Related papers: Correspondences with split polynomial equations

Let $G$ denote a finite group and $\pi: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash…

We define a generalized Jacobian $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and a generalized Picard group $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ of a graph $\mathit{Gr}$ with respect to a modulus $ \mathfrak{m}=\sum_{i=1}^s m_iw_i$ with $w_i$…

The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of…

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic…

Given a polynomial $W$ with an isolated singularity, we can consider the Jacobian ring as an invariant of the singularity. If in addition we have a group action on the polynomial ring with $W$ fixed, we are led to consider the twisted…

Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic…

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating…

Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the…

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of…

We explore connections between the category of tropical abelian varieties (tav), $\mathbb{T}\mathcal{A}$, and the the category of tropical curves, $\mathbb{T}\mathcal{C}$, first in a broader context and then specifically by studying the…

In this paper we derive new identities satisfied by Chebyshev polynomials of the first kind and big q-Jacobi polynomials. An immediate benefit of the derived identities is the achievement of closed-form expressions for the Laurent…

We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both…

This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric…

We examine \'etale covers of genus two curves that occur in the linear system of a polarizing line bundle of type $(1,d)$ on a complex abelian surface. We give results counting fixed points of involutions on such curves as well as…

We give the explicit equations for a P^3 x P^3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of…

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…

In this paper, we construct certain algebraic correspondences between genus three curves and certain type of Calabi-Yau threefolds which is double coverings of three dimensional projective space. Via this correspondences, the first…

Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness…

Using an approach to the Jacobian Conjecture by L.M. Dru\.zkowski and K. Rusek 12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev[27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of…

Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not…