1,427 research outputs found
Maps between curves and arithmetic obstructions
Let X and Y be curves over a finite field. In this article we explore methods
to determine whether there is a rational map from Y to X by considering
L-functions of certain covers of X and Y and propose a specific family of
covers to address the special case of determining when X and Y are isomorphic.
We also discuss an application to factoring polynomials over finite fields.Comment: 8 page
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Examples of CM curves of genus two defined over the reflex field
In "Proving that a genus 2 curve has complex multiplication", van Wamelen
lists 19 curves of genus two over with complex multiplication
(CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois
over . The generic case of non-Galois quartic CM-fields did not
feature in this list, as the field of definition in that case always contains a
real quadratic field, known as the real quadratic subfield of the reflex field.
We extend van Wamelen's list to include curves of genus two defined over this
real quadratic field. Our list therefore contains the smallest "generic"
examples of CM curves of genus two.
We explain our methods for obtaining this list, including a new
height-reduction algorithm for arbitrary hyperelliptic curves over totally real
number fields. Unlike Van Wamelen, we also give a proof of our list, which is
made possible by our implementation of denominator bounds of Lauter and Viray
for Igusa class polynomials.Comment: 31 pages; Updated some reference
On the reducibility type of trinomials
Say a trinomial x^n+A x^m+B \in \Q[x] has reducibility type
if there exists a factorization of the trinomial into
irreducible polynomials in \Q[x] of degrees , ,...,, ordered
so that . Specifying the reducibility type of a
monic polynomial of fixed degree is equivalent to specifying rational points on
an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable
hope that all its rational points may be described; and techniques are
available that may also find all points when the genus is 2. Thus all
corresponding reducibility types may be described. These low genus instances
are the ones studied in this paper.Comment: to appear in Acta Arithmetic
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
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