Let $p\gt 2$ be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod $p$ representations of the absolute Galois group of ${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to $n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of ${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain $R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$ unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.

Given an intersection of two quadrics $X\subset { \mathbb{P} }^{m- 1} $, with $m\geq 9$, the quantitative arithmetic of the set $X( \mathbb{Q} )$ is investigated under the assumption that the singular locus of $X$ consists of a pair of conjugate singular points defined over $ \mathbb{Q} (i)$.

Let $q$ be a prime and $- D\lt - 4$ be an odd fundamental discriminant such that $q$ splits in $ \mathbb{Q} ( \sqrt{- D} )$. For $f$ a weight-zero Hecke–Maass newform of level $q$ and ${\Theta }_{\chi } $ the weight-one theta series of level $D$ corresponding to an ideal class group character $\chi $ of $ \mathbb{Q} ( \sqrt{- D} )$, we establish a hybrid subconvexity bound for $L(f\times {\Theta }_{\chi } , s)$ at $s= 1/ 2$ when $q\asymp {D}^{\eta } $ for $0\lt \eta \lt 1$. With this circle of ideas, we show that the Heegner points of level $q$ and discriminant $D$ become equidistributed, in a natural sense, as $q, D\rightarrow \infty $ for $q\leq {D}^{1/ 20- \varepsilon } $. Our approach to these problems is connected to estimating the ${L}^{2} $-restriction norm of a Maass form of large level $q$ when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke–Maass $L$-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet $L$-functions in certain ranges.

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

It is known that $\zeta (1+ it)\ll \mathop{(\log t)}\nolimits ^{2/ 3} $ when $t\gg 1$. This paper provides a new explicit estimate $\vert \zeta (1+ it)\vert \leq \frac{3}{4} \log t$, for $t\geq 3$. This gives the best upper bound on $\vert \zeta (1+ it)\vert $ for $t\leq 1{0}^{2\cdot 1{0}^{5} } $.

We use a generalisation of Vinogradov’s mean value theorem of Parsell et al. [‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331] and ideas of Schmidt [‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396] to give nontrivial bounds for the number of solutions to polynomial congruences, when the solutions lie in a very general class of sets, including all convex sets.

In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

The main result in the earlier paper (by the first author) is improved as follows. The number of odd multiperfect numbers with at most $r$ distinct prime factors is bounded by ${4}^{{r}^{2} } / {2}^{r+ 2} (r- 1)!$.

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$ . Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$ . Let $\delta = v(\mathrm{Disc} (F))$ . The Montes algorithm computes an OM factorization of $F$ . The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , for a prescribed precision $\nu $ . In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , assuming that the residue field of $k$ is small.

A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $.

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Let $p\gt 2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a $\mathrm{mod} \hspace{0.167em} p$ Galois representation to arise from a $\mathrm{mod} \hspace{0.167em} p$ Hilbert modular form of parallel weight one, by proving a ‘companion forms’ theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre’s conjecture for $F$ implies Artin’s conjecture for totally odd two-dimensional representations over $F$.

Let $N/ F$ be an odd-degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak{O}}_{N} $ and ${\mathfrak{O}}_{F} = \mathfrak{O}$. Let $ \mathcal{A} $ be the unique fractional ${\mathfrak{O}}_{N} $-ideal with square equal to the inverse different of $N/ F$. B. Erez showed that $ \mathcal{A} $ is a locally free $\mathfrak{O}[G] $-module if and only if $N/ F$ is a so-called weakly ramified extension. Although a number of results have been proved regarding the freeness of $ \mathcal{A} $ as a $ \mathbb{Z} [G] $-module, the question remains open. In this paper we prove that $ \mathcal{A} $ is free as a $ \mathbb{Z} [G] $-module provided that $N/ F$ is weakly ramified and under the hypothesis that for every prime $\wp $ of $\mathfrak{O}$ which ramifies wildly in $N/ F$, the decomposition group is abelian, the ramification group is cyclic and $\wp $ is unramified in $F/ \mathbb{Q} $. We make crucial use of a construction due to the first author which uses Dwork’s exponential power series to describe self-dual integral normal bases in Lubin–Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and the Galois Gauss sum involved. Our results generalise work of the second author concerning the case of base field $ \mathbb{Q} $.

Let $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

Let $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.

In this article we show that the Czech mathematician Václav Šimerka discovered the factorization of $\frac{1}{9} (1{0}^{17} - 1)$ using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed similar algorithms. Šimerka also gave the first examples of what later became known as Carmichael numbers.

For each solvable Galois group which appears in degree $9$ and each allowable signature, we find polynomials which define the fields of minimum absolute discriminant.

Gama and Nguyen [‘Finding short lattice vectors within Mordell’s inequality’, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, 2008, 257–278] have presented slide reduction which is currently the best SVP approximation algorithm in theory. In this paper, we prove the upper and lower bounds for the ratios $\Vert { \mathbf{b} }_{i}^{\ast } \Vert / {\lambda }_{i} (\mathbf{L} )$ and $\Vert {\mathbf{b} }_{i} \Vert / {\lambda }_{i} (\mathbf{L} )$, where ${\mathbf{b} }_{1} , \ldots , {\mathbf{b} }_{n} $ is a slide reduced basis and ${\lambda }_{1} (\mathbf{L} ), \ldots , {\lambda }_{n} (\mathbf{L} )$ denote the successive minima of the lattice $\mathbf{L} $. We define generalised slide reduction and use slide reduction to approximate $i$-SIVP, SMP and CVP. We also present a critical slide reduced basis for blocksize 2.

In this paper we study Lp−Lr estimates of both the extension operator and the averaging operator associated with the algebraic variety S = {x ∈ : Q(x) = 0}, where Q(x) is a non-degenerate quadratic form over the finite field with q elements. We show that the Fourier decay estimate on S is good enough to establish the sharp averaging estimates in odd dimensions. In addition, the Fourier decay estimate enables us to simply extend the sharp L2−L4 conical extension result in , due to Mockenhaupt and Tao, to the L2−L2(d+1)/(d−1) estimate in all odd dimensions d ≥ 3. We also establish a sharp estimate of the mapping properties of the average operators in the case when the variety S in even dimensions d ≥ 4 contains a d/2-dimensional subspace.