The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\mathrm{\Psi }$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. When $\mathrm{\Psi }$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on ${\mathbb{Z}}_p^{\times }$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the Asai representation of a quadratic Hilbert modular form.

$p$-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of ${\mathbb{P}}^1\setminus \left\{0,{\mu }_N,\infty \right\}$. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of $p$-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro-unipotent harmonic action.

Soit $K$ un corps valué de hauteur $1$ et d’inégales caractéristiques $\left(0,p\right)$, et soit $k$ son corps résiduel. Dans cet article, nous construisons une nouvelle cohomologie de Weil pour les $k$-schémas de type fini à valeurs dans les $A_K$-modules, avec $A_K$ une $K$-algèbre de « périodes abstraites $p$-adiques » qui admet une description explicite par générateurs et relations. Nous démontrons des théorèmes de comparaison reliant cette nouvelle cohomologie de Weil aux cohomologies de Weil classiques : la cohomologie rigide de Berthelot et les cohomologies $\ell$-adiques, pour $\ell \ne p$. Nous énonçons également des conjectures sur l’anneau $A_K$ dont l’une d’elles entraîne l’indépendance de $\ell$ en cohomologie.

Let $K$ be a valued field of height $1$ and unequal characteristics $\left(0,p\right)$, and let $k$ be its residue field. In this article, we construct a new Weil cohomology for finite type $k$-schemes with values in $A_K$-modules, where $A_K$ is a $K$-algebra of “$p$-adic abstract periods” admitting an explicit description by generators and relations. We establish comparison theorems relating this new Weil cohomology to the classical ones: Berthelot’s rigid cohomology and the $\ell$-adic cohomologies, for $\ell \ne p$. We also state some conjectures on the ring $A_K$. One of these conjectures implies the independence of $\ell$ in cohomology.

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck–Witt ring of the base field. Previous work of the first author and recent work of Déglise, Jin and Khan established a motivic Gauss–Bonnet formula relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. We apply this formula to $SL$-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a uniqueness result for pushforward maps in $SL$-oriented theories, identifying these maps concretely in examples of interest.

We calculate the first and second moments of $L$-functions in the family of quadratic twists of a fixed elliptic curve $E$ over ${\mathbb{𝔽}}_q\left[x\right]$, asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of $L$-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

Fixing $t\in ℝ$ and a finite field ${\mathbb{𝔽}}_q$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over ${\mathbb{𝔽}}_q$ whose zeta function vanishes at $\frac{1}{2}+it$. Our upper bound is independent of $g$ and tends to $0$ as $q$ grows.

We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This $n$-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least $q^{\beta }$, with $\beta >1∕2-1∕\left(2\left(n+1\right)\right)$. This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the second author.

We characterize finite groups $G$ having a cyclic Sylow $p$-subgroup in terms of the action of a specific Galois automorphism on the principal $p$-block of $G$, for $p=2,3$. