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The subject of this paper is the study of small complete arcs in PG(2,q), for q odd, with at least (q+1)∕2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample to a misconception in the literature. [ABSTRACT FROM AUTHOR]

We study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in Z/ℓ$\mathbb {Z}/\ell$. We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish. [ABSTRACT FROM AUTHOR]

Let K be a complete, discretely valued field with finite residue field and GK its absolute Galois group. The subject of this note is the study of the set of positive integers d for which there exists an absolutely irreducible ℓ‐adic representation of GK of dimension d with rational traces on inertia. Our main result is that non‐Sophie Germain primes are not in this set when the residue characteristic of K is >3. The result stated in the title is a special case. [ABSTRACT FROM AUTHOR]

In a series of papers we have analysed the embedding of certain groups H as normal subgroups of absolutely irreducible skew linear groups G, see [7], [8], [9] and [10]. Here we drop the absolutely irreducibility assumption on G. If H is locally finite we derive relatively strong conditions on G, although not as strong as when G is absolutely irreducible. If H is abelian, very much in contrast to the absolutely irreducible case, we show that nothing can be said. The phrase “bounded by an integer-valued function of n only” we abbreviate to “n-bounded”. [ABSTRACT FROM PUBLISHER]

We study equations in groups G with unique mth roots for each positive integer m. A word equation in two letters is an expression of the form w(X, A)=B, where w is a finite word in the alphabet {X, A}. We think of A, B∈G as fixed coefficients, and X∈G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals: while others such as X2 A X=B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification.To a word w, we associate a polynomial Pw∈ℤ[x, y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if Pw(x2, y2) has an absolutely irreducible factor in ℤ[x, y], then the equation w(X, A)=B is not solvable in terms of radicals. [ABSTRACT FROM PUBLISHER]

Let p be an odd prime and denote by [p], the finite field of residue classes, mod p. In Euclidean n-space, let n denote the lattice of points x = (x1, …, xn) with integral coordinates and C = C(n, p), the set of points of n satisfying [ABSTRACT FROM PUBLISHER]

It is shown that, given continuous, absolutely irreducible representations of ${\rm Gal}(\mathbb{Q}^{\rm ac}/\mathbb{Q})$ with values in ${\rm GL}_2(\mathbb{F}_2)$ and ${\rm GL}_2(\mathbb{F}_3)$, and having cyclotomic determinant, there is a weight 2 newform of some level whose mod $2$ and mod $3$ representations are equivalent to those given. [ABSTRACT FROM AUTHOR]

The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq. [ABSTRACT FROM PUBLISHER]

Let F$F$ be a totally real field and n⩽4$n\leqslant 4$ a natural number. We study the monodromy groups of any n$n$‐dimensional strictly compatible system {ρλ}λ$\lbrace \rho _\lambda \rbrace _\lambda$ of λ$\lambda$‐adic representations of F$F$ with distinct Hodge–Tate numbers such that ρλ0$\rho _{\lambda _0}$ is irreducible for some λ0$\lambda _0$. When F=Q$F=\mathbb {Q}$, n=4$n=4$, and ρλ0$\rho _{\lambda _0}$ is fully symplectic, the following assertions are obtained. (i)The representation ρλ$\rho _\lambda$ is fully symplectic for almost all λ$\lambda$.(ii)If in addition the similitude character μλ0$\mu _{\lambda _0}$ of ρλ0$\rho _{\lambda _0}$ is odd, then the system {ρλ}λ$\lbrace \rho _\lambda \rbrace _\lambda$ is potentially automorphic and the residual image ρ¯λ(GalQ)$\bar{\rho }_\lambda (\operatorname{Gal}_\mathbb {Q})$ has a subgroup conjugate to Sp4(Fℓ)$\mathrm{Sp}_4(\mathbb {F}_\ell)$ for almost all λ$\lambda$. [ABSTRACT FROM AUTHOR]

This paper is part of an effort to determine a certain class of simple 2‐fusion systems, and to use that result to simplify the proof of the classification of the finite simple groups. The main theorem proves that the 2‐fusion system of the Monster is the unique simple system with a fully centralized involution whose centralizer is the fusion system of the universal covering group of the Baby Monster. [ABSTRACT FROM AUTHOR]

Assuming finiteness of the Tate–Shafarevich group, we prove that the Birch–Swinnerton–Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2‐adic places. [ABSTRACT FROM AUTHOR]

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non‐Galois extensions whose Galois closure has a Galois group permutation‐isomorphic to a prescribed group G (in short, "G‐extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) Q$\mathbb {Q}$ gain rank over infinitely many G‐extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that "many" elliptic curves gain rank over infinitely many G‐extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties. [ABSTRACT FROM AUTHOR]

