Related papers: On complex explicit formulae connected with the M\…
In this paper we prove the analytic continuation of a two variable zeta function defined using the vector space of binary forms of degree $d$ to the entire two dimensional complex space as a meromorphic function.
By using exclusively real analysis, we give explicit estimates of some classical summatory functions involving the M\"obius function.
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of…
We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of Helmholtz equation by the WKBJ or EWT methods. We obtain the complex-valued solutions (called…
The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues.…
Let C be real-analytic simple closed curve in the complex plane which is symmetric with respect to the real axis. Let r>0 be such that C+ir misses C-ir. We prove that if a continuous function f extends holomorphically from C+it for each t…
In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions…
We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension $2$ closed subset in a homogeneous space under a semisimple algebraic group, and for…
The space of Lam\'e functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe…
Using the formalism of bar complexes and their relative versions, we give a new, purely algebraic, construction of the so-called universal elliptic KZB connection in arbitrary level. We compute explicit analytic formulae, and we compare our…
Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
In the article we outline the set of Matlab functions that enable the computation of elliptic Integrals and Jacobian elliptic functions for real arguments. Correctness, robustness, efficiency and accuracy of the functions are discussed in…
We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this…
Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}/\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+At^6+B$. An algorithm to determine $\mathrm{rank } E_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented…
By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are…
We study complex Lie algebras spanned by pairs \left(Z,Y\right) of germs of a meromorphic vector field of the complex plane satisfying \left[Z,Y\right]=\delta Y for some \delta\in\ww C . This topic relates to Liouville-integrability of the…