Related papers: Counting newforms with prescribed ramified supercu…

Given a newform with the Fourier expansion $\sum_{n=1}^\infty b(n)q^n\in\mathbb Z[[q]]$, a prime $p$ is said to be non-ordinary if $p\mid b(p)$. We exemplify several newforms of weight 4 for which the latter divisibility implies a stronger…

For an odd prime $p$ and a positive integer $n$, let ${_n}G_n[\cdots]_p$ denote McCarthy's $p$-adic hypergeometric function. In this article, we prove $p$-adic analogue of certain classical hypergeometric identities and using these…

Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$…

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate…

Let $k$ be an algebraically closed field of positive characteristic $p$. We describe the full lattice of subfunctors of the diagonal $p$-permutation functor $kR_k$ obtained by $k$-linear extension from the functor $R_k$ of linear…

Let $\mathcal{O}$ be a maximal order in the quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$. We prove two theorems that allow us to recover the structure of $\mathcal{O}$ from limited information. The first says that for…

Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the rational prime $p$. We completely classify the valuations of the $p^n$-torsion points…

The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in…

Let $S_k(\Gamma^{\mathrm{para}}(N))$ be the space of Siegel paramodular forms of level $N$ and weight $k$. Let $p\nmid N$ and let $\chi$ be a nontrivial quadratic Dirichlet character mod $p$. Based on our previous work, we define a linear…

We establish a connection between the superconformal index of $\mathcal{N}=4$ $U(N)$ SYM and the elliptic Ruijsenaars-Schneider integrable system. The index admits an expression in terms of elliptic Macdonald polynomials, which leads to a…

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special…

We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.

Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite.…

We use the Omnes representation to obtain the q-squared dependence of the form factors f+ and f0 for semileptonic H -> pi decays from the elastic pi H -> pi H scattering amplitudes, where H denotes a B or D meson. The scattering amplitudes…

The restriction of a supercuspidal representation of SL_2(k), for k a local nonarchimedean field, to a maximal compact subgroup decomposes as a multiplicity-free direct sum of irreducible representations. We explicitly describe this…

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

Let $\text{E}/\text{F}$ be a quadratic extension of non-Archimedean local fields with odd residual characteristic. In this paper, we give equivalent conditions for a simple supercuspidal representation $\pi$ of $\text{GL}(n, \text{E})$ to…

For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…