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In the present article, we formulate a conjectural uniform error term in the Chebotarev–Sato–Tate distribution for abelian surfaces Q -isogenous to a product of not Q ¯ -isogenous non-CM-elliptic curves, established by the author in Amri (Eur J Math, 2023. https://doi.org/10.1007/s40879-023-00682-5, Theorem 1.1). As a consequence, we provide a conditional direct proof to the generalized Lang–Trotter conjecture recently formulated and studied in Chen et al. (Ramanujan J, 2022). [ABSTRACT FROM AUTHOR]

We present new examples of PGL 2 (F 7) and PSL 2 (F 7) number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's PGL 2 (F 7) polynomial, (ii) Modular method: computation of Katz modular forms of weight one over F ‾ 7 with prime level, and (iii) Searching for polynomials with prescribed ramification. Method (i) quickly generates many PGL 2 (F 7) number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of PGL 2 (F 7) or PSL 2 (F 7) number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases. [ABSTRACT FROM AUTHOR]

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]

We study the Weil representation ρ of a curve over a p -adic field with potential reduction of compact type. We show that ρ can be reconstructed from its stable reduction. For superelliptic curves of the form y n = f (x) at primes p whose residue characteristic is prime to the exponent n we make this explicit. [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]

It is known that any Galois representation ρ : G Q → GL (2 , F p) with determinant equal to the mod- p cyclotomic character, arises from the p -torsion of an elliptic curve over Q , if and only if p ≤ 5. In dimension g = 2 , when p ≤ 3 , it is again known that any Galois representation valued in GSp (4 , F p) with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes p and dimensions g ≥ 2. When g ≥ 2 and (g , p) ≠ (2 , 2) , (2 , 3) , (3 , 2) , we prove the existence of a Galois representation over Q valued in GSp (2 g , F p) with cyclotomic similitude character, that cannot arise as the p -torsion representation of any g -dimensional abelian variety over Q. [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 give a general conjecture concerning the existence of Eisenstein congruences between weight k ≥ 3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ of conductor N. The special cases M = 1 and M = p prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number. We classify the subgroups G of Sp 4 (F ℓ) and GSp 4 (F ℓ) that act irreducibly on F ℓ 4 , but such that every element of G fixes an F ℓ -vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ℓ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ for which some abelian surface A / Q fails the local-global principle for isogenies of degree ℓ . [ABSTRACT FROM AUTHOR]

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 1 2 -torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 1 2 -torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2 , 3 or 4 -torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves. [ABSTRACT FROM AUTHOR]

In this paper, we show that the conjectural mod p local Langlands correspondence can be realized in the mod p cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field F and of mod l (≠ p) local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13]. [ABSTRACT FROM AUTHOR]

This paper investigates congruences of -adic representations arising from effective A -motives defined over a global function field K. We give a criterion for two congruent -adic representations coming from strongly semi-stable effective A -motives to be isomorphic up to semi-simplification when restricted to decomposition groups of suitable places of K. This is a function field analog of Ozeki–Taguchi's criterion for ℓ -adic representations of number fields. Motivated by a non-existence conjecture on abelian varieties over number fields stated by Rasmussen and Tamagawa, we also show that there exist no strongly semi-stable effective A -motives with some constrained. [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].

Let $G$ be a split reductive group over the ring of integers in a $p$ -adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$ , valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$ -adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for $G$ -valued representations. In particular, we give a root theoretic condition on the $p$ -adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. [ABSTRACT FROM AUTHOR]

Let ℓ ≥ 5 be prime. For the partition function p (n) and 5 ≤ ℓ ≤ 31 , Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c ∈ F ℓ , we prove congruences of the form p (ℓ Q 3 n + β 0) ≡ c ⋅ p (ℓ Q n + β 1) (mod ℓ) for infinitely many primes Q. For a positive integer r , let p r (n) be the r -colored partition function. Our methods yield similar congruences for p r (n). In particular, if r is an odd positive integer for which ℓ > 5 r + 19 and 2 r + 2 ≢ 2 ± 1 (mod ℓ) , then we show that there are infinitely many congruences of the form p r (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Our methods involve the theory of modular Galois representations. [ABSTRACT FROM AUTHOR]

Let f 1 and f 2 be nonzero Hecke eigenforms on SL 2 (Z). For X > 0 , let B (f 1 , f 2 , X) be the number of ratios of λ 1 (p) to λ 2 (p) for primes p ≤ X , where λ i (p) denotes the Hecke eigenvalue of f i at p. In this paper, we prove that if f 1 is not a constant multiple of f 2 , then B (f 1 , f 2 , X) ≫ f 1 , f 2 (log ⁡ X) 1 15 . [ABSTRACT FROM AUTHOR]

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]

Two abelian varieties A_1, A_2 over a number field K are strongly iso-Kummerian if the torsion fields K(A_1[d]) and K(A_2[d]) coincide for all d \geq 1. For all g \geq 4 we construct geometrically simple, strongly iso-Kummerian g-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair. [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]

Let p be an odd prime and e p be its irregularity index. If 4 e p + 8 < p − 1 2 we construct a Galois representation with image in the diagonal torus of GSp 4 (F p) that lifts to a characteristic 0 representation unramified at all primes ℓ ≠ p with image containing a finite index subgroup of GSp 4 (Z p). [ABSTRACT FROM AUTHOR]

Let E be a Q -isogeny class of elliptic curves defined over Q. The isogeny graph associated to E is a graph which has a vertex for each element of E and an edge for each Q -isogeny of prime degree that maps one element of E to another element of E , with the degree recorded as a label of the edge. The isogeny-torsion graph associated to E is the isogeny graph associated to E where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. The main result of the article is a determination of which isogeny-torsion graphs associated to Q -isogeny classes of elliptic curves defined over Q correspond to infinitely many j -invariants. [ABSTRACT FROM AUTHOR]

We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in ℙ q 2 n + 1 of degree at least three. We also consider the L -functions of quadratic twists of fixed degree of an elliptic curve over a function field q (t). To determine the typical Galois group in the elliptic curve setting requires using some known cases of the Birch and Swinnerton-Dyer conjecture. This extends and generalizes work of Chavdarov, Katz and Jouve. [ABSTRACT FROM AUTHOR]

For a given elliptic curve E defined over the rationals, we study the density of primes p satisfying gcd (# E (F p) , p − 1) = 1 and give a conjectural value for this density with strong heuristic evidence for most elliptic curves, in an appropriate sense. [ABSTRACT FROM AUTHOR]

Let G be a split reductive group with dim ⁡ Z (G) ≤ 1. We show that for any prime p that is large enough relative to G , there is a finitely ramified Galois representation ρ : Γ Q → G (Z p) with open image. We also show that for any given integer e , if the index of irregularity of p is at most e and if p is large enough relative to G and e , then there is a Galois representation ρ : Γ Q → G (Z p) ramified only at p with open image, generalizing a theorem of Ray [8]. The first type of Galois representation is constructed by lifting a suitable Galois representation into G (F p) using a lifting theorem of Fakhruddin–Khare–Patrikis [4] , and the second type of Galois representation is constructed using a variant of the argument in Ray's work [8]. [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]

Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over Q and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over Q that are not Q ¯ -isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture. [ABSTRACT FROM AUTHOR]

Keating and Rudnick studied the variance of the polynomial von Mangoldt function Λ : F q [ t ] → C in arithmetic progressions and short intervals using two equidistribution results by Katz. Hall, Keating and Roditty-Gershon then generalised the result for arithmetic progressions for a von Mangoldt function Λ ρ attached to a Galois representation ρ : Gal ( F q (t) ‾ / F q (t)) → GL m ( Q ‾ ℓ). We employ a recent equidistribution result by Sawin in order to generalise the corresponding result for short intervals for Λ ρ. [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]

Let ρ ¯ f , λ be the residual Galois representation attached to a newform f and a prime ideal λ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals λ such that ρ ¯ f , λ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to 4 , 4 or 5 . We also develop explicit criteria to check the reducibility of ρ ¯ f , λ , leading to an algorithm that computes the exact set of such λ 's. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz' operator in characteristic zero, and we prove a new Sturm bound theorem. [ABSTRACT FROM AUTHOR]

In a recent paper of Dokchitser-Dokchitser-Maistret-Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first étale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture. [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 determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$ -adic field, when $l \neq p$ , relating them to the space of $q$ -power-stable semisimple conjugacy classes in the dual group. As a consequence, we give a local proof of the $l \neq p$ Breuil–Mézard conjecture of the author, in the tame case. [ABSTRACT FROM AUTHOR]