We develop a global version of Heath‐Brown's p‐adic determinant method to study the asymptotic behaviour of the number N(W; B) of rational points of height at most B on certain subvarieties W of Pn defined over Q. The most important application is a proof of the dimension growth conjecture of Heath‐Brown and Serre for all integral projective varieties of degree d ≥ 2 over Q. For projective varieties of degree d ≥ 4, we prove a uniform version N(W; B) = Od,n,ε(Bdim W+ε) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi‐projective surfaces. If, for example, X́$X\acute{\ }$ is the complement of the lines on a non‐singular surface X ⊂ P3 of degree d, then we show that N(X́;B)=Od(B3/√d(logB)4+B)$N(X\acute{\ };B) = {O}_d({{B}^{3/\surd d }{{({\log B})}}^4 + B})$. For surfaces defined by forms a0x0d+a1x1d+a2x2d+a3x3d${a}_0x_0^d + {a}_1x_1^d + {a}_2x_2^d + {a}_3x_3^d$ with non‐zero coefficients, then we use a new geometric result for Fermat surfaces to show that N(X́;B)=Od(B3/√d(logB)4)$N({X\acute{\ };B}) = {O}_d({B}^{3/\surd d}{({\log B})}^4)$ for B ≥ e. [ABSTRACT FROM AUTHOR]

We give a new local proof of the Breuil-Mézard conjecture in the case of a reducible representation of the absolute Galois group of Qp, p>2, that has scalar semi-simplification, via a formalism of Paškūnas. [ABSTRACT FROM AUTHOR]

A 2‐dimensional version of Farey dissection for function fields K=Fq(t)$K=\mathbb {F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics X⊂PKn−1$X\subset \mathbb {P}^{n-1}_{K}$, under the assumption that q$q$ is odd and n⩾9$n\geqslant 9$. [ABSTRACT FROM AUTHOR]

Tverberg's theorem states that any set of t(r,d)=(r−1)(d+1)+1$t(r,d)=(r-1)(d+1)+1$ points in Rd$\mathbb {R}^d$ can be partitioned into r subsets whose convex hulls have non‐empty r‐fold intersection. Moreover, generic collections of fewer points cannot be so divided. Extending earlier work of the first author, we show that one can nonetheless guarantee inscribed "polytopal partitions" with specified symmetry conditions in many such circumstances. Namely, for any faithful and full‐dimensional orthogonal representation ρ:G→O(d)$\rho \colon G\rightarrow O(d)$ of any order r group G, we show that a generic set of t(r,d)−d$t(r,d)-d$ points in Rd$\mathbb {R}^d$ can be partitioned into r subsets so that there are r points, one from each of the resulting convex hulls, which are the vertices of a convex d‐polytope whose isometry group contains G via the regular action afforded by the representation. As with Tverberg's theorem, the number of points is optimal for this. At one extreme, this gives polytopal partitions for all regular r‐gons in the plane, as well as for three of the six regular 4‐polytopes in R4$\mathbb {R}^4$. At the other extreme, one has polytopal partitions for d‐polytopes on r vertices with isometry group equal to G whenever G is the isometry group of a vertex‐transitive d‐polytope. [ABSTRACT FROM AUTHOR]

We introduce the notion of a coordinatek${\bf k}$‐algebra scheme and the corresponding notion of a B$\mathcal {B}$‐operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it generalizes the operators related to D$\mathcal {D}$‐rings from Moosa and Scanlon (J. Math. Log. 14 (2014), no. 02, 1450009) as well. We classify the (coordinate) k${\bf k}$‐algebra schemes for a perfect field k${\bf k}$ and we also discuss the model‐theoretic properties of fields with B$\mathcal {B}$‐operators. [ABSTRACT FROM AUTHOR]

In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over Fq[t]$\mathbb {F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q$q$ and the degree d$d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree d⩾64$d\geqslant 64$, building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q$q$ and d$d$, and it is this dependence which simplifies the treatment of the dimension growth conjecture. [ABSTRACT FROM AUTHOR]

We compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative subgroup of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic. [ABSTRACT FROM AUTHOR]

We define modular equations in the setting of Shimura varieties of PEL type as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number‐theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces. [ABSTRACT FROM AUTHOR]

We verify a case of Artin's conjecture for a certain range by showing that every cubic and quadratic form over a -adic field with at least 14 variables has a non-trivial common zero, provided the cardinality of the residue class field exceeds 293. The proof involves generalizing a -adic minimization procedure due to Schmidt, to hold for systems of forms of differing degrees. [ABSTRACT FROM AUTHOR]

We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of S integral points on integral models of twists of modular curves over Q, for any finite set of primes S. We deduce this from an existence theorem for elliptic curves over Q satisfying certain local conditions. [ABSTRACT FROM PUBLISHER]

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials. [ABSTRACT FROM AUTHOR]

Kobayashi recently proved that the generalized Heegner cycles of Bertolini–Darmon–Prasanna can be interpolated along the anticyclotomic tower, giving rise to distribution valued cohomology classes with expected growth rate. We interpolate these classes along Coleman families. This construction plays a role in the proofs of p‐adic Gross–Zagier formulae at for non‐ordinary eigenforms and consequentially, also in the proof of a conjecture of Perrin‐Riou. [ABSTRACT FROM AUTHOR]

Suppose F is a finite extension of Qp, G is the group of F‐points of a connected reductive F‐group, and I1 is a pro‐p‐Iwahori subgroup of G. We construct two spectral sequences relating derived functors on mod‐p representations of G to the analogous functors on Hecke modules coming from pro‐p‐Iwahori cohomology. More specifically: (1) using results of Ollivier–Vignéras, we provide a link between the right adjoint of parabolic induction on pro‐p‐Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a 'Poincaré duality spectral sequence' relating duality on pro‐p‐Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules Hi(I1,π). [ABSTRACT FROM AUTHOR]

Let fnewbe a classical new form of weight ≥ 2 and prime to p level. We study the arithmetic of fnewand its unique p-stabilisation f when f… is p-irregular, that is, when its Hecke polynomial at p admits a single repeated root. In particular, we study p-adic weight families through f and its base-change to an imaginary quadratic field F where p splits, and prove that the respective eigencurves are both Gorenstein at f. We use this to construct a two-variable p-adic L-function over a Coleman family through f, and a three-variable p-adic L-function over the base-change of this family to F. We relate the two- and three-variable p-adic L-functions via p-adic Artin formalism. These results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in this case. In an appendix, we prove results towards Hida duality for modular symbols, constructing a pairing between Hecke algebras and families of over convergent modular symbols and proving that it is non-degenerate locally around any cusp form. This allows us to control the sizes of (classical and Bianchi) Hecke algebras in families. [ABSTRACT FROM AUTHOR]

A partial linear space is a pair (P,L) where P is a non‐empty set of points and L is a collection of subsets of P called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms G of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non‐collinear points precisely when G is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of AΓL1(q). Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Dedicated to the memory of Professor A. I. KostrikinThe main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations φ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations φ that can be realized over an unramified extension of Qp, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and G≠ G2(5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1

A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d2 variables whose coefficients are integers of k has a nontrivial zero. In this paper, we consider the case of this conjecture in which k is a p-adic field. Although a counterexample due to Terjanian [16] proved Artin's conjecture false in this situation, Ax and Kochen [2] have shown when [k:Qp] = n is finite, that given d, there exists a number p(d, n) such that Artin's conjecture is true provided that p is larger than p(d, n). Unfortunately, the methods of Ax and Kochen do not lead to explicit estimates for p(d, n). Cohen [5] found a method which determines the possible cardinalities of the residue class fields of all p-adic fields for which Artin's conjecture is false, and Brown [3] has used this to bound p(d, 1), but this bound is so large that one feels that it must be possible to do better. Hence, it is still an interesting problem to obtain estimates on the size of p(d, n). [ABSTRACT FROM PUBLISHER]

We are interested in Weil representations of Sp(2n, R), where R is the ring Z/plZ, p is an odd prime and l is a positive integer, or, more generally, R = 5/5l, where 5 is the ring of integers of a local field, 5 is the maximal ideal of 5 and 5/5 has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2n, 5) has to factor through a representation of Sp(2n, 5/5l) for some l. [ABSTRACT FROM AUTHOR]

Given a smooth cubic hypersurface X over a finite field of characteristic greater than 3 and two generic points on X, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number of degree d k-rational curves on X passing through those two points. We use this to deduce the dimension and irreducibility of the moduli space parametrising such curves, for large enough d. [ABSTRACT FROM AUTHOR]

We introduce the notion of joint spectrum of a compact set of matrices S⊂GLd(C), which is a multi‐dimensional generalization of the joint spectral radius. We begin with a thorough study of its properties (under various assumptions: irreducibility, Zariski‐density, and domination). Several classical properties of the joint spectral radius are shown to hold in this generalized setting and an analogue of the Lagarias–Wang finiteness conjecture is discussed. Then we relate the joint spectrum to matrix valued random processes and study what points of it can be realized as Lyapunov vectors. We also show how the joint spectrum encodes all word metrics on reductive groups. Several examples are worked out in detail. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures. [ABSTRACT FROM AUTHOR]

The number (up to isomorphism) of positive-definite, even, unimodular lattices of rank 8r grows rapidly with r. However, Bannai [1] has shown that, when counted according to weight, those with non-trivial automorphisms make up a fraction of the whole, which goes rapidly to zero as r’. Therefore it is of some interest to produce families of positive-definite, even, unimodular lattices with large automorphism groups and unbounded ranks.Suppose that G is a finite group and V is an irreducible [G]-module such that V— is still irreducible. Then, as observed by Gross [8], the space of G-invariant symmetric bilinear forms on V is one-dimensional and is necessarily generated by a positive-definite form, unique up to scaling by non-zero positive rationals. Thompson [23] showed that, if V is also irreducible modp for all primes p, then it contains an invariant lattice (unique up to scaling) which is even and unimodular with appropriate scaling of the quadratic form. Examples arising in this manner are the E8-lattice of rank 8, the Leech lattice of rank 24 and the Thompson-Smith lattice of rank 248. Gow [6] has also constructed some examples associated with the basic spin representations of 2An and 2Sn. [ABSTRACT FROM AUTHOR]

This paper originated with the observation that while all of the known stable lattice packings of spheres are highly symmetric, it is futile to try to prove a converse statement: the ordinary integer-lattice provides a distinctly unstable packing of spheres, but admits a large group of orthogonal symmetries nonetheless. The integerlattice is in fact very unstable—the slightest perturbation places the spheres in a more efficient configuration. We will call such a lattice fragile. The purpose of this note is to prove that a highly symmetric lattice must be either stable or fragile. [ABSTRACT FROM PUBLISHER]

We investigate local–global principles for Galois cohomology, in the context of function fields of curves over semi‐global fields. This extends work of Kato's on the case of function fields of curves over global fields. [ABSTRACT FROM AUTHOR]

We use Galois cohomology methods to produce optimal mod pd level lowering congruences to a p‐adic Galois representation that we construct as a well‐chosen lift of a given residual mod p representation. Using our explicit Galois cohomology methods, for F a number field, ΓF its absolute Galois group and G a reductive group, k a finite field and a suitable representation ρ¯:ΓF→G(k), ramified at a finite set of primes S, we construct under favorable conditions lifts ρ, {ρq} of ρ¯ to G(W(k)) for q∈Q with Q a finite set of places of F. The lifts {ρq} have the following properties: ρ:ΓF→G(W(k)) is ramified precisely at S∪Q; for q∈Q, ρq:GF→G(W(k)) is unramified outside S∪Q∖{q} and ρ and ρq are congruent mod pd if ρ mod pd is unramified at q. Furthermore, the Galois representations {ρq} are 'independent'. [ABSTRACT FROM AUTHOR]

It is established that every cubic form in at least eight variables which can be realised as a diagonal form on a hyperplane has a non‐trivial p‐adic solution for all primes p. [ABSTRACT FROM AUTHOR]

Fix an odd prime p, and suppose that ... to a non-commutative Iwasawa theory setting. [ABSTRACT FROM AUTHOR]

New series of 4m‐dimensional universally strongly perfect lattices ΛI and ΓJ are constructed with 2BW2m#⊆ΓJ⊆BW2m⊆ΛI⊆BW2m#.The lattices are found by restricting the half‐spin representations of the automorphism group of the Barnes–Wall lattice to its subgroup Um:=Cm(41H). The group Um is the Clifford–Weil group associated to the Hermitian self‐dual codes over F4 containing 1, so the ring of polynomial invariants of Um is spanned by the genus m complete weight enumerators of such codes. This allows us to show that all the Um‐invariant lattices are universally strongly perfect. We introduce a new construction, D(cyc), for chains of (extended) cyclic codes to obtain (bounds on) the minimum of the new lattices. [ABSTRACT FROM AUTHOR]

We use a variant of Bukh's random algebraic method to show that for every natural number k⩾2 there exists a natural number ℓ such that, for every n, there is a graph with n vertices and Ωk(n1+1/k) edges with at most ℓ paths of length k between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant. [ABSTRACT FROM AUTHOR]

This paper develops the theory of KLR algebras with a Dynkin diagram automorphism. This is foundational material intended to allow folding techniques in the theory of KLR algebras. [ABSTRACT FROM AUTHOR]

For a finite group G, let diam (G) denote the maximum diameter of a connected Cayley graph of G. A well‐known conjecture of Babai states that diam (G) is bounded by (log2|G|)O(1) in case G is a non‐abelian finite simple group. Let G be a finite simple group of Lie type of Lie rank n over the field Fq. Babai's conjecture has been verified in case n is bounded, but it is wide open in case n is unbounded. Recently, Biswas and Yang proved that diam (G) is bounded by qO(n(log2n+log2q)3). We show that in fact diam (G)