6,169 research outputs found
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
F-Theory Vacua with Z_3 Gauge Symmetry
Discrete gauge groups naturally arise in F-theory compactifications on
genus-one fibered Calabi-Yau manifolds. Such geometries appear in families that
are parameterized by the Tate-Shafarevich group of the genus-one fibration.
While the F-theory compactification on any element of this family gives rise to
the same physics, the corresponding M-theory compactifications on these
geometries differ and are obtained by a fluxed circle reduction of the former.
In this note, we focus on an element of order three in the Tate-Shafarevich
group of the general cubic. We discuss how the different M-theory vacua and the
associated discrete gauge groups can be obtained by Higgsing of a pair of
five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles
in -fibers that appear at certain codimension two loci in the base. We
explicitly identify all three curves that give rise to the corresponding Higgs
fields. In this analysis the investigation of different resolved phases of the
underlying geometry plays a crucial r\^ole.Comment: 13 page
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