462 research outputs found
Dynamics on supersingular K3 surfaces
For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on
the supersingular K3 surface of Artin invariant one which does not lift to any
characteristic zero model. Our construction builds on elliptic fibrations to
produce a closed formula for the automorphism's characteristic polynomial on
second cohomology, which turns out to be an irreducible Salem polynomial of
degree 22 with coefficients varying with p.Comment: 12 pages, 3 figures; v2: main result improved to Salem degree 2
On the dihedral Euler characteristics of Selmer groups of abelian varieties
This note shows how to use the framework of Euler characteristic formulae to
study Selmer groups of abelian varieties in certain dihedral or anticyclotomic
extensions of CM fields via Iwasawa main conjectures, and in particular how to
verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this
setting. When the Selmer group is cotorsion with respect to the associated
Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch
and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with
respect to the associated Iwasawa algebra, we give a conjectural description of
the Euler characteristic of the cotorsion submodule, and explain how to deduce
inequalities from the associated main conjecture divisibilities of Perrin-Riou
and Howard.Comment: 26 pages. Previous discussion of two-variable setting removed, and
discussion of the indefinite setting modified accordingly. To appear in the
HIM "Arithmetic and Geometry" conference proceeding
Pairing computation on hyperelliptic curves of genus 2
Bilinear pairings have been recently used to construct cryptographic schemes with new and novel properties, the most celebrated example being the Identity Based Encryption scheme of Boneh and Franklin. As pairing computation is generally the most computationally intensive part of any painng-based cryptosystem, it is essential to investigate new ways in which to compute pairings efficiently.
The vast majority of the literature on pairing computation focuscs solely on using elliptic curves. In this thesis we investigate pairing computation on supersingular hyperelliptic curves of genus 2 Our aim is to provide a practical alternative to using elliptic curves for pairing based cryptography. Specifically, we illustrate how to implement pairings efficiently using genus 2 curves, and how to attain performance comparable to using elliptic curves.
We show that pairing computation on genus 2 curves over F2m can outperform elliptic curves by using a new variant of the Tate pairing, called the r¡j pairing, to compute the fastest pairing implementation in the literature to date We also show for the first time how the final exponentiation required to compute the Tate pairing can be avoided for certain hyperelliptic curves.
We investigate pairing computation using genus 2 curves over large prime fields, and detail various techniques that lead to an efficient implementation, thus showing that these curves are a viable candidate for practical use
Mirror symmetry on K3 surfaces via Fourier-Mukai transform
We use a relative Fourier-Mukai transform on elliptic K3 surfaces to
describe mirror symmetry. The action of this Fourier-Mukai transform on the
cohomology ring of reproduces relative T-duality and provides an
infinitesimal isometry of the moduli space of algebraic structures on
which, in view of the triviality of the quantum cohomology of K3 surfaces, can
be interpreted as mirror symmetry.Comment: 15 pages, AMS-LaTeX v1.2. Final version to appear in Commun. Math.
Phy
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