We construct a mod $\ell $ congruence between a Klingen Eisenstein series (associated with a classical newform $\phi $ of weight k) and a Siegel cusp form f with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch–Kato Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0\rho _{\phi }(2-k)\otimes \mathbf {Q}_{\ell }/\mathbf {Z}_{\ell })$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine–Laffaille universal deformation ring of ${\overline {\rho }}_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is cyclic. For this, we prove a result about extensions of Fontaine–Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

Let $\pi $ be a cuspidal, cohomological automorphic representation of an inner form G of $\operatorname {{PGL}}_2$ over a number field F of arbitrary signature. Further, let $\mathfrak {p}$ be a prime of F such that G is split at $\mathfrak {p}$ and the local component $\pi _{\mathfrak {p}}$ of $\pi $ at $\mathfrak {p}$ is the Steinberg representation. Assuming that the representation is noncritical at $\mathfrak {p}$, we construct automorphic $\mathcal {L}$-invariants for the representation $\pi $. If the number field F is totally real, we show that these automorphic $\mathcal {L}$-invariants agree with the Fontaine–Mazur $\mathcal {L}$-invariant of the associated p-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.

In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections.

We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$: It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$. In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$, and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

We use the theory of trianguline $(\varphi ,\Gamma )$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at p, including those with characteristic p coefficients. The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17], [Che13] after bounding the slope, and we carry out a Taylor–Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.

We investigate Eisenstein congruences between the so-called Euler systems of Garrett–Rankin–Selberg type. This includes the cohomology classes of Beilinson–Kato, Beilinson–Flach, and diagonal cycles. The proofs crucially rely on different known versions of the Bloch–Kato conjecture, and are based on the study of the Perrin-Riou formalism and the comparison between the different p-adic L-functions.

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 weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.

A (folklore?) conjecture states that no holomorphic modular form $F(\tau )=\sum _{n=1}^{\infty } a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau }$, such that its anti-derivative $\sum _{n=1}^{\infty } a_nq^n/n$ has integral coefficients in the q-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note, we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.

Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is $\leq 2$. In this paper, we show that the eigenvariety for $\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$, constructed by Andreatta, Iovita, and Pilloni, is proper at integral weights for $p\geq 3$. We also prove a weaker result for $p=2$.

The formula of the title relates p-adic heights of Heegner points and derivatives of p-adic L-functions. It was originally proved by Perrin-Riou for p-ordinary elliptic curves over the rationals, under the assumption that p splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$-invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $-operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb {P}^1$, landing in the compactly supported completed $\mathbb {C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$, and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb {P}^1$. Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.

We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.

We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

We find and prove a class of congruences modulo 4 for eta-products associated with certain ternary quadratic forms. This study was inspired by similar conjectured congruences modulo 4 for certain mock theta functions.

It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.