We give a natural construction of framed mixed Tate motives unramified over $\mathbb{Z}$ whose periods are the multiple $\zeta$-values. Namely, for each convergent multiple $\zeta$-value we define two boundary divisors A and B in the moduli space $\overline{\mathcal{M}}_{0,n+3}$ of stable curves of genus zero. The corresponding multiple zeta-motive is the nth cohomology of the pair $(\overline{\mathcal{M}}_{0,n+3}-A,B)$.

We give estimates for exponential sums of the form $\sum_{n \leq N}\Lambda(n)\exp(2 \pi i a g^n/m)$, where m is a positive integer, a and g are integers relatively prime to m, and $\Lambda$ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form $\sum_{p \leq N}\exp(2 \pi i a M_p/m)$, where Mp is the Mersenne number; $M_p=2^p-1$ for any prime p. We also estimate some closely related sums, including $\sum_{n \leq N}\mu(n)\exp(2 \pi i a g^n/m)$ and $\sum_{n \leq N}\mu^2(n)\exp(2 \pi i a g^n/m)$, where $\mu$ is the Möbius function.

We prove a portion of a conjecture of Conrad, Diamond, and Taylor, yielding some new cases of the Fontaine–Mazur conjectures, specifically, the modularity of certain potentially Barsotti–Tate Galois representations. The proof follows the template of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered $\phi_1$-modules of Breuil.

We give an elementary proof of the function field (or geometric) case of the Mordell–Lang conjecture in characteristic zero.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Rédei's triple symbol.

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\mathbb{C}^*$ (or, more generally, with coefficients in the complex points of an algebraic torus over $\mathbb{C}$) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.

Gross and Rubin have made conjectures about special values of equivariant L-functions associated to abelian extensions of global fields. We describe a common refinement, due to Burns, and give evidence in favour of this conjecture for quadratic extensions and cyclotomic fields. We also note that the statement provides a new interpretation of further conjectures of Darmon and Gross.

We study the local topological zeta function associated to a complex function that is holomorphic at the origin of $\mathbb{C}^2$ (respectively $\mathbb{C}^3$). We determine all possible poles less than −1/2 (respectively −1). On $\mathbb{C}^2$ our result is a generalization of the fact that the log canonical threshold is never in ]5/6,1[. Similar statements are true for the motivic zeta function.

Let X/S be a smooth morphism of schemes in characteristic p and let $(E,\nabla)$ be a sheaf of $\mathcal{O}_{X}$-modules with integrable connection on X. We give a formula for the cohomology sheaves of the de Rham complex of $(E,\nabla)$ in terms of a Higgs complex constructed from the p-curvature of $(E,\nabla)$. This formula generalizes the classical Cartier isomorphism, with which it agrees when $(E,\nabla)$ is the constant connection.

Let X be a smooth projective variety. Using modified psi classes on the stack of genus-zero stable maps to X, a new associative quantum product is constructed on the cohomology space of X. When X is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in X, and specialises to the usual quantum product upon resetting the parameters corresponding to the modified psi classes. For $X=\mathbb{P}^2$, the product is equivalent to that of the contact cohomology of Ernström and Kennedy.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group $GL_r(\mathbb{C}[z]/z^n)$. We prove that they carry complete hyper-Kähler metrics.

Let G be a real special or general linear group and $\sigma_{0}$ be the transpose-inverse involution. We characterize the image of $f \mapsto {\rm tr}(\pi(f) \pi(\sigma_{0}) )$ for irreducible representations $\pi$ of $G \rtimes \langle \sigma_{0}\rangle$, and K-finite $f \in C_{c}^{\infty}(G)$.

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.