Related papers: On a modular Fermat equation

We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin…

In this short note we apply methods of Aspinwall-Katz to compute superpotentials of D-branes wrapped on more general obstructed rational curves in Calabi-Yau threefolds. We find an a priori unexpected match between superpotentials from…

Around 2000 Kudla presented conjectures about deep relations between arithmetic intersection theory, Eisenstein series and their derivatives, and special values of Rankin $L-$series. The aim of this text is to work out the details of an old…

This talk reviews Feynman integrals, which are associated to elliptic curves. The talk will give an introduction into the mathematics behind them, covering the topics of elliptic curves, elliptic integrals, modular forms and the moduli…

We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one…

The present paper addresses a semimodule counting conjecture of Moreno-Fr\'{i}as and Rosales for numerical semigroups. Applying Pfister and Steenbrink's Theory for punctual Hilbert schemes of curve singularities, we show that this…

Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions,…

We prove that the Eynard-Orantin symplectic invariants of the curve xy-y^2=1 are the orbifold Euler characteristics of the moduli spaces of genus g curves. We do this by associating to the Eynard-Orantin invariants of xy-y^2=1 a problem of…

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive…

We consider certain CM elliptic curves which are related to Fermat curves, and express the values of $L$-functions at $s=2$ in terms of special values of generalized hypergeometric functions. We compare them and a similar result of…

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal…

Using the Davenport-Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree $k$ in more than $2k$ variables the expected asymptotic formula for the density of solutions holds true. This appears to be…

The classical theory of elliptic curves with complex multiplication is a fundamental tool for studying the arithmetic of abelian extensions of imaginary quadratic fields. While no direct analogue is available for real quadratic fields, a…

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…

We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number…

To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups.…

In the paper we show the equality between the Lempert function and the Green function with two poles with equal weights in the bidisc thus giving the positive answer to a conjecture of Coman in the simplest unknown case. Actually, a…

The Poincar\'e torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of…

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an…

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of $\Q$, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite…