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Let N be a positive integer. Let H be a subgroup of GL (2 , Z / N Z) of level N and let E be an elliptic curve defined over the rationals with j E ≠ 0 , 1728 . Then the image ρ ¯ E , N Gal Q ¯ / Q , of the mod-N Galois representation attached to E, is conjugate to a subgroup of H if and only if E corresponds to a non-cuspidal rational point on the modular curve X H generated by H. In this article, we are interested when ρ ¯ E , N Gal Q ¯ / Q is conjugate to H. More precisely, we classify all subgroups H of GL (2 , Z / N Z) that are direct products of groups of distinct prime-power level for which X H contains infinitely many non-cuspidal rational points but there is no elliptic curve E / Q such that ρ ¯ E , N Gal Q ¯ / Q is conjugate to H itself. [ABSTRACT FROM AUTHOR]

Let F q be the finite field with q ≥ 5 elements, A : = F q [ T ] and F : = F q (T) . Assume that q is odd and take | · | to be the absolute value at ∞ that is normalized by | T | = q . Given a pair w = (g 1 , g 2) ∈ A 2 with g 2 ≠ 0 , consider the associated Drinfeld module ϕ w : A → A { τ } of rank 2 defined by ϕ T w = T + g 1 τ + g 2 τ 2 . Fix integers c 1 , c 2 ≥ 1 and define | w | : = max { | g 1 | 1 c 1 , | g 2 | 1 c 2 } . I show that when ordered by height, there is a positive density of pairs w = (g 1 , g 2) , such that the T-adic Galois representation attached to ϕ w is surjective. [ABSTRACT FROM AUTHOR]

We survey the connections between modular forms and representations of Galois groups that are predicted by the Langlands programme. We focus in particular on the applications of congruences between modular forms (through automorphy lifting theorems) to an improved understanding of these connections, including the author's recent joint work with James Newton on the existence of the symmetric power liftings of Hilbert modular forms. [ABSTRACT FROM AUTHOR]

We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

Consider a family of modular forms of weight 2, all of whose residual (mod p) Galois representations are isomorphic. It is well known that their corresponding Iwasawa λ -invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these λ -invariants grow or remain stable. [ABSTRACT FROM AUTHOR]

We study the Eisenstein ideal for modular forms of even weight k >2 and prime level N. We pay special attention to the phenomenon of extra reducibility: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur--Tate L-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we recall Kato's formulation of this main conjecture in the case of a family of motives given by twists by characters of conductor N and p-power order and its relation to other formulations of the equivariant main conjecture. [ABSTRACT FROM AUTHOR]

Let kk be a nonnegative integer. Let KK be a number field and OK{{\mathcal{O}}}_{K} be the ring of integers of KK. Let ℓ≥5\ell \ge 5 be a prime and vv be a prime ideal of OK{{\mathcal{O}}}_{K} over ℓ\ell . Let ff be a modular form of weight k+12k+\frac{1}{2} on Γ0{\Gamma }_{0}(4) such that its Fourier coefficients are in OK{{\mathcal{O}}}_{K}. In this article, we study sufficient conditions that if ff has the form f(z)≡∑n=1∞∑i=1taf(sin2)qsin2(modv)f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers si{s}_{i}, then ff is congruent to a linear combination of iterated derivatives of a single theta function modulo vv.

Let p be the p-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of GLn of unitary type. Under very mild hypotheses on p, we prove the vanishing of the (Bloch-Kato) adjoint Selmer group of p. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields. [ABSTRACT FROM AUTHOR]

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a p-adic field. We also show that these points are dense in the subspace parameterizing deformations with the determinant equal to a fixed crystalline character. Our proof is purely local and works for all p-adic fields and all residual Galois representations. [ABSTRACT FROM AUTHOR]

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank 7 motive whose Galois group is of type G2 in the cohomology of Siegel modular varieties of genus 3. Under some additional local hypotheses we also show automorphic multiplicity 1 as well as meromorphic continuation of the spin L-functions. [ABSTRACT FROM AUTHOR]

We use a classification result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with ℓ-adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of M3,n and M3,n for n ≤ 14 and of local systems 핍λ on A3 for |λ| ≤ 16. [ABSTRACT FROM AUTHOR]

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]

The classification of the local Galois representations using (φ , Γ) -modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of (φ q , Γ LT) -modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from (ψ q , Γ LT) -modules. We also compute the Iwasawa cohomology over the Lubin-Tate extensions in terms of the ψ q -operator acting on the étale (φ q , Γ LT) -module attached to the local Galois representation. Moreover, we generalize the notion of (φ q , Γ LT) -modules over the coefficient ring R and show that the equivalence given by Kisin and Ren extends to the Galois representations over R. This equivalence allows us to generalize our results to the case of coefficient rings. [ABSTRACT FROM AUTHOR]

Let F be a newform of weight 2k on Γ 0 (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ , N) = 1 . For each fundamental discriminant D, let χ D be a quadratic character associated with quadratic field Q (D) . Assume that for each D, the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is non-negative. Let W ℓ + (resp. W ℓ - ) be the set of positive (resp. negative) fundamental discriminants D with (D , N) = 1 such that the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is zero. We prove that for each sign ϵ , if W ℓ ϵ is a non-empty finite set, then W ℓ ϵ ⊂ 1 , (- 1) ℓ - 1 2 ℓ. By this result, we prove that if ϵ is the sign of (- 1) k , then k ≥ ℓ - 1 or k = ℓ - 1 2. These are applied to obtain a lower bound for # { D ∈ W ℓ ϵ : | D | ≤ X } and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q . To prove these results, first we refine Waldspurger's formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations. [ABSTRACT FROM AUTHOR]

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford–Tate group, ℓ -adic monodromy groups, and the Sato–Tate group. Assuming the Mumford–Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree. [ABSTRACT FROM AUTHOR]

In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms. [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 investigate a question of Burns and Sano concerning the structure of the module of Euler systems for a general p-adic representation. Assuming the weak Leopoldt conjecture, and the vanishing of µ-invariants of natural Iwasawa modules, we obtain an Iwasawa-theoretic classification criterion for Euler systems which can be used to study this module. This criterion, taken together with Coleman's conjecture on circular distributions, leads us to pose a refinement of the aforementioned question for which we provide strong, and unconditional, evidence. We furthermore answer this question in the affirmative in many interesting cases in the setting of the multiplicative group over number fields. As a consequence of these results, we derive explicit descriptions of the structure of the full collection of Euler systems for the situations in consideration. [ABSTRACT FROM AUTHOR]

We describe a systematic investigation into the existence of congruences between the mod p torsion modules of elliptic curves defined over Q, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod p congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes p≥7 to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than 500000. We also show that while such congruences exist for each p≤17, there are none for p≥19 in the database, in line with a strong form of the Frey-Mazur conjecture. [ABSTRACT FROM AUTHOR]

We study an analog of Serre's modularity conjecture for projective representations ...:Gal(.../K)→PGL2(k), where K is a totally real number field. We prove cases of this conjecture when k=F5. [ABSTRACT FROM AUTHOR]

We provide examples of abelian surfaces over number fields K whose reductions at almost all good primes possess an isogeny of prime degree ℓ rational over the residue field, but which themselves do not admit a K-rational ℓ -isogeny. This builds on work of Cullinan and Sutherland. When K = Q , we identify certain weight-2 newforms f with quadratic Fourier coefficients whose associated modular abelian surfaces A f exhibit such a failure of a local–global principle for isogenies. [ABSTRACT FROM AUTHOR]

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 Q → GL 4 (Q ¯ p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 ⊕ ρ ⊕ 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 → 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. [ABSTRACT FROM AUTHOR]

In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms.

Crystabelline representations are representations of the absolute Galois group G Q p over Q ¯ p that become crystalline on G F for some abelian extension F / Q p . Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by p 2 is crystabelline. In this article, we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo p reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the 2 -adic eigencurve of tame level 1 are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the 3 -adic eigencurve; Coleman's halo conjecture and the ghost conjecture which give predictions about the slopes on the p -adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of two-dimensional crystabelline representations whose reductions modulo p are reducible. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in (0 , p - 1 2) \ Z are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur. [ABSTRACT FROM AUTHOR]

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 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]

We extend the computations in [2–4] to find the cohomology in degree five of a congruence subgroup of SL 4 (Z) with coefficients in a field twisted by a nebentype character, along with the action of the Hecke algebra on the cohomology. This is the top cuspidal degree. For each Hecke eigenclass we find, we produce the unique Galois representation that appears to be attached to it. The computations require serious modifications to our previous algorithms. Nontrivial coefficients add a layer of complication to our data structures. New possibilities must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder to report when the attached Galois representation is uniquely determined by our data. [ABSTRACT FROM AUTHOR]

For every prime p ≥ 5 for which a certain condition on the class group Cl (Q (μ p)) is satisfied, we construct a p -adic analytic Galois extension of the infinite cyclotomic extension Q (μ p ∞ ) with some special ramification properties. In greater detail, this extension is unramified at primes above p and tamely ramified above finitely many rational primes and its Galois group over Q (μ p ∞ ) is isomorphic to a finite index subgroup of SL 2 (Z p) which contains the principal congruence subgroup. For the primes 107, 139, 271 and 379 such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to p. [ABSTRACT FROM AUTHOR]

In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for the supercuspidal prime p = 2. The same for odd supercuspidal primes are determined by Bhattacharya-Ghate. We also treat the case of odd unramified supercuspidal primes of level zero also removing a mild hypothesis of them. As an intermediate step, we write down a description of the inertial Galois representation even for p = 2 generalizing the construction of Ghate-Mézard. Some numerical examples using Sage and LMFDB are provided supporting some of our theorems. [ABSTRACT FROM AUTHOR]

Abstract We prove that irreducible residual crystalline representations of the absolute Galois group of an unramified extension of Q p have smooth representable crystalline framed deformation problems, provided that the Hodge–Tate weights lie in the Fontaine–Laffaille range. We then extend this result to the flagged lifting problem associated to any Fontaine–Laffaille upper triangular representation whose flag is of maximal length. We calculate the relative dimension of these various crystalline lifting functors in terms of the underlying Hodge–Tate weight structures, and also apply these results to give an alternative proof of the fact that every such residual representation admits a so-called "universally twistable lift". Finally we give some brief indications as to the various directions in which these results might be generalised. [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]

Abstract Let A be an abelian variety defined over a number field K. The number of torsion points that are rational over a finite extension L is bounded polynomially in terms of the degree [ L : K ] of L over K. Under the following three conditions, we compute the optimal exponent for this bound in terms of the dimension of abelian subvarieties and their endomorphism rings: (1) A is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; (2) A is of "Lefschetz type", that is, the Mumford–Tate group is the group of symplectic or orthogonal similitudes which commute with the endomorphism ring; (3) A satisfies the Mumford–Tate Conjecture. This result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Krasoń, we also prove the Mumford–Tate Conjecture for a few new cases of abelian varieties of Lefschetz type. [ABSTRACT FROM AUTHOR]

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 GL2, 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 GL2 to an eigenvariety for GSP4 , while the remainder appear as isolated points on the eigenvariety. [ABSTRACT FROM AUTHOR]

Abstract Let F / Q p be a finite extension. We explore the universal supersingular mod p representations of GL 2 (F) by computing a basis for their spaces of invariants under the pro- p Iwahori subgroup. This generalizes works of Breuil and Schein (from Q p and the totally ramified cases to an arbitrary extension F / Q p). Using these results we then construct, for an unramified F / Q p , quotients of the universal supersingular modules which have as quotients all the supersingular representations of GL 2 (F) with a GL 2 (O F) -socle that is expected to appear in the mod p local Langlands correspondence. A construction in the case of an extension of Q p with inertia degree 2 and suitable ramification index is also presented. [ABSTRACT FROM AUTHOR]

Abstract Given two pure representations of the absolute Galois group of an ℓ -adic number field with coefficients in Q ‾ p (with ℓ ≠ p), we show that the Frobenius-semisimplifications of the associated Weil–Deligne representations are twists of each other by an integral power of a certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations. [ABSTRACT FROM AUTHOR]

Abstract We present a strategy to obtain explicit equations for the modular double covers associated respectively to both a split and a non-split Cartan subgroup of GL 2 (F p) with p prime. Then we apply it successfully to the level 13 case. [ABSTRACT FROM AUTHOR]

Abstract For a representation of the absolute Galois group of the rationals over a finite field of characteristic p , we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine–Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at p arising from Fontaine–Laffaille theory. [ABSTRACT FROM AUTHOR]

Let p and l be distinct primes, and let p be an orthogonal or symplectic representation of the absolute Galois group of an l-adic field over a finite field of characteristic p. We define and study a liftable deformation condition of lifts of p 'ramified no worse than p', generalizing the minimally ramified deformation condition for GLn studied in Clozel et al. [ Automorphy for some l-adic lifts of automorphic mod l Galois representations , Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields. [ABSTRACT FROM AUTHOR]

We develop methods to study 2-dimensional 2-adic Galois representations ρ of the absolute Galois group of a number field K , unramified outside a known finite set of primes S of K , which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on K and S , we show how to determine the determinant det ⁡ ρ , whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K = Q , and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis [2] . [ABSTRACT FROM AUTHOR]

This article is an introduction to the early life and work of Robert P. Langlands, creator and founder of the Langlands program. The story is, to a large extent, told by Langlands himself, in his own words. Our focus is on two of Langlands' major discoveries: automorphic L-functions and the principle of functoriality. It was Langlands' desire to communicate his excitement about his newly discovered objects that resulted in his famous letter to Andr'e Weil. This article is aimed at a general mathematical audience and we have purposely not included the more technical aspects of Langlands' work. [ABSTRACT FROM AUTHOR]