On isogenies among certain abelian surfaces

We construct a three-parameter family of non-hyperelliptic and bielliptic
plane genus-three curves whose associated Prym variety is two-isogenous to the
Jacobian variety of a general hyperelliptic gen ...

Isogenous components of Jacobian surfaces

Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K
= p \geq 0$, and $\mbox{Jac} (\mathcal X, \iota)$ its Jacobian, where $\iota$
is the principal polarization of $\mbox{Jac} ...

Arithmetic statistics of Prym surfaces

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym
varieties of double covers of genus-$1$ curves by genus-$3$ curves. These
abelian surfaces carry a polarization of type $(1,2 ...

Period matrices of some hyperelliptic Riemann surfaces

In this paper, we calculate period matrices of algebraic curves defined by
$$w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2)\cdots (z^2-a_{g-1}^2)$$ for any $g\geq 2$
and $a_1, a_2, \dots, a_{g-1}\in \mathbb{R}$ w ...

Geometric decomposition of abelian varieties of order 1

Since the 1970s, the complete classification (up to isogeny) of abelian
varieties over finite fields with trivial group of rational points has been
known from results of Madan--Pal and Robinson; with ...

Slice conformality: Riemann manifolds and logarithm on quaternions and octonions

We establish the quaternionic and octonionic analogs of the classical Riemann
surfaces of the complex logarithm and $n$-th root function, and give a unifying
definition of such functions in the quater ...

The arithmetic of a twist of the Fermat quartic

We study the arithmetic of the twist of the Fermat quartic defined by $X^4 +
Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the
Mordell--Weil group of the Jacobian variety expli ...

Action of the automorphism group on the Jacobian of Klein's quartic curve

Klein's simple group $H$ of order $168$ is the automorphism group of the
plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full
automorphism group $G$ of the Jacobian $J=J(C)$ is ...

Singular curves and their compactified Jacobians

We survey the theory of the compactified Jacobian associated to a singular
curve. We focus on describing low genus examples using the Abel map.

How many quasiplatonic curves?

We show that the number of isomorphism classes of quasiplatonic Riemann
surfaces of genus $\,\le g\,$ has a growth of type $\,g^{\log g}\,$. The number
of non--isomorphic regular dessins of genus $\,\ ...