The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

An integer $n$ is said to be $y$-friable if its largest prime factor $P^{+}(n)$ is less than $y$. In this paper, it is shown that the $y$-friable integers less than $x$ have a weak exponent of distribution at least $3/5-{\it\varepsilon}$ when $(\log x)^{c}\leqslant x\leqslant x^{1/c}$ for some $c=c({\it\varepsilon})\geqslant 1$, that is to say, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli ${\leqslant}x^{3/5-{\it\varepsilon}}$ for each fixed $a\neq 0$ and ${\it\varepsilon}>0$. We apply this to the estimation of the sum $\sum _{2\leqslant n\leqslant x,P^{+}(n)\leqslant y}{\it\tau}(n-1)$ when $(\log x)^{c}\leqslant y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec with recent work of Harper on friable integers in arithmetic progressions.

We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.

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.

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is of finite type over a field, then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and Knuth–Bendix’s completions into a homotopical completion–reduction, applied to Artin’s and Garside’s presentations. The main result of the paper states that the so-called Tits–Zamolodchikov 3-cells extend Artin’s presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.