Related papers: Asymptotic period relations for Jacobian elliptic …
This text is a presentation (without proofs) of some of my recent results on the singular terms of asymptotic expansions of period-integrals using (a,b)-modules. I try to explain why this simple algebraic structure is interesting and…
We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle…
We describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.
We investigate a periodic quantum graph in form of a square lattice with a general self-adjoint coupling at the vertices. We analyze the spectrum, in particular, its high-energy behaviour. Depending on the coupling type, bands and gaps have…
An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are…
An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic…
We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a two-dimensional non-compact algebraic group -- the generalized Jacobian of an elliptic curve with two points identified. This leads to a transparent description of its…
We give real Jacobian elliptic function parametrizations for a genuinely asymmetric biquadratic curve where the variables and parameters are real.
Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer…
Every fibration of a projective hyper-K\"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the…
We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication.
It is shown in this paper that there is a fine correlation of the third order between the values of the functions $Z[\vp_1(t)]$ and $\tilde{Z}^2(t)$ which corresponds to two collections of disconnected sets. The corresponding new asymptotic…
We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.
In this paper we obtain conditions on the divisors of the group order of the Jacobian of a hyperelliptic genus 2 curve, generated by the complex multiplication method described by Weng (2003) and Gaudry (2005). Examples, where these…
This paper is concerned with investigating the asymptotic behavior of the gradients of solutions to a class of elliptic systems with general boundary data, especially covering the Lam\'{e} systems, in a narrow region. The novelty of this…
Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and…
We study the behavior of Galois periods under the local theta correspondence for even orthogonal and symplectic groups. Specifically, we compare their multiplicities and construct explicit transfer maps. Furthermore, we establish both an…
The coefficient of the log term in the entanglement entropy associated with hyperspherical surfaces in flat space-time is shown to equal the conformal anomaly by conformally transforming Euclideanised space--time to a sphere and using…
We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate…
Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $\mathbb{Q}$ and by the subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic abelian varieties,…