arXiv subjects

Krzysztof Klosin

Publications and source records attributed to Krzysztof Klosin.

At least 19 recordsLinked to original sources

arXiv2025

Klingen Eisenstein series congruences and modularity

We construct a mod $\ell$ congruence between a Klingen Eisenstein series (associated to a classical newform $ϕ$ 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 $H^1_f(\mathbf{Q}, \textrm{ad}^0ρ_ϕ(2-k)\otimes \mathbf{Q}_{\ell}/\mathbf{Z}_{\ell})$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine-Laffaille universal deformation ring of $\overlineρ_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\overlineρ_ϕ(k-2))$ is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf{Q}, \textrm{ad}^0\overlineρ_ϕ(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

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arXiv2022

$R=T$ theorems for weight one modular forms

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $χ$. For certain non-quadratic $χ$ we prove an $R=T$ result for $T$ the weight 1 specialisation of the Hida Hecke algebra acting on non-classical weight 1 forms, under the assumption that no two Hida families congruent to an Eisenstein series cross in weight 1. For quadratic $χ$ we prove that the quotient of $R$ corresponding to deformations split at $p$ is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.

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arXiv2020

Irreducibility of limits of Galois representations of Saito-Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit $σ_2$ of a sequence of irreducible essentially self-dual Galois representations $σ_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus ρ\oplus χ$ with $ρ$ irreducible, two-dimensional of determinant $χ$, where $χ$ is the mod $p$ cyclotomic character. More precisely, we assume that $σ_k$ are crystalline (with a particular choice of weights) and Siegel-ordinary at $p$. Such representations arise in the study of $p$-adic families of Siegel modular forms and properties of their limits as $k\to 2$ appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by $p$-adic $L$-values of an elliptic modular form (giving rise to $ρ$) which we assume are non-zero.

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arXiv2019

Deformations of Saito-Kurokawa type and the Paramodular Conjecture (with an appendix by Cris Poor, Jerry Shurman, and David S. Yuen)

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\barσ$ with $\barσ^{\rm ss}=χ^{k-2} \oplus ρ\oplus χ^{k-1}$, where $χ$ is the mod $p$ cyclotomic character and $ρ$ is an absolutely irreducible reduction of the Galois representation $ρ_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f(\mathbf{Q}, ρ_f(1-k)\otimes \mathbf{Q}_p/\mathbf{Z}_p)$ and $H^1_f(\mathbf{Q}, ρ(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\barσ$ then the universal deformation ring is a dvr. When $k=2$ this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of the abelian surface of conductor 731. When $k>2$, we obtain an $R^{\rm red}=T$ theorem showing modularity of all such deformations of $\barσ$.

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arXiv2019

On Ribet's isogeny for $J_0(65)$

Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.

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arXiv2019

Galois extensions and a Conjecture of Ogg

Let $N=pq$ be a product of two distinct primes. There is an isogeny $J_0(N)^{\rm new}\to J^N$ defined over $\mathbf{Q}$ between the new quotient of $J_0(N)$ and the Jacobian of the Shimura curve attached to the indefinite quaternion algebra of discriminant $N$. In the case when $p=2,3,5,7,13$, Ogg made predictions about the kernels of these isogenies. We show that Ogg's conjecture is not true in general. Afterwards, we propose a strategy for proving results toward Ogg's conjecture in certain situations. Finally, we discuss this strategy in detail for $N=5\cdot 13$.

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arXiv2018

Modularity of residual Galois extensions and the Eisenstein ideal

For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal $J$ is non-principal, then $d >n$. When $F=\mathbf{Q}$ we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the non-principality of $J$ that can be checked in practice, allowing us to furnish an example where $d>n$.

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arXiv2018

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $U(n,n)(A_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_ϕ$ (of an elliptic modular form $ϕ$) on $U(n,n)(A_{\mathbf{Q}})$ proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence prime for $I_ϕ$ is controlled by the $\ell$-divisibility of a product of special values of the symmetric square $L$-function of $ϕ$. We close the paper by computing an example when our main theorem applies.

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arXiv2016

On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

In this paper we study deformations of mod $p$ Galois representations $τ$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $τ_1$ and $τ_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_Σ(F, {\rm Hom}(τ_2, τ_1)) \subset {\rm Ext}^1(τ_2, τ_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_Σ(F, {\rm Hom}(τ_2, τ_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{B}$ admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if $τ$ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).

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arXiv2016

A $p$-adic Hermitian Maass lift

For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $χ_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $χ_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $χ_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a $p$-adic analytic family of automorphic forms in the Maass space of level $p$.

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arXiv2015

On the norms of $p$-stabilized elliptic newforms (with an appendix by Keith Conrad)

Let $f \in S_κ(Γ_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $λ_f(p)$ for a prime $p$ not dividing $N$. Let $α_p$ and $β_p$ be complex numbers satisfying $α_p + β_p = λ_f(p)$ and $α_p β_p = p^{κ-1}$. We calculate the norm of $f_{p}^{α_p}(z) = f(z) - β_{p} f(pz)$ as well as the norm of $U_p f$, both classically and adelically. We use these results along with some convergence properties of the Euler product defining the symmetric square L-function of $f$ to give a `local' factorization of the Petersson norm of $f$.

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arXiv2013

On Ihara's lemma for degree one and two cohomology over imaginary quadratic fields

We prove a version of Ihara's Lemma for degree q=1,2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k\geq 2) over an imaginary quadratic field with torsion (prime power) coefficients. This extends an earlier result of the author which concerned the case k=2, q=1. Our method here is different and uses results of Diamond and Blasius-Franke-Grunewald. We discuss the relationship of our main theorem to the problem of the existence of level-raising congruences.

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arXiv2013

On higher congruences between automorphic forms

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π_0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π_0 and other automorphic forms. We apply this result to several situations including the congruences described by Mazur's Eisenstein ideal.

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arXiv2011

Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form

Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the action of all Hecke operators. When D_K is prime we obtain a Hecke-equivariant descent from S_{k,-k/2}^M to the space of elliptic cusp forms S_{k-1}(D_K, χ_K), where χ_K is the quadratic character of K. For a given ϕ\in S_{k-1}(D_K, χ_K), a prime l >k, we then construct (mod l) congruences between the Maass form corresponding to ϕand hermitian modular forms orthogonal to S_{k,-k/2}^M whenever the l-adic valuation of L^{alg}(\Sym^2 ϕ, k) is positive. This gives a proof of the holomorphic analogue of the unitary version of Harder's conjecture. Finally, we use these congruences to provide evidence for the Bloch-Kato conjecture for the motives \Sym^2 ρ_ϕ(k-3) and \Sym^2 ρ_ϕ(k), where ρ_ϕ denotes the Galois representation attached to ϕ.

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arXiv2011

On deformation rings of residually reducible Galois representations and R=T theorems

We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer groups associated with rho_1 and rho_2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaiche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R=T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over the rationals.

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arXiv2010

A deformation problem for Galois representations over imaginary quadratic fields

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL_2(A_F) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

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arXiv2007

Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let ρ_f be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρ_f in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attached to the Galois module \ad^0ρ_f|_{G_K} (-1), and K= Q(i). Our method uses an idea of Ribet in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U(2,2).

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arXiv2007

Ihara's lemma for imaginary quadratic fields

An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal P of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard P-degeneracy maps between the cuspidal sheaf cohomology H^1_!(X_0, M_0)^2 --> H^1_!(X_1, M_1) is Eisenstein. Here X_0 and X_1 are analogues over F of the modular curves X_0(N) and X_0(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL(2) which is due to Serre.

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