Related papers: Congruences between modular forms modulo prime pow…

In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence--modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two…

In this article we prove level raising for cuspidal eigenforms modulo prime powers (for odd primes) of weight $k\geq 2$ and arbitrary character, extending the result in weight two established by the work of Tsaknias and Wiese and…

Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result…

Among topological modular forms with level structure, $TMF_0(7)$ at the prime $3$ is the first example that had not been understood yet. We provide a splitting of $TMF_0(7)$ at the prime 3 as $TMF$-module into two shifted copies of $TMF$…

Let $F$ be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ associated to Hilbert modular forms of regular weight.

Let $p$ be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_p)$. We also determine the structure of certain…

Let $N$ be a positive integer and let $f$ be a meromorphic modular function of level $N$ with rational Fourier coefficients. For a prime $p$, define a function $f_p$ on the complex upper half-plane $\mathbb{H}$ by \begin{equation*}…

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a…

A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\times}, where D is a definite quaternion algebra…

Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under r is an m^th power…

We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $\sigma=(\sigma_1,\dots,\sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map…

It is shown that each complex conjugate of a meromorphic modular form for $\mathrm{SL}_2(\mathbb{Z})$ of any complex weight $p$ occurs as the image of a harmonic modular form under the operator $2i y^p \, \partial_{\bar z}$. These harmonic…

We compare modular forms of characteristic $p>0$ (i.e. Drinfeld's modular forms) and automorphic forms. We prove that spaces of these modular forms (which are of characteristic $p$) can be described by function spaces of characteristic…

We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine…

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in…

We use Galois cohomology methods to produce optimal mod $p^d$ level lowering congruences to a $p$-adic Galois representation that we construct as a well chosen lift of a given residual mod $p$ representation. Using our explicit Galois…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\rho$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained…

We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…