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If in a given rank r, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number \ell, there is an absolutely irreducible \ell-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r if once there is an irreducible rank r complex linear representation with determinant isomorphic to this torsion character, then for any \ell, there is an absolutely irreducible one of rank r and determinant this given character, which is defined over \bar { \mathbb {Z}}_\ell. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for \ell \ge 3). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. [ABSTRACT FROM AUTHOR]

Let \sigma and \tau denote a pair of absolutely irreducible p-ordinary and p-distinguished Galois representations into \operatorname {GL}_2(\overline {\mathbb {F}}_p). Given two primitive forms (f,g) such that \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where \overline {\rho }_f\cong \sigma and \overline {\rho }_g\cong \tau, we show that the Iwasawa Main Conjecture for the double product \rho _f\otimes \rho _g depends only on the residual Galois representation \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p). More precisely, if IMC(f\otimes g) is true for one pair (f,g) with \overline {\rho }_f \cong \sigma and \overline {\rho }_g\cong \tau and whose \mu-invariant equals zero, then it is true for every congruent pair too. [ABSTRACT FROM AUTHOR]

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [-H,H], is O(H^{3.91}). More generally, we show that if n \geqslant 3 and n \notin \{ 7, 8, 10 \} then there are O(H^{n-1.017}) monic, irreducible polynomials of degree n with integer coefficients in [-H,H] and Galois group not containing A_n. Save for the alternating group and degrees 7,8,10, this establishes a 1936 conjecture of van der Waerden. [ABSTRACT FROM AUTHOR]

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. [ABSTRACT FROM AUTHOR]

We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves X over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many n-th order Weil bounds, and ending asymptotically at the Drinfeld-Vlădut, bound. We relate this set of bounds to those of Oesterl'e, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve X can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface X × X together with the general theory of symmetric Toeplitz matrices. We also give some interpretation for the defect of asymptotically exact towers. This is achieved by pushing further the classical Weil proof in terms of euclidean relationships between classes in the euclidean part FX of the numerical group Num(X × X) generated by classes of graphs of iterations of the Frobenius morphism. The noteworthy Toeplitz shape of their intersection matrix takes a central place by implying a very strong cyclic structure on FX. [ABSTRACT FROM AUTHOR]

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy. [ABSTRACT FROM AUTHOR]

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E over an imaginary quadratic field K is -1. When the root number of E over K is 1, we first establish the structure theorem of the p^\infty-Selmer group of E over K. The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various "Kolyvagin systems" and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p^\infty-Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p-converse to the theorem of Gross–Zagier and Kolyvagin. [ABSTRACT FROM AUTHOR]

Let p > 3 be a prime number and let GQp be the absolute Galois group of Qp. In this paper, we find Galois stable lattices in the 3-dimensional irreducible semi-stable non-crystalline representations of GQp with Hodge-Tate weights (0, 1, 2) by constructing the corresponding strongly divisible modules. We also compute the Breuil modules corresponding to the mod p reductions of these strongly divisible modules and determine which of the original representations has an absolutely irreducible mod p reduction. [ABSTRACT FROM AUTHOR]

Let q be a prime power and let G be an absolutely irreducible subgroup of GLd(F), where F is a finite field of the same characteristic as Fq, the field of q elements. Assume that G ≅ G(q), a quasisimple group of exceptional Lie type over Fq which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from G to the standard copy of G(q). If G ≅ ³D4(q) with q even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle. [ABSTRACT FROM AUTHOR]

Let X be a normal geometrically connected variety over a finite field κ of characteristic p. Let (ρλ : π1(X) → GLn(Eλ))λ be any semisimple E-rational compatible system, where E is a number field and λ ranges over the finite places of E not above p. We derive new properties on the monodromy groups of such systems for almost all λ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. Key inputs to our results are automorphic methods and the Langlands correspondence over global function fields proved by L. Lafforgue. To say more, let (̅ρλ : π1(X) → GLn(kλ))λ be the corresponding mod-λ system, where for every λ by Oλ and kλ we denote the valuation ring and the residue field of Eλ, and where the reduction is done with respect to some π1(X)-stable Oλ-lattice Æ'©λ of En λ. Let also Ggeo λ be the Zariski closure of ρλ(π1(X̅κ)) in GLn,E, and let Ggeo λ be its schematic closure in AutOλ(Æ'©λ). Assume in the following that the algebraic groups Ggeo λ are connected. We prove that for almost all λ the group scheme Ggeo λ is semisimple over Oλ, and its special fiber agrees with the Nori envelope of ρλ(π1(X̅κ)). A comparable result under different hypotheses was proved by A. Cadoret, C.-Y. Hui, and A. Tamagawa using other methods. As an intermediate result, we show for X a curve that any potentially tame compatible system of mod-λ representations can be lifted to a compatible system over a number field; this implies for almost all λ the semisimplicity of the restriction ρλ | π1(Xκ). Finally, we establish adelic openness for (ρλ| π1(Xκ))λ in the sense of C. Y. Hui and M. Larsen, for E = Q in general, and for E ⊋ Q under additional hypotheses. [ABSTRACT FROM AUTHOR]

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles''s famous paper (1995). Internal to group theory, $1$-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology $H^1_{operatorname {gen}}(G,L) :=underset {qto infty }{lim } H^1(G(q),L)$ (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group $G(q)$ of Lie type, with absolutely irreducible coefficients $L$ (in the defining characteristic of $G$), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on $H^1(G(q),L)$ itself, still depending only on the root system. The generic $H^1$ result, and related results for $operatorname {Ext}^1$, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules $Delta ^{text {rm red}}(lambda ), nabla _{text {rm red}}(lambda )$, indexed by dominant weights $lambda $, for a reductive group $G$. The modules $Delta ^{text {rm red}}(lambda )$ and $nabla _{text {rm red}}(lambda )$ arise naturally from irreducible representations of the quantum enveloping algebra $U_zeta $ (of the same type as $G$) at a $p$th root of unity, where $p>0$ is the characteristic of the defining field for $G$. Finally, we apply our $operatorname {Ext}^1$-bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on $H^1(G(q),L)$. [ABSTRACT FROM AUTHOR]

We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L-functions: the ordinary, polynomial (the p-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model. [ABSTRACT FROM AUTHOR]

We study the twisted knot module for the universal deformation of an SL2-representation of a knot group and introduce an associated L-function, which may be seen as an analogue of the algebraic p-adic L-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we compute the L-function by some concrete examples for 2-bridge knots. [ABSTRACT FROM AUTHOR]

Majorana representations have been introduced by Ivanov [ Cambridge Tracts in Mathematics , Cambridge University Press, Cambridge, 2009] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A_{12} (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]). In this paper we prove that A_{12} has two possible Majorana sets, one of which is the set \mathcal X_b of involutions of cycle type 2^2, the other is the union of \mathcal X_b with the set \mathcal X_s of involutions of cycle type 2^6. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_{12} that fall into the class of Fischer involutions when A_{12} is embedded in the Monster. We prove that A_{12} has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_{12}, satisfies the Straight Flush Conjecture (see A. A. Ivanov [ Contemp. Math. , Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_{12}. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8\leq n\leq 12. [ABSTRACT FROM AUTHOR]

Let G be a finite group and K a number field. We construct a G-extension E/F, with F of transcendence degree 2 over K, that specializes to all G-extensions of K_\mathfrak {p}, where \mathfrak {p} runs over all but finitely many primes of K. If furthermore G has a generic extension over K, we show that the extension E/F has the so-called Hilbert–Grunwald property. These results are compared to the notion of essential dimension of G over K, and its arithmetic analogue. [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be a smooth quasi-projective complex variety. We prove that any G-irreducible, G-cohomologically rigid local system on X with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S.) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when G= \mathrm {GL}_n, and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for G-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity. [ABSTRACT FROM AUTHOR]

Let $E$ be an extraspecial-type group and $V$ a faithful, absolutely irreducible $k[E]$-module, where $k$ is a finite field. Let $G$ be the normalizer in $GL(V)$ of $E$. We show that, with few exceptions, there exists a $v\in V$ such that the restriction of $V$ to $C_H(v)$ is self-dual whenever $H\le G$ and $(\vert H\vert, \vert V\vert)=1$. [ABSTRACT FROM AUTHOR]

Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are: \par {\bf I.} {\em Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant.} \par {\bf II.} {\em If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group).} \par {\bf III.} {\em Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically'' equivalent.} \par This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent. \par We assume throughout that the characteristic of $K$ does not divide $2|G|$. \par Solutions to {\bf I} and {\bf II} are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is ``split''. The results for {\bf III} are strongest when the degrees of the absolutely irreducible representations of $G$ are odd -- for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index -- and, in the case that $K$ is a local or global field, when the representations are split. [ABSTRACT FROM AUTHOR]

This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for $G$ an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group $A$ is the additive group of a finite-dimensional vector space $V$ over a locally finite field $F$ of prime characteristic $p$, and the automorphism group $G$ is a simple infinite absolutely irreducible subgroup of $GL(V)$. Thus $G$ is isomorphic to an infinite simple periodic group of Lie type, and $G$ is realized in $GL(V)$ via a twisted tensor product $\phi$ of infinitesimally irreducible representations. If $S$ is a Sylow $p$-subgroup of $G$ and if $\langle v\rangle$ is the unique line in $V$ stabilized by $S$, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus $T_G$ on $\langle v\rangle$. At present, we are able to handle the case where $\phi$ is a rational representation with character field equal to $F$. [ABSTRACT FROM AUTHOR]

We consider systems of partial differential equations equivariant under the Euclidean group $\mathbf{E}(n)$ and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when $n=1$ and $n=2$ and for reaction-diffusion equations with general $n$, reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of $\mathbf{E}(n)$. The representation theory of $\mathbf{E}(n)$ is driven by the irreducible representations of $\mathbf{O}(n-1)$. For $n=1$, this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When $n=2$, there are precisely two significantly different types of reduced equation: \textit{scalar} and \textit{pseudoscalar}, corresponding to the trivial and nontrivial one-dimensional representations of $\mathbf{O}(1)$. There are infinitely many possibilities for each $n\ge3$. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $ \mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $ \sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $ \mathbb{Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $ \mathbb{Q}$ by $ \mathbb{Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $ \mathcal{V}_{M}$ (resp., $ \mathcal{S}_{M}$). If $v\in \mathcal{V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all $ {\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{S}_{M}$. Then $V(O_{M})\not =\varnothing $. We also prove two approximation theorems for $M$. [ABSTRACT FROM AUTHOR]

In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo p-adically closed fields. [ABSTRACT FROM AUTHOR]

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let k be an algebraically closed field of characteristic zero and let X be an irreducible affine algebraic variety over k. Consider the linear difference equation σ (Y) = AY, where A ∈ GLn(k(X)(x)) and σ is the shift operator σ (x) = x + 1. Assume that the Galois group G of the above equation over k(X)(x) is defined over k(X), i.e., the vanishing ideal of G is generated by a finite set S ⊂ k(X)[X,1/det (X)]. For a c ∈ X, denote by vc the map from k[X] to k given by vc(ƒ) = ƒ(c) for any ƒ∈ k[X]. We prove that the set of c ∈ X satisfying that vc(A) and vc(S) are well-defined and the affine variety in GLn(k) defined by vc(S) is the Galois group of σ (Y) = vc(A)Y over k(x) is Zariski dense in X. We apply our result to van der Put-Singer's conjecture which asserts that an algebraic subgroup G of GLn(k) is the Galois group of a linear difference equation over k(x) if and only if the quotient G/G° by the identity component is cyclic. We show that if van der Put-Singer's conjecture is true for k = C, then it will be true for any algebraically closed field k of characteristic zero. [ABSTRACT FROM AUTHOR]

We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank 1 and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5, and 6, compute the parameters of the associated anticanonical codes, and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters, and some of them turn out to beat the best known codes listed on the database codetable. [ABSTRACT FROM AUTHOR]

We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig. Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases. Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G. E. Wall. [ABSTRACT FROM AUTHOR]

In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by the Christoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring's problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations. [ABSTRACT FROM AUTHOR]

Let p ≥ 7 be a prime number, and letf be a normalized eigen-newform with good reduction at p such that its pth Fourier coefficient vanishes. We construct a rank-two Euler system attached to the p-adic realization of the symmetric square motive of f. Furthermore, we show that the nontriviality is guaranteed by the nonvanishing of the leading term of the relevant L-value and the nonvanishing of a certain p-adic period modulo p. [ABSTRACT FROM AUTHOR]