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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 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.
Let
$n$
be either
$2$
or an odd integer greater than
$1$
, and fix a prime
$p>2(n+1)$
. Under standard ‘adequate image’ assumptions, we show that the set of components of
$n$
-dimensional
$p$
-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on
$n$
) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2)179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.
By making use of the ‘creative microscoping’ method, Guo and Zudilin [‘Dwork-type supercongruences through a creative $q$-microscope’, Preprint, 2020, arXiv:2001.02311] proved several Dwork-type supercongruences, including some conjectures of Swisher. In this paper, we apply the Guo–Zudilin method to prove a new Dwork-type supercongruence, which uniformly generalises several conjectures of Swisher.
We sharpen earlier work of Dabrowski on near-perfect power values of the quartic form $x^{4}-y^{4}$, through appeal to Frey curves of various signatures and related techniques.
Let
$F$
be a totally real field in which
$p$
is unramified. Let
$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$
be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place
$v$
above
$p$
. Let
$\mathfrak{m}$
be the corresponding Hecke eigensystem. We describe the
$\mathfrak{m}$
-torsion in the
$\text{mod}\,p$
cohomology of Shimura curves with full congruence level at
$v$
as a
$\text{GL}_{2}(k_{v})$
-representation. In particular, it only depends on
$\overline{r}|_{I_{F_{v}}}$
and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic
$\text{GL}_{2}(\mathbf{F}_{q})$
-projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.200(1) (2015), 1–96].
We adapt a technique of Kisin to construct and study crystalline deformation rings of
$G_{K}$
for a finite extension
$K/\mathbb{Q}_{p}$
. This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For
$K$
unramified over
$\mathbb{Q}_{p}$
and Hodge–Tate weights in
$[0,p]$
, we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of
$\mathbb{Q}_{p}$
, with Hodge–Tate weights in
$[0,p]$
, are potentially diagonalizable.
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 prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$. This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations’, Duke Math. J.149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$.
Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$. Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant
\begin{equation}
\lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of }
\text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big)
\end{equation}
at all $f\in\mathcal{H}(\overline{\rho})$, under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$, generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$.
For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.
which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A $q$-analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl.466 (2018), 749–761].
In this article we construct a p-adic three-dimensional eigenvariety for the group
$U$
(2,1)(
$E$
), where
$E$
is a quadratic imaginary field and
$p$
is inert in
$E$
. The eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta, Iovita and Stevens [
$p$
-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch–Kato conjecture for some Galois characters of
$E$
, extending the results of Bellaiche and Chenevier to the case of a positive sign.
We consider families of Siegel eigenforms of genus
$2$
and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for
$\operatorname{GL}_{2}$
, via a
$p$
-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the
$p$
-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for
$\operatorname{GL}_{2}$
to an eigenvariety for
$\operatorname{GSp}_{4}$
, while the remainder appear as isolated points on the eigenvariety.
J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
We construct the
$p$
-adic standard
$L$
-functions for ordinary families of Hecke eigensystems of the symplectic group
$\operatorname{Sp}(2n)_{/\mathbb{Q}}$
using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group
$\operatorname{Sp}(4n)_{/\mathbb{Q}}$
, which guarantees the nonvanishing of local zeta integrals and allows us to
$p$
-adically interpolate the restrictions of the Siegel Eisenstein series to
$\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$
.
Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.
We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over
$\mathbb{Q}$
of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use
$R=T$
theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from
$\text{GSp}(4)$
to
$U(4)$
. In that case we obtain a corollary for abelian surfaces.
A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of
$G$
-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type
$A_{1}^{n}$
,
$C_{2}$
, and
$\mathbf{F}_{p}$
-split groups of type
$A_{2}$
(this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular
$3$
-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.