1,127 research outputs found
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
Examples of abelian surfaces with everywhere good reduction
We describe several explicit examples of simple abelian surfaces over real
quadratic fields with real multiplication and everywhere good reduction. These
examples provide evidence for the Eichler-Shimura conjecture for Hilbert
modular forms over a real quadratic field. Several of the examples also support
a conjecture of Brumer and Kramer on abelian varieties associated to Siegel
modular forms with paramodular level structures.Comment: 26 pages. Final version (to appear in Mathematische Annalen
Multiplicative excellent families of elliptic surfaces of type E_7 or E_8
We describe explicit multiplicative excellent families of rational elliptic
surfaces with Galois group isomorphic to the Weyl group of the root lattices
E_7 or E_8. The Weierstrass coefficients of each family are related by an
invertible polynomial transformation to the generators of the multiplicative
invariant ring of the associated Weyl group, given by the fundamental
characters of the corresponding Lie group. As an application, we give examples
of elliptic surfaces with multiplicative reduction and all sections defined
over Q for most of the entries of fiber configurations and Mordell-Weil
lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials
with Galois group W(E_7) or W(E_8).Comment: 23 pages. Final versio
Algorithmic design of self-assembling structures
We study inverse statistical mechanics: how can one design a potential
function so as to produce a specified ground state? In this paper, we show that
unexpectedly simple potential functions suffice for certain symmetrical
configurations, and we apply techniques from coding and information theory to
provide mathematical proof that the ground state has been achieved. These
potential functions are required to be decreasing and convex, which rules out
the use of potential wells. Furthermore, we give an algorithm for constructing
a potential function with a desired ground state.Comment: 8 pages, 5 figure
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