3,055 research outputs found
Maximal hyperelliptic curves of genus three
AbstractThis note contains general remarks concerning finite fields over which a so-called maximal, hyperelliptic curve of genus 3 exists. Moreover, the geometry of some specific hyperelliptic curves of genus 3 arising as quotients of Fermat curves, is studied. In particular, this results in a description of the finite fields over which a curve as studied here, is maximal
Genus 3 curves with many involutions and application to maximal curves in characteristic 2
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many
involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H
(not containing the hyperelliptic involution if C is hyperelliptic). Then C is
an Artin-Schreier cover of the three elliptic curves obtained as the quotient
of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous
to the product of these three elliptic curves. In this paper we exhibit
explicit models for genus 3 curves with many involutions, and we compute
explicit equations for the elliptic quotients. We then characterize when a
triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a
genus 3 curve, and we apply this result to the construction of maximal curves.
As a consequence, when q is nonsquare and m=\lfloor 2 sqrt(q) \rfloor = 1,5,7
mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an
infinite number of values of q nonsquare.Comment: 18 page
On the linear bounds on genera of pointless hyperelliptic curves
An irreducible smooth projective curve over is called
\textit{pointless} if it has no -rational points. In this paper
we study the lower existence bound on the genus of such a curve over a fixed
finite field . Using some explicit constructions of
hyperelliptic curves, we establish two new bounds that depend linearly on the
number . In the case of odd characteristic this improves upon a result of R.
Becker and D. Glass. We also provide a similar new bound when is even
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
We show that for a class of two-loop diagrams, the on-shell part of the
integration-by-parts (IBP) relations correspond to exact meromorphic one-forms
on algebraic curves. Since it is easy to find such exact meromorphic one-forms
from algebraic geometry, this idea provides a new highly efficient algorithm
for integral reduction. We demonstrate the power of this method via several
complicated two-loop diagrams with internal massive legs. No explicit elliptic
or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde
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