Related papers: Bilinear forms in Weyl sums for modular square roo…

Let $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference…

We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our main result enables the derivation of elementary arithmetic…

Local Weyl modules over two-dimensional currents with values in $gl_r$ are deformed into spaces with bases related to parking functions. Using this construction we 1) propose a simple proof that dimension of the space of diagonal…

The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl $m$-function for several self-adjoint extensions with interesting boundary conditions: separated,…

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…

We study tree-level biadjoint scalar amplitudes in the language of $D$-modules. We construct left ideals in the Weyl algebra $D$ that allow a holonomic representation of $n$-point amplitudes in terms of the linear partial differential…

We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. Under the obvious restriction map, the space of biinvariants is proved isomorphic to the Weyl group…

Let $q$ be an odd power of a prime $p\in \mathbb{N}$, and $\mathrm{PPSP}(\sqrt{q})$ be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over $\mathbb{F}_q$…

We prove a spectral summation formula for the product of four Fourier coefficients of half-integral weight cusp forms in Kohnen's subspace. The other side of the formula involves certain generalized class numbers of pairs of quadratic forms…

We establish the first moment bound $$ \sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon} $$ for triple product $L$-functions, where $\Psi$ is a fixed Hecke-Maass form on…

The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier…

We study the splitting fields of the family of polynomials $f_n(X)= X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties. Serre related the function on primes $N_p(f_n)$, for a…

Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q)…

We prove a power saving upper bound for the sum of Fourier coefficients $\rho_f(\cdot)$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 Theorem of Duke and Iwaniec, and the…

The Springer correspondence makes a link between the characters of a Weyl group and the geometry of the nilpotent cone of the corresponding semisimple Lie algebra. In this article, we consider a modular version of the theory, and show that…

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative…

We develop further the theory of $q$-deformations of real numbers introduced by Morier-Genoud and Ovsienko, and focus in particular on the class of real quadratic irrationals. Our key tool is a $q$-deformation of the modular group…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that…

We study the representation theory of the Soergel calculus algebra $ A_w := \mbox{End}_{{\mathcal D}_{(W,S)}} (\underline{w}) $ over $\mathbb C$ in type $\tilde{A}_1$. We generalize the recent isomorphism between the nil-blob algebra…