Related papers: Generic Solutions of Equations Involving the Modul…
An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group S_N is presented for integer values of the parameter. The corresponding integrals…
Jacobi-Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.
We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular…
In this article we solve a general class of sextic equations. The solution follows if we consider the $j$-invariant and relate it with the polynomial equation's coefficients. The form of the solution is a relation of Rogers-Ramanujan…
In this article, we study the combinatorics of congruence subgroups of the modular group. More precisely, we consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of…
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…
We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there…
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
We explain an experimental method to find CY-type differential equations of order $3$ related to modular functions of genus zero. We introduce a similar class of Calabi-Yau differential equations of order $5$, show several examples and make…
The purpose of this paper is to establish the existence of solutions with prescribed norm to a class of nonlinear equations involving the mixed fractional Laplacians. This type of equations arises in various fields ranging from biophysics…
It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for the…
We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often…
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…
In this paper, we study the combinatorics of congruence subgroups of the modular group by generalizing results obtained in the non-modular case. For this, we define a notion of irreducible solutions from which we can build all the…
In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…
By employing polynomial-reduced KP integrability, combined with the string equation, this work establishes explicit relationships between the generalized Kontsevich model, the topological recursion of the spectral curve, and the geometry of…
In a previous paper (joint with Hausel and Rodriguez-Villegas) we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the…
In this paper we prove that assuming Schanuel's conjecture, an exponential polynomial in one variable over the algebraic numbers has only finitely many algebraic solutions. This implies a positive answer to Shapiro's conjecture for…
The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions and equivariant forms.