Related papers: Congruences between modular forms and related modu…

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod…

In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to…

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…

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the…

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…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

We say that a normalized modular form is of CM type modulo $\ell$ by an imaginary quadratic field $K$ if its Fourier coefficients $a_p$ are congruent to $0$ modulo a prime $\mathcal L\mid \ell$ for every prime $p$ that is inert in $K$. In…

The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to…

Let N be a positive integer and let f be a newform of weight 2 on \Gamma_0(N). In earlier joint work with K. Ribet and W. Stein, we introduced the notions of the modular number and the congruence number of the quotient abelian variety A_f…

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

We generalize a result of Galatius and Venkatesh which relates the graded module of cohomology of locally symmetric spaces to the graded homotopy ring of the derived Galois deformation rings, by removing certain assumptions, and in…

In this paper we prove a general theorem about congruences between automorphic forms on a reductive group G which is compact at infinity modulo the center. If the rank is one, this essentially reduces to Ribet's level-raising theorem. We…

We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $\lambda_f:T \to O$ measures the failure of $R,T$ to…

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…

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of…

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor--Wiles hypotheses and is tamely ramified…

Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of…

Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(\mathbb…