425 research outputs found
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves
This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute
Computing N\'eron-Tate heights of points on hyperelliptic Jacobians
It was shown by Faltings and Hriljac that the N\'eron-Tate height of a point
on the Jacobian of a curve can be expressed as the self-intersection of a
corresponding divisor on a regular model of the curve. We make this explicit
and use it to give an algorithm for computing N\'eron-Tate heights on Jacobians
of hyperelliptic curves. To demonstrate the practicality of our algorithm, we
illustrate it by computing N\'eron-Tate heights on Jacobians of hyperelliptic
curves of genus from 1 to 9.Comment: 13 pages. v5: As kindly pointed out by Raymond van Bommel, the height
is computed in this paper with respect to twice the theta divisor, not the
theta divisor itself (as written in v4
Mahler's measure and elliptic curves with potential complex multiplication
Given an elliptic curve defined over which has potential
complex multiplication by the ring of integers of an imaginary
quadratic field we construct a polynomial which
is a planar model of and such that the Mahler measure is related to the special value of the -function at .Comment: 24 pages. Comments are very welcome
CM cycles on Shimura curves, and p-adic L-functions
Let f be a modular form of weight k>=2 and level N, let K be a quadratic
imaginary field, and assume that there is a prime p exactly dividing N. Under
certain arithmetic conditions on the level and the field K, one can attach to
this data a p-adic L-function L_p(f,K,s), as done by
Bertolini-Darmon-Iovita-Spiess. In the case of p being inert in K, this
analytic function of a p-adic variable s vanishes in the critical range
s=1,...,k-1, and therefore one is interested in the values of its derivative in
this range. We construct, for k>=4, a Chow motive endowed with a distinguished
collection of algebraic cycles which encode these values, via the p-adic
Abel-Jacobi map.
Our main result generalizes the result obtained by Iovita-Spiess, which gives
a similar formula for the central value s=k/2. Even in this case our
construction is different from the one found by Iovita-Spiess
Explicit Methods in Number Theory
The aim of the series of Oberwolfach meetings on ‘Explicit methods in number theory’ is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and -functions and their special values, and modular forms and functions
Algebraische Zahlentheorie
The workshop brought together researchers from Europe, the US and Japan, who reported on various recent developments in algebraic number theory and related fields. Dominant themes were p-adic methods, L-functions and automorphic forms but other topics covered a very wide range of algebraic number theory
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