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In this paper we investigate the (classical) weights of mod p Siegel modular forms of degree 2 toward studying Serre’s conjecture for G S p 4 / Q . We first construct various theta operators on the space of such forms à la Katz and define the theta cycles for the specific theta operators. Secondly, we study the partial Hasse invariants on each Ekedahl–Oort stratum and their local behaviors. This enables us to obtain a kind of weight reduction theorem for mod p Siegel modular forms of degree 2 without increasing the level. [ABSTRACT FROM AUTHOR]

We show that the André motive of a hyper-Kähler variety X over a field K ⊂ C with b 2 (X) > 6 is governed by its component in degree 2. More precisely, we prove that if X 1 and X 2 are deformation equivalent hyper-Kähler varieties with b 2 (X i) > 6 and if there exists a Hodge isometry f : H 2 (X 1 , Q) → H 2 (X 2 , Q) , then the André motives of X 1 and X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of X 1 and X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true. [ABSTRACT FROM AUTHOR]

We prove a potential automorphy theorem for suitable Galois representations Γ F + → GSpin 2 n + 1 (F ¯ p) and Γ F + → GSpin 2 n + 1 (Q ¯ p) , where Γ F + is the absolute Galois group of a totally real field F + . We also prove results on solvable descent for GSp 2 n (A F +) and use these to put representations Γ F + → GSpin 2 n + 1 (Q ¯ p) into compatible systems of GSpin 2 n + 1 (Q ¯ l) -valued representations. [ABSTRACT FROM AUTHOR]

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell–Weil lattice and the geometric Picard number. [ABSTRACT FROM AUTHOR]

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious"—all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective ℓ -adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis' result should hold, giving Galois representations into certain groups intermediate between GL 2 and PGL 2 ; and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with an example. [ABSTRACT FROM AUTHOR]

Let Runiv be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(Runiv[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of Runiv. However, his method gives little information about the quotient ring before inverting p. We give a method for studying this quotient in certain cases, and carry it out in the simplest non-trivial case. Precisely, suppose that V0 is the trivial two dimensional representation and let R be the unique Zp-flat and reduced quotient of Runiv such that MaxSpec(R[1/p]) consists of ordinary representations with Hodge-Tate weights 0 and 1. We describe the functor of points of (a slightly modified version of) R and show that the irreducible components of Spec(R) are normal and Cohen-Macaulay, but not Gorenstein. As a consequence, we find that certain global deformation rings are torsion-free and Cohen-Macaulay, but not Gorenstein. [ABSTRACT FROM AUTHOR]

In this paper we construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre’s modularity conjecture, to totally real fields. [ABSTRACT FROM AUTHOR]

Given a compatible subsystem $$\{\rho _\ell \}_\ell $$ of n-dimensional $$\ell $$ -adic Galois representations arising from étale cohomology of any complete, non-singular variety over a number field K, we define $$\Gamma _\ell := \rho _\ell ({\text {Gal}}(\bar{K}/K))$$ and let $$\mathbf {G}_\ell $$ denote the Zariski closure of $$\Gamma _\ell $$ in $$\mathrm {GL}_n$$ . If $$\mathbf {G}_\ell $$ is of Type A in the sense that all simple composition factors are of type A in the Cartan-Killing classification, then $$\Gamma _\ell $$ is, in a suitable sense, maximal in $$\mathbf {G}_\ell $$ for all $$\ell \gg 0$$ . As a corollary, if $$\rho _\ell $$ is semisimple and $$\ell $$ is sufficiently large, then $$\mathbf {G}_\ell $$ is unramified. [ABSTRACT FROM AUTHOR]

The Galois representations associated to weight 1 eigenforms over $$\bar{\mathbb {F}}_{p}$$ are remarkable in that they are unramified at $$p$$ , but the effective computation of these modular forms presents challenges. One complication in this setting is that a weight 1 cusp form over $$\bar{\mathbb {F}}_{p}$$ need not arise from reducing a weight 1 cusp form over $$\bar{\mathbb {Q}}$$ . In this article we propose a unified Hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method that are motivated by the refined inverse Galois problem. [ABSTRACT FROM AUTHOR]

We prove a Néron-Ogg-Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $$l$$ -adic cohomology group is unramified. We also prove a $$p$$ -adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic. [ABSTRACT FROM AUTHOR]

Let $$\kappa \ge 6$$ be an even integer, $$M$$ an odd square-free integer, and $$f \in S_{2\kappa -2}(\Gamma _0(M))$$ a newform. We prove that under some reasonable assumptions that half of the $$\lambda $$ -part of the Bloch-Kato conjecture for the near central critical value $$L(\kappa ,f)$$ is true. We do this by bounding the $$\ell $$ -valuation of the order of the appropriate Bloch-Kato Selmer group below by the $$\ell $$ -valuation of algebraic part of $$L(\kappa ,f)$$ . We prove this by constructing a congruence between the Saito-Kurokawa lift of $$f$$ and a cuspidal Siegel modular form. [ABSTRACT FROM AUTHOR]