Let $\mathfrak{R}$ be a complete discrete valuation ring, $S=\mathfrak{R}[[u]]$ and $d$ a positive integer. The aim of this paper is to explain how to efficiently compute usual operations such as sum and intersection of sub- $S$ -modules of $S^d$ . As $S$ is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the $p$ -adic and $u$ -adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with $S$ -modules has applications in Iwasawa theory and $p$ -adic Hodge theory. It is used in particular in Caruso and Lubicz (Preprint, 2013,arXiv:1309.4194) to compute the semi-simplified modulo $p$ of a semi-stable representation.

We develop algorithms to turn quotients of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.

In this paper we study the discrete logarithm problem in medium- and high-characteristic finite fields. We propose a variant of the number field sieve (NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_{p^n}$ for the whole range of applicability of the NFS and lowers the asymptotic complexity from $L_{p^n}({1/3},({128/9})^{1/3})$ to $L_{p^n}({1/3},(2^{13}/3^6)^{1/3})$ in the medium-characteristic case, and from $L_{p^n}({1/3},({64/9})^{1/3})$ to $L_{p^n}({1/3},((92 + 26 \sqrt{13})/27)^{1/3})$ in the high-characteristic case.

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields $K$ for the quartic del Pezzo surface $S$ of singularity type ${\boldsymbol{A}}_{3}$ with five lines given in ${\mathbb{P}}_{K}^{4}$ by the equations ${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

We extend the modularity lifting result of P. Kassaei (‘Modularity lifting in parallel weight one’,J. Amer. Math. Soc.26 (1) (2013), 199–225) to allow Galois representations with some ramification at $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ . We also prove modularity mod 5 of certain Galois representations. We use these results to prove new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent $p$ -adic Hilbert modular forms of finite slope which substantially generalizes a similar result in P. Kassaei (‘Modularity lifting in parallel weight one’, J. Amer. Math. Soc.26 (1) (2013), 199–225).

In this note we shall prove the existence of an uncountable subset of Liouville numbers (which we call the set of ultra-Liouville numbers) for which there exist uncountably many transcendental analytic functions mapping the subset into itself.

We prove that there is a correspondence between Ramanujan-type formulas for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet $L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients ${\it\eta}^{13}({\it\tau})/{\it\eta}(13{\it\tau})$ and ${\it\eta}^{2}(13{\it\tau})/{\it\eta}^{2}({\it\tau})$. The method can also be used to give short and simple proofs for the analogous cubic, quintic and septic theories.

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.

We prove modularity of some two-dimensional, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above $2$ and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the $2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

We prove a simple level-raising result for regular algebraic, conjugate self-dual automorphic forms on $\mathrm{GL}_n$ . This gives a systematic way to construct irreducible Galois representations whose residual representation is reducible.

Weinvestigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

We show that the exponent of distribution of the ternary divisor function $d_{3}$ in arithmetic progressions to prime moduli is at least $1/2+1/46$, improving results of Friedlander–Iwaniec and Heath-Brown. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to $1/2+1/34$.

We address the problem of evaluating an $L$ -function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$ -function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

We construct an elliptic curve over the field of rational functions with torsion group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and rank equal to four, and an elliptic curve over $\mathbb{Q}$ with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.