In this paper, we determine mod 2 Galois representations ρ ¯ ψ , 2 : G K : = Gal ( K ¯ / K ) → GSp 4 ( 𝔽 2 ) associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family X 0 5 + X 1 5 + X 2 5 + X 3 5 + X 4 5 - 5 ψ X 0 X 1 X 2 X 3 X 4 = 0 , ψ ∈ K defined over a number field K under the irreducibility condition of the quintic trinomial f ψ below. Applying this result, when K = F is a totally real field, for some at most quadratic totally real extension M / F , we prove that ρ ¯ ψ , 2 | G M is associated to a Hilbert–Siegel modular Hecke eigen cusp form for GSp 4 ( 𝔸 M ) of parallel weight three. In the course of the proof, we observe that the image of such a mod 2 representation is governed by reciprocity of the quintic trinomial f ψ ( x ) = 4 x 5 - 5 ψ x 4 + 1 , ψ ∈ K whose decomposition field is generically of type 5-th symmetric group S 5 . This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of Gal ( F ¯ / F ) due to Shu Sasaki or Pilloni–Shu and several Langlands functorial lifts for Hilbert cusp forms. Then, it guarantees the existence of a desired Hilbert–Siegel modular cusp form of parallel weight three matching with the Hodge type of the compatible system in question. A twisted version is also discussed and it is related to general quintic trinomials.

Abstract Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak{g}$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda $-independence results on the algebraic monodromy groups of compatible systems of $\lambda $-adic Galois representations of number fields.

Let ρ be an n-dimensional odd direct sum of irreducible mod p odd Galois representations, each of dimension 1, 2, or 3. Assume that the 3-dimensional constituents satisfy the ADP conjecture, p is sufficiently large, the Serre conductor N of ρ is squarefree, and the Serre conductors of the irreducible constituents of ρ are all greater than 1. We prove that ρ is attached to a Hecke eigenclass in the cohomology of Γ0(n,N) with coefficients in M, where M is a finite-dimensional irreducible F−p[GLn(Fp)]-representation F tensored with a nebentype character ϵ. The level N, the weight F, and the character ϵ are those predicted by a conjecture of Ash, Doud, and Pollack, which generalizes Serre’s conjecture for GL2. The proof uses modular symbols and their restriction to the Borel–Serre boundary.

Let p p be a fixed odd prime, and let K K be a finite extension of Q p \mathbf {Q}_p with ring of integers O K \mathcal {O}_K . The Emerton-Gee stack for G L 2 GL_2 is a stack of ( φ , Γ ) (\varphi , \Gamma ) -modules. The stack, denoted X 2 \mathcal {X}_2 , can be interpreted as a moduli stack of representations of the absolute Galois group of K K with p p -adic coefficients. The reduced part of the Emerton-Gee stack, denoted X 2 , r e d \mathcal {X}_{2, red} , is an algebraic stack defined over a finite field of characteristic p p and can be viewed as a moduli stack of Galois representations with mod p p coefficients. The irreducible components of X 2 , r e d \mathcal {X}_{2, red} are labelled in a natural way by Serre weights, which are the irreducible mod p p representations of G L 2 ( O K ) GL_2(\mathcal {O}_K) . Each irreducible component of X 2 , r e d \mathcal {X}_{2, red} has dimension [ K : Q p ] [K:\mathbf {Q}_p] . In this article, we compute G L 2 ( O K ) GL_2(\mathcal {O}_K) -extensions of pairs of Serre weights and, motivated by the conjectural categorical p p -adic Langlands programme, we show that a non-trivial extension of a pair of non-isomorphic Serre weights implies a codimension 1 1 intersection of the corresponding irreducible components. The converse of this statement is true if the Serre weights are chosen to be sufficiently generic. Furthermore, we show that the number of top-dimensional components in a codimension 1 1 intersection is related to the depth of the extensions of the corresponding Serre weights.

Abstract Under some technical assumptions of a global nature, we establish the weight part of Serre’s conjecture for mod p Galois representations for CM fields that are tamely ramified and sufficiently generic at p .

Inspired by [Pan26], we give a new proof that for an overconvergent modular eigenform f of weight 1 + k with k ∈ ℤ ≥ 1 , assuming that its associated Galois representation ρ f : Gal ℚ → GL 2 ( ℚ ¯ p ) is irreducible, then f is classical if and only if the associated Galois representation ρ f is de Rham at p . For the proof, we prove that theta operator θ k coincides with Fontaine operator in a suitable sense.

Bogomolov property and Galois representations

The lifting problem for Galois representations

Abstract We classify all instances of the condition $$a_{p}(f) \equiv x \bmod \lambda $$ a p ( f ) ≡ x mod λ being related to a congruence on the prime p , where $$a_{p}(f)$$ a p ( f ) denotes the p th Fourier coefficient of a classical normalised cuspidal eigenform f and $$\lambda $$ λ is a prime in the number field generated by the Fourier coefficients of f . This classification is done in terms of the (projective) image of the mod $$\lambda $$ λ Galois representation associated with f and extends work by Swinnerton-Dyer. We highlight that for $$x = 0$$ x = 0 , this condition is more often implied by a congruence on the prime p than the general value of $$a_{p}(f) \bmod \lambda $$ a p ( f ) mod λ . Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra’s numerical criterion. In this paper, we relate L(1,pi ,{{,textrm{Ad},}}^circ ) to the congruence ideals for cohomological cuspidal automorphic representations pi of textrm{GL}_n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of L(1,pi ,{{,textrm{Ad},}}^circ ). This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of textrm{GL}_n, its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.

Abstract Let be an odd prime, and suppose that and are weight two newforms sharing the same irreducible Galois representation modulo . We establish a transition formula relating the ‐invariants and in the case where their underlying modular varieties have the same dimension. For abelian extensions , this essentially removes the condition present in the earlier work of Greenberg–Vatsal and Emerton–Pollack–Weston.

Abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

We construct a new Euler system for the Galois representation V_{f,\chi} attached to a newform f of weight 2r\geq 2 twisted by an anticyclotomic Hecke character \chi . The Euler system is anticyclotomic in the sense of Jetchev–Nekovář–Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch–Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa–Greenberg main conjecture for V_{f,\chi} . In particular, in the case where the base-change of f to our imaginary quadratic field has root number +1 and \chi has higher weight (which implies that the complex L -function L(V_{f,\chi},s) vanishes at the center), our results show that the Bloch–Kato Selmer group of V_{f,\chi} is nonzero, as predicted by the Bloch–Kato conjecture; and if in addition a certain distinguished class \kappa_{f,\chi} is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch–Kato conjecture for V_{f,\chi} were left wide open by the earlier approaches using Heegner cycles and/or Beilinson–Flach elements. Our construction is based instead on a generalization of the Gross–Kudla–Schoen diagonal cycles.

Abstract We compare the Iwasawa invariants of fine Selmer groups of p -adic Galois representations over admissible p -adic Lie extensions of a number field K to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let K be a number field, let V be a p -adic representation of the absolute Galois group $$G_K$$ of K , and choose a $$G_K$$ -invariant lattice $${T \subseteq V}$$ . We study the fine Selmer groups of $${A = V/T}$$ over suitable p -adic Lie extensions $$K_\infty /K$$ , comparing their corank and $$\mu $$ -invariant to the corank and the $$\mu $$ -invariant of the Iwasawa module of ideal class groups in $$K_\infty /K$$ . In the second part of the article, we compare the Iwasawa $$\mu $$ - and $$l_0$$ -invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $$\mathbb {Z}_p$$ -extensions of K .

One of the fundamental properties of automorphic forms is that their periods – integrals against certain distinguished cycles or distributions – give special values of L -functions. The Langlands program posits that automorphic representations for a reductive group G correspond to (generalizations of) Galois representations into its Langlands dual group \check G . Periods and L -functions are specific ways to extract numerical invariants from the two sides of the Langlands program; in interesting cases, they match with one another.Relative Langlands Duality is the systematic study of the manifestations of this matching at all “tiers” of the Langlands program (global, local, geometric, arithmetic, etc.). A key point is a symmetric conceptualization of both sides: periods arise from suitable Hamiltonian G -actions G\circlearrowright M and L -functions from suitable Hamiltonian {\check G} -actions {\check G}\circlearrowright \check M .

Abstract In the 1960s, Atkin discovered congruences modulo primes for the partition function in arithmetic progressions modulo , where is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes . Here we consider (for primes ) the ‐colored generalized Frobenius partition functions ; these are natural level analogues of . For each such we prove that there are similar congruences for for all primes outside of an explicit finite set depending on . To prove the result we first construct, using both theoretical and computational methods, cusp forms of half‐integral weight on which capture the relevant values of modulo . We then apply previous work of the authors on the Shimura lift for modular forms with the eta multiplier together with tools from the theory of modular Galois representations.

Locally induced Galois representations with exceptional residual images

abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

Let C / ℚ be a genus 2 curve whose Jacobian J / ℚ has real multiplication by a quadratic order in which 7 splits. We describe an algorithm which outputs twists of the Klein quartic curve which parametrise elliptic curves whose mod 7 Galois representations are isomorphic to a sub-representation of the mod 7 Galois representation attached to J / ℚ . Applying this algorithm to genus 2 curves of small conductor in families of Bending and Elkies–Kumar we exhibit a number of genus 2 Jacobians whose Tate–Shafarevich groups (unconditionally) contain a non-trivial element of order 7 which is visible in an abelian three-fold.

Let E be an elliptic curve over a number field L and for a finite set S of primes, let ρ E,S :Gal(L ¯/L)→GL 2 (ℤ S ) be the S-adic Galois representation. If L∩ℚ(ζ n )=ℚ for all positive integers n whose prime factors are in S, then detρ E,S :Gal(L ¯/L)→ℤ S × is surjective. We say that a finite index subgroup H⊆GL 2 (ℤ S ) is minimal if det:H→ℤ S × is surjective, but det:K→ℤ S × is not surjective for any proper closed subgroup K of H. We show that there are no minimal subgroups of GL 2 (ℤ S ) unless S={2}, while minimal subgroups of GL 2 (ℤ 2 ) are plentiful. We give models for all the genus 0 modular curves associated to minimal subgroups of GL 2 (ℤ 2 ), and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at 2 and with minimal 2-adic image.