Let $K$ be a number field. For any system of semisimple mod $\ell$ Galois representations $\{{\it\phi}_{\ell }:\text{Gal}(\bar{\mathbb{Q}}/K)\rightarrow \text{GL}_{N}(\mathbb{F}_{\ell })\}_{\ell }$ arising from étale cohomology (Definition 1), there exists a finite normal extension $L$ of $K$ such that if we denote ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/K))$ and ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/L))$ by $\bar{{\rm\Gamma}}_{\ell }$ and $\bar{{\it\gamma}}_{\ell }$, respectively, for all $\ell$ and let $\bar{\mathbf{S}}_{\ell }$ be the $\mathbb{F}_{\ell }$-semisimple subgroup of $\text{GL}_{N,\mathbb{F}_{\ell }}$ associated to $\bar{{\it\gamma}}_{\ell }$ (or $\bar{{\rm\Gamma}}_{\ell }$) by Nori’s theory [On subgroups of$\text{GL}_{n}(\mathbb{F}_{p})$, Invent. Math. 88 (1987), 257–275] for sufficiently large $\ell$, then the following statements hold for all sufficiently large $\ell$.

A(i) The formal character of $\bar{\mathbf{S}}_{\ell }{\hookrightarrow}\text{GL}_{N,\mathbb{F}_{\ell }}$ (Definition 1) is independent of $\ell$ and equal to the formal character of $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}{\hookrightarrow}\text{GL}_{N,\mathbb{Q}_{\ell }}$, where $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}$ is the derived group of the identity component of $\mathbf{G}_{\ell }$, the monodromy group of the corresponding semi-simplified $\ell$-adic Galois representation ${\rm\Phi}_{\ell }^{\text{ss}}$.

A(ii) The non-cyclic composition factors of $\bar{{\it\gamma}}_{\ell }$ and $\bar{\mathbf{S}}_{\ell }(\mathbb{F}_{\ell })$ are identical. Therefore, the composition factors of $\bar{{\it\gamma}}_{\ell }$ are finite simple groups of Lie type of characteristic $\ell$ and are cyclic groups.

B(i) The total $\ell$-rank $\text{rk}_{\ell }\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) is equal to the rank of $\bar{\mathbf{S}}_{\ell }$ and is therefore independent of $\ell$.

B(ii) The $A_{n}$-type $\ell$-rank $\text{rk}_{\ell }^{A_{n}}\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) for $n\in \mathbb{N}\setminus \{1,2,3,4,5,7,8\}$ and the parity of $(\text{rk}_{\ell }^{A_{4}}\bar{{\rm\Gamma}}_{\ell })/4$ are independent of $\ell$.

In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.

We obtain uniqueness theorems for L-functions in the extended Selberg class when the functions share values in a finite set and share values weighted by multiplicities.

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of ‘topological automorphic forms’ associated to this curve, as well as one associated to a quotient by an Atkin–Lehner involution.

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization ${\it\phi}:A\rightarrow \hat{A}$ defined over a field $k$ . Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell ]$ which is maximal isotropic for the Riemann form associated to ${\it\phi}$ . We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with ${\it\phi}$ . Then $B$ , as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$ . In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$ . We obtain a complexity of $\tilde{O} (\ell ^{(rg)/2})$ operations in $k$ where $r=2$ (respectively, $r=4$ ) if $\ell$ is a sum of two (respectively, four) squares which constitutes an improvement over the algorithm described in Cosset and Robert (Math. Comput. (2013) accepted for publication). We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$ .

We study the problem of efficiently constructing a curve $C$ of genus  $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$ -rational points.

In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus- $2$ curves necessarily takes time exponential in the size of its input.

On the other hand, we provide an algorithm for producing a genus- $2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus- $2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus- $2$ curves each having exactly $10^{2013}$ points.

In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus- $2$ curves over $k$ that have $k$ -rational degree- $3$ maps to elliptic curves, including formulas for the genus- $2$ curves, the associated elliptic curves, and the degree- $3$ maps.

Supplementary materials are available with this article.

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota–Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

The discriminant of a trinomial of the form $x^{n}\pm \,x^{m}\pm \,1$ has the form $\pm n^{n}\pm (n-m)^{n-m}m^{m}$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have that $((n^{2}-n+1)/3)^{2}$ always divides $n^{n}-(n-1)^{n-1}$ . In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as $n$ varies seems to be independent of $m$ , and this set can be seen as a generalization of the Wieferich primes, those primes $p$ such that $2^{p}$ is congruent to 2 (mod $p^{2}$ ). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.

The Frobenius number $F(\boldsymbol{a})$ of a lattice point $\boldsymbol{a}$ in $\mathbb{R}^{d}$ with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of $\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math. 181 (2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when $\boldsymbol{a}$ is taken to be random in an enlarging domain in $\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$, $K$ a finite extension of $\mathbb{Q}_{p}$, for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

This work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation(1)

$\begin{equation*}r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty).\end{equation*}$

${\mathfrak s}$

Let p be a prime and K a number field of degree p. We determine the finiteness of the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup or that it has a cyclic isogeny of prescribed degree.

Suppose that $G$ is a connected reductive algebraic group defined over $\mathbf{R}$, $G(\mathbf{R})$ is its group of real points, ${\it\theta}$ is an automorphism of $G$, and ${\it\omega}$ is a quasicharacter of $G(\mathbf{R})$. Kottwitz and Shelstad defined endoscopic data associated to $(G,{\it\theta},{\it\omega})$, and conjectured a matching of orbital integrals between functions on $G(\mathbf{R})$ and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on $G$ and ${\it\theta}$.