ C defined over Q for some positive integer N. We present an algorithm to compute explicitly equations for modular non-hyperelliptic curves defined over Q of genus 3. Let C be a modular curve of level N , we say that C is new if the corresponding morphism between J1(N) and Jac(C) factorizes through the new part of J1(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves.

Journal of Symbolic Computation, 2001

Let C be a curve of genus 2 and ψ1 : C −→ E1 a map of degree n, from C to an elliptic curve E1, both curves defined over C. This map induces a degree n map φ1 : P 1 −→ P 1 which we call a Frey-Kani covering. We determine all possible ramifications for φ1. If ψ1 : C −→ E1 is maximal then there exists a maximal map ψ2 : C −→ E2, of degree n, to some elliptic curve E2 such that there is an isogeny of degree n 2 from the Jacobian JC to E1 × E2. We say that JC is (n, n)-decomposable. If the degree n is odd the pair (ψ2, E2) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.

1997

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(f0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(f0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the onevariable case, we let f denote the p-ordinary Galois representation with values ...

2006

Proposition 1.0.2. i.e., φ 1 is the q-expansion of E 1 (τ, 1 , B). 2 pending on B. 1 Alternatively, φ1(τ ) can be obtained by calculating the integral over M(C) of a theta function valued in (1, 1) forms; this amounts to a very special case of the results of . The analogous computation in the case of modular curves was done by Funke .

Advances in Geometry, 2001

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 modelsZandY, respectively. In this paper we apply the Weil conjectures to show thatYandZare rigid and we prove that theLfunction of their common third étale 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~Γ1(6). By Tate's conjecture, this should imply thatY, the fibred square of the universal elliptic curve~S1(6), and Verrill's rigid Calabi-Yau~𝒵A3, which all have the sameLfunction, are in correspondence overℚ. We show that this is indeed the case by giving explicit maps.

Mathematische Zeitschrift, 2005

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the number of k-isomorphism classes of curves of genus two over a finite field. Moreover, we prove that the field of moduli of any curve coincides with its field of definition, by exhibiting rational models of curves with any prescribed value of their Igusa invariants. Finally, we use cohomological methods to find, for each rational model, an explicit description of its twists. In this way, we obtain a parameterization of all k-isomorphism classes of curves of genus two in terms of geometric and arithmetic invariants.

Pacific Journal of Mathematics, 1997

2012

Abstract. We present explicit equations of semi-stable elliptic surfaces (i.e., having only type In singular fibers) which are associated to the torsion-free genus zero congruence subgroups of the modular group as classified by A. Sebbar. 1.