5,938 research outputs found
Elliptic Curves over Real Quadratic Fields are Modular
We prove that all elliptic curves defined over real quadratic fields are
modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv
versio
On finiteness conjectures for modular quaternion algebras
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 GL-type 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
Counting hyperelliptic curves that admit a Koblitz model
Let k be a finite field of odd characteristic. We find a closed formula for
the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic
curves of genus g over k, admitting a Koblitz model. These numbers are
expressed as a polynomial in the cardinality q of k, with integer coefficients
(for pointed curves) and rational coefficients (for non-pointed curves). The
coefficients depend on g and the set of divisors of q-1 and q+1. These formulas
show that the number of hyperelliptic curves of genus g suitable (in principle)
of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not
2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more
resistant to the attacks to the DLP; for these values of g the number of curves
is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)
Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields
We compute explicit rational models for some Hilbert modular surfaces
corresponding to square discriminants, by connecting them to moduli spaces of
elliptic K3 surfaces. Since they parametrize decomposable principally polarized
abelian surfaces, they are also moduli spaces for genus-2 curves covering
elliptic curves via a map of fixed degree. We thereby extend classical work of
Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska,
V\"olklein, Magaard and others, producing explicit families of reducible
Jacobians. In particular, we produce a birational model for the moduli space of
pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n
from C to E, as well as a tautological family over the base, for 2 <= n <= 11.
We also analyze the resulting models from the point of view of arithmetic
geometry, and produce several interesting curves on them.Comment: 36 pages. Final versio
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