Related papers: On Chudnovsky-Ramanujan Type Formulae
The classical theory of $p$-adic (elliptic) modular forms arose in the 1970's from the work of J.-P.\ Serre \cite{se1} who took $p$-adic limits of the $q$-expansions of these forms. It was soon expanded by N.\ Katz \cite{ka1} with a more…
We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber-Pagani and van Zelm…
In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of…
In this article we prove that for all primes $p\not=2,3$, the Ramanujan vector field has an invariant algebraic curve and then we give a moduli space interpretation of this curve in terms of Cartier operator acting on the de Rham cohomology…
By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalsch{\"u}tz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of…
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the…
Motivated by a conjecture of Lian and Yau concerning the mirror map in string theory, we determine when the mirror map q-series of certain elliptic curve and K3 surface families are Hauptmoduln (genus zero modular functions). Our geometric…
In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic…
Beginning in 2006, G. Gentili and D.C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball centered at 0 the set of regular…
We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses…
In this talk we discuss Feynman integrals which are related to elliptic curves. We show with the help of an explicit example that in the set of master integrals more than one elliptic curve may occur. The technique of maximal cuts is a…
This note attempts to study lacunary trigonometric products with values in the matrix group SU(1,1) in analogy with lacunary trigonometric series. The central questions are the characterization of their convergence in an appropriately…
For every prime $p \geq 5$, we compute the $p$-th power of the Ramanujan vector field that arises from the differential relations discovered by Ramanujan for the Eisenstein series $E_2,E_4$ and $E_6$. Our method results in explicit…
The Milnor formula $\mu=2\delta-r+1$ relates the Milnor number $\mu$, the double point number $\delta$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a…
Kaneko and Sakai recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and…
Modular curves like X_0(N) and X_1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL_2(Z), they allow for a more arithmetic description as…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
By introducing a class of meromorphic functions with certain ramification structures on $\Bbb{CP}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These…
The perimeter of an ellipse has no exact closed-form expression in terms of elementary functions, and numerous approximations have been proposed since the eighteenth century. Classical formulas by Fagnano, Euler, and Ramanujan, as well as…
Recently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become…