18,035 research outputs found
Modular Curves Of Genus 2
We prove that there is only a finite number of genus 2 curves C defined over
Q such that there exists a nonconstant morphism pi:X_1(N) --->C defined over Q
and the jacobian of C, J(C), is a Q-factor of the new part of the jacobian of
X_1(N), J_1(N)^{new}. Moreover, we prove that there are only 149 genus two
curves of this kind with the additional requeriment that their jacobians are
Q-simple. We determine the corresponding newforms and present equations for all
these curves
Modular polynomials for genus 2
Modular polynomials are an important tool in many algorithms involving
elliptic curves. In this article we investigate their generalization to the
genus 2 case following pioneering work by Gaudry and Dupont. We prove various
properties of these genus 2 modular polynomials and give an improved way to
explicitly compute them
Abelian Surfaces over totally real fields are Potentially Modular
We show that abelian surfaces (and consequently curves of genus 2) over
totally real fields are potentially modular. As a consequence, we obtain the
expected meromorphic continuation and functional equations of their Hasse--Weil
zeta functions. We furthermore show the modularity of infinitely many abelian
surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential
modularity results for genus one curves over (not necessarily CM) quadratic
extensions of totally real fields.Comment: 285 page
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