Efficient computation of BSD invariants in genus 2

Recently, all Birch and Swinnerton-Dyer invariants, except for the order of Sha, have been computed for all curves of genus 2 contained in the L-functions and Modular Forms Database. This report explains the improvements made to the implementation of the algorithm described in arXiv:1711.10409 that were needed to do the computation of the Tamagawa numbers and the real period in reasonable time. We also explain some of the more technical details of the algorithm, and give a brief overview of the methods used to compute the special value of the $L$-function and the regulator.Comment: Source code included as ancillary files. Comments always welcome

Computing L-series of hyperelliptic curves

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.Comment: 15 pages, corrected minor typo

Néron-Tate heights on the Jacobians of high-genus hyperelliptic curves

We use Arakelov intersection theory to study heights on the Jacobians of high-genus hyperelliptic curves. The main results in this thesis are: 1) new algorithms for computing Neron-Tate heights of points on hyperelliptic Jacobians of arbitrary dimension, together with worked examples in genera up to 9 (pre-existing methods are restricted to genus at most 2 or 3). 2) a new definition of a naive height of a point on a hyperelliptic Jacobian of arbitrary dimension, which does not make use of a projective embedding of the Jacobian or a quotient thereof. 3) an explicit bound on the difference between the Neron-Tate height and this new naive height. 4) a new algorithm to compute sets of points of Neron-Tate height up to a given bound on a hyperelliptic Jacobian of arbitrary dimension, again without making use of a projective embedding of the Jacobian or a quotient thereof

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction

We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.Comment: 24 page