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Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of…
This article shows explicit relation between fractional expressions of Schottky-Klein type for hyperelliptic $\sigma$-functions and a product of differences of the algebraic coordinates on each stratum of natural stratification in a…
We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian…
Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a…
We explicitly find an equation and a projective embedding of the Kummer surface associated to the Jacobian of a curve of genus 2 given by an equation of the form y^2 + h(x)y = f(x) over an arbitrary ground field as well as several maps that…
This paper is the second in a series of two papers which study the phenomenon of tropical split Jacobians. The first paper is a contemplative study, embedded in the broader context of exploring connections between the category of tropical…
We prove that the generic point of a Hilbert modular four-fold is not a Jacobian. The proof uses degeneration techniques and is independent of properties of the mapping class group used in preceding papers on locally symmetric subvarieties…
We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is…
In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic $\wp$ functions, also called Kleinian $\wp$ functions. This result is based on the recently developed theory of multivariable sigma…
We show that if $C$ is a smooth projective curve and $\mathfrak{d}$ is a $\mathfrak{g}^{n}_{2n}$ on $C$, then we obtain a rational map $\mathrm{Sym}^{n}(C)\dashrightarrow\mathfrak{d}$ whose fibers can be related in an interesting way to…
Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this…
The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories - recently introduced by the authors - which share and generalise relevant…
The division of compact Riemann surfaces into 3 cases K_C<0, g=0, or K_C=0, g=1, or K_C>0, g>=2 is well known, and corresponds to the familiar trichotomy of spherical, Euclidean and hyperbolic non-Euclidean plane geometry. Classification…
We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other…
We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko--Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the…
An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…
A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related…
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…
This article extends relations of Mumford's UVW-expressions to those in subvarieties in a hyperelliptic Jacobian using Baker's method.