Related papers: Paramodular forms coming from elliptic curves

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$…

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce…

We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of…

This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithmetic of the Fourier coefficients of modular forms and their generalisations. In Chapter 2, we compute lower…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to…

This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series…

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric…

The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients…

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients…

We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

We investigate the ramification of modular parametrizations of elliptic curves over Q at the cusps. We prove that if the modular form associated to the elliptic curve has minimal level among its twists by Dirichlet characters, then the…

This is a continuation of our previous paper 1502.01744. We examine a class of non-commutative algebras A that depend on an elliptic curve and a translation automorphism of it. They may be defined in terms of the 4-dimensional Sklyanin…

With the method of the relative trace formula and the classification of simple supercuspidal representations, we establish some Fourier trace formulas for automorphic forms on $PGL(2)$ of cubic level. As applications, we obtain a…

In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura…

Generically, one can attach to a Q-curve C octahedral representations Gal(Qbar/Q) --> GL(2,Fbar_3) coming from the Galois action on the 3-torsion of those abelian varieties of GL_2-type whose building block is C. When C is defined over a…

We describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze…

We give closed formulas for the first few expansion coefficients of the basic modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\). Here the rank \(r\) is larger or equal to \(3\), and the forms in question include the coefficient forms…

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic…

In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the…