Related papers: Some congruences for traces of singular moduli

We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric…

In the recent article arXiv:1606.03351, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end they…

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials…

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then…

Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this…

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are…

We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli.

In this short note, we prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka,…

We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.

In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and…

Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to…

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur ("Modular Curves and the Eisenstein Ideal") from a…

We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of…

Graded Hecke algebras can be constructed in terms of equivariant cohomology and constructible sheaves on nilpotent cones. In earlier work, their standard modules and their irreducible modules where realized with such geometric methods. We…

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois…

Fix a prime number p and choose, once and for all, an embedding of the algebraic closure of Q into Qp. Let k and N be integers, and suppose N is not divisible by p. If f is a modular form of weight k, level N, and trivial character which is…

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\\i+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ where $B_n$ is the $n$-th…

We formulate some properties of a conjectural object $X_{fun}(r,n)$ parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give formulas for $\goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its reductions at $\goth…