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We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation ρ ¯ {\bar{\rho}} of a p-adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K is totally ramified with n = 3 {n=3} or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ ¯ {\bar{\rho}} of given Hodge–Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general. [ABSTRACT FROM AUTHOR]

It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3. [ABSTRACT FROM AUTHOR]

Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve X 1 {X_{1}} defined over k, there exists a lifting X of the curve to the ring W ⁢ (k) {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X / W ⁢ (k) {X/W(k)}. These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π 1 ⁢ (X K) {\pi_{1}(X_{K})} of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson's uniformization theory of hyperbolic Riemann surfaces via Higgs bundles. [ABSTRACT FROM AUTHOR]

Let Fq be a finite field with q elements, F = Fq an algebraic closure of Fq, X1 an absolutely irreducible projective smooth curve over Fq, and S1 a finite set of closed points of X1. Let N1 be the cardinality of S1. Erasing the index 1 indicates extension of scalars to F. Replacing it by m indicates extension of scalars to Fqm ⊂ F. Let Fr be the Frobenius endomorphism of the curve X over F, deduced from the Fq-form X1 of X. For ℓ a prime not dividing q, the pullback by Fr is an autoequivalence of the category of smooth ℓ-adic sheaves (= Qℓ-local systems) on X - S. It induces a permutation Fr* of the set of isomorphism classes of irreducible smooth ℓ-adic sheaves of rank n on X - S, whose local monodromy at each s∈S is unipotent with a single Jordan block (= principal unipotent). Let T(X1,S1,n) be the number of fixed points of Fr* acting on this set. We will also consider the number of fixed points T(X1,S1,n,m) of the iterate Fr*,m of Fr*. As Fr*,m : X → X is the Frobenius, for X1/Fq replaced by Xm/Fqm, one has T(X1,S1,n,m) = T(Xm,Sm,n). The number T(X1,S1,n) is computed when N1 = 2 in [4] where it is shown that the T(X1,S1,n,m) are given by a formula reminiscent of a Lefschetz fixed point formula. The automorphic-Galois reciprocity is used there to reduce the computation of T(X1,S1,n) to counting automorphic representations on GL(n), and the assumption N1 ≥ 2 to move the counting to a division algebra, where the trace formula is easier to use. The trace formula used in Section 4 of [4] is based on a new, categorical approach to computing the trace formula in the very special case of counting that is needed there. It is not yet known if it extends to the use of Hecke correspondences and in the non-compact quotient case. Our aim here is to give an alternative proof of Section 4 of [4] using the usual trace formula in the compact quotient case. As stated in [4], after Lemma 4.9, the equivalence between the two approaches seems to us interesting. [ABSTRACT FROM AUTHOR]

Let G be a finite group. For each integral representation ρ of G we consider ρ-decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties ( X, H) with -action, of dimension equal to the degree of ρ, which admit a decomposition of the lattice for X into two G-invariant sublattices isotropic with respect to ℑ H, with one of the sublattices ℤ G-isomorphic to ρ. We give a construction for ρ-decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves. Conversely, if ρ is absolutely irreducible, we show that each ρ-decomposable p.p.a.v. is (isomorphic to) one of those constructed above, thereby characterizing them. In the case of irreducible, reduced root systems, we consider the natural representation of its associated Weyl group, apply the preceding general construction, and characterize completely the associated families of principally polarized abelian varieties, which correspond to modular curves. [ABSTRACT FROM AUTHOR]

We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a generalization of Wiman's sextics [38] and Edge's sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in the table www.manypoints.org, see [37]. [ABSTRACT FROM AUTHOR]

Let C be a linear code of length n and dimension k over the finite field F q m . The trace code Tr (C) is a linear code of the same length n over the subfield F q . The obvious upper bound for the dimension of the trace code over F q is m k. If equality holds, then we say that C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a denote the code obtained from C and a multiplier vector a ∈ ( F q m ) n . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m (k + h) , where h ≥ 0 is the Singleton defect of C. For the extremal case n = m (h + k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank. [ABSTRACT FROM AUTHOR]

In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered. [ABSTRACT FROM AUTHOR]

Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition . Let K be a special parahoric subgroup of G, in good position relative to ( Z, U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from to , we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 픽 p. We show that in the general case need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers. [ABSTRACT FROM AUTHOR]

An even unimodular 72-dimensional lattice Γ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant −7. The automorphism group of Γ contains the absolutely irreducible rational matrix group (2(7) × 2(25)) : 2. [ABSTRACT FROM AUTHOR]

We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups 3 D4( q) in characteristic coprime to q to any proper subgroup is reducible. [ABSTRACT FROM AUTHOR]

We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases. [ABSTRACT FROM AUTHOR]

We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 ⁢ M 12 , M 11 , and A 6 = O 2 ⁢ (M 10) over F 3 , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway's sporadic simple groups. [ABSTRACT FROM AUTHOR]

Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers p m , where p ≥ 5 and m ≥ 2 . In this paper, we study mod p m filtrations and mod p m theta cycles. We give a bound on some elements in the mod p m theta cycle ( m ≥ 2 ), and we exactly compute those values in the case that m = 2 . [ABSTRACT FROM AUTHOR]

We construct (2 , 3) -generators for the finite 8-dimensional orthogonal groups, proving the following results: the groups Ω 8 + ⁢ (q) and P ⁢ Ω 8 + ⁢ (q) are (2 , 3) -generated if and only if q ≥ 4 ; the groups Ω 8 - ⁢ (q) and P ⁢ Ω 8 - ⁢ (q) are (2 , 3) -generated for all q ≥ 2 . [ABSTRACT FROM AUTHOR]

In this article, we study compact Riemann surfaces of genus 𝑔 with an automorphism of prime order g + 1 . The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine and realise their full automorphism groups and compute their fields of moduli. We also study some aspects of their Jacobian varieties such as isogeny decompositions and complex multiplication. Finally, we determine the period matrix of the Accola–Maclachlan curve of genus four. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime, k a finitely generated field of characteristic different from ℓ , and X a smooth geometrically connected curve over k. Say a semisimple representation of π 1 ét ⁢ (X k ¯) is arithmetic if it extends to a finite index subgroup of π 1 ét ⁢ (X) . We show that there exists an effective constant N = N ⁢ (X , ℓ) such that any semisimple arithmetic representation of π 1 ét ⁢ (X k ¯) into GL n ⁡ (ℤ ℓ ¯) , which is trivial mod ℓ N , is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel's linearization theorem and the ℓ -adic form of Baker's theorem on linear forms in logarithms. [ABSTRACT FROM AUTHOR]

We prove Deligne's conjecture for symmetric fifth L-functions of elliptic newforms of weight greater than 5. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti–Whittaker periods of its symmetric cube functorial lift to GL 4 {\operatorname{GL}_{4}}. [ABSTRACT FROM AUTHOR]

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by two invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet–Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes. [ABSTRACT FROM AUTHOR]

We clarify the definition of the divisorial hull and recollect some basic facts. Then we correct Lemma 4.2 and Theorem 11.2 (1)–(2) in the original article. [ABSTRACT FROM AUTHOR]

We complete the description of automorphism groups of all nonsplit extensions of elementary abelian 2-groups by PSL 2 ⁢ (q) \mathrm{PSL}_{2}(q) , with 푞 odd, for an irreducible induced action. An application of this result to the theory of 휋-submaximal subgroups is given. [ABSTRACT FROM AUTHOR]

We study some embeddings of finite classical groups defined over field extensions, focusing on their geometry. The embedded groups are subgroups of classical groups lying outside the main Aschbacher classes. We concentrate on PG(8, q) where the embedded groups can be seen as automorphism groups of natural geometric objects: Hermitian Veroneseans, Twisted Hermitian Veroneseans and rational curves. [ABSTRACT FROM AUTHOR]

We prove strong multiplicity one results for Siegel eigenforms of degree two for the symplectic group Sp4(ℤ). [ABSTRACT FROM AUTHOR]

Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type y n = x ℓ (x m + 1). ${{y}^{n}}={{x}^{\ell }}\left({{x}^{m}}+1 \right).$ New results are obtained via certain subcovers of the nonsingular model of v N = u t 2 − u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$ where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve. [ABSTRACT FROM AUTHOR]

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a ℤ-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley's parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G}. In this paper, we determine the finite simple groups of type E 6 ⁢ (q) {E_{6}(q)} and E 6 2 ⁢ (q) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal. [ABSTRACT FROM AUTHOR]

Héthelyi and Külshammer showed that the number of conjugacy classes k(G) of any solvable finite group G whose order is divisible by the square of a prime p is at least (49 ⁢ p+1)/60. Here an asymptotic generalization of this result is established. It is proved that there exists a constant c > 0 such that, for any finite group G whose order is divisible by the square of a prime p, we have k ⁢ (G) ≥ cp. [ABSTRACT FROM AUTHOR]

G. Navarro raised the question of when two vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup. The present note provides a sufficient criterion for this to happen. This generalises a result by Navarro for simple modules over finite p-solvable groups, which is the main motivation for this note. [ABSTRACT FROM AUTHOR]

We compute an upper bound for the dimension of the tangent spaces at classical points of certain eigenvarieties associated with definite unitary groups, especially including the so-called critically refined cases. Our bound is given in terms of "critical types" and when our bound is minimized it matches the dimension of the eigenvariety. In those cases, which we explicitly determine, the eigenvariety is necessarily smooth and our proof also shows that the completed local ring on the eigenvariety is naturally a certain universal Galois deformation ring. [ABSTRACT FROM AUTHOR]

The minimal base size b(G) for a permutation group G is a widely studied topic in permutation group theory. Z. Halasi and K. Podoski [Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 2016, 8, 5857–5887] proved that b(G) ≤ 2 for coprime linear groups. Motivated by this result and the probabilistic method used by T. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber [Personal communication, Bielefeld, 2017] whether or not, for coprime linear groups G ≤ GL(V), there exists a constant c such that the probability that a random c-tuple is a base for G tends to 1 as |V| → ∞. While the answer to this question is negative in general, it is positive under the additional assumption that G is primitive as a linear group. In this paper, we show that almost all 11-tuples are bases for coprime primitive linear groups. [ABSTRACT FROM AUTHOR]

Let V be a vector space over a field k. We call a subgroup G ⊂ GL(V) a fixed-point subgroup if det(1−g) = 0 for all g ∈ G. Let q be a power of a prime. In this paper, we classify the fixed-point subgroups of GL3(q). [ABSTRACT FROM AUTHOR]

We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists. [ABSTRACT FROM AUTHOR]

We relate p-adic families of Jacobi forms to big Heegner points constructed by B. Howard, in the spirit of the Gross–Kohnen–Zagier theorem. We view this as a GL ⁢ (2) {\mathrm{GL}(2)} instance of a p-adic Kudla program. [ABSTRACT FROM AUTHOR]

Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = xℓ(xm+1). [ABSTRACT FROM AUTHOR]

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In this article, we generalize some works of Bertolini–Darmon and Vatsal on anticyclotomic L-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p-adic L-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists. [ABSTRACT FROM AUTHOR]

We prove an integral R = 𝐓 {R=\mathbf{T}} theorem for odd two-dimensional p-adic representations of G 𝐐 {G_{\mathbf{Q}}} which are unramified at p, extending results of [5] to the non-minimal case. We prove, for any p, the existence of Katz modular forms modulo p of weight one which do not lift to characteristic zero. [ABSTRACT FROM AUTHOR]

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in eight variables. The argument develops work of Colliot-Thélène, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off three hyperbolic planes. [ABSTRACT FROM AUTHOR]

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ⁡ ( 5 , q 2 ) {\operatorname{PG}(5,q^{2})} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, whose associated semifield has center containing 𝔽 q {\mathbb{F}_{q}}. Equivalently, we classify, up to isotopy, commutative semifields of order q 6 {q^{6}} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, with middle nucleus containing 𝔽 q 2 {\mathbb{F}_{q^{2}}} and center containing 𝔽 q {\mathbb{F}_{q}}. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, if Deligne's “petits camarades conjecture” holds, then a Langlands type correspondence holds also for p-adic coefficients on a smooth curve over a finite field. As an application, we prove that any overconvergent F -isocrystal of rank less than or equal to 2 on a smooth curve is ı-mixed. [ABSTRACT FROM AUTHOR]

Suppose that B is a Brauer p-block of a finite group G with a unique modular character φ .We prove that φ is liftable to an ordinary character of G (which moreover is p-rational for odd p). This confirms the basic set conjecture for these blocks. [ABSTRACT FROM AUTHOR]

The Iwahori--Hecke algebras of type A are deformations of the group algebras of the symmetric groups, and can be defined as quotients of the group algebras of the braid groups. We determine the image of the braid groups inside these algebras, when defined over a finite field, in the semisimple case, and for suitably large (but controllable) order of the defining (quantum) parameter. [ABSTRACT FROM AUTHOR]

Using the overconvergent cohomology modules introduced by Ash--Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash--Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of p-adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier's interpolation theorem. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that if y'' = f (y, y', t, α, β,…) is a "generic" Painlevé equation (that is, an equation in one of the families PI--PVI but with the relevant complex parameters α, β…. algebraically independent), and if y1,… yn are solutions such that y1; y'1; y2; y'2,… yn, y'n are algebraically dependent over C (t), then already for some 1 ≤ i < j ≤ n, yi, y'i, yj, yj are algebraically dependent over C(t). The proof combines results by the Japanese school on "irreducibility" of the Painlevé equations, with the trichotomy theorem for strongly minimal sets in differentially closed fields. [ABSTRACT FROM AUTHOR]

Let be a finite Galois CM-extension of number fields with Galois group G. In an earlier paper, the author has defined a module over the center of the group ring which coincides with the Sinnott-Kurihara ideal if G is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of implies the minus part of the equivariant Tamagawa number conjecture at an odd prime p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that Iwasawa's μ-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the minus- p-part of the equivariant Tamagawa number conjecture from the vanishing of μ for the same class of extensions. As an application we prove the non-abelian Brumer and Brumer-Stark conjecture outside the 2-primary part for every monomial Galois extension of provided that certain μ-invariants vanish. [ABSTRACT FROM AUTHOR]

We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and genus 2 curves. As an application, we construct an absolutely simple 2-dimensional abelian variety over Q all of whose quadratic twists must have positive rank, according to the Birch-Swinnerton-Dyer conjecture. [ABSTRACT FROM AUTHOR]

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties. [ABSTRACT FROM AUTHOR]

We propose two optimal representations for the elements of trace zero subgroups of twisted Edwards curves. For both representations, we provide efficient compression and decompression algorithms. The efficiency of the algorithm is compared with the efficiency of similar algorithms on elliptic curves in Weierstrass form. [ABSTRACT FROM AUTHOR]

We show that the sporadic simple group M(22), the exceptional group of Lie type ²E6(2) and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group. [ABSTRACT FROM AUTHOR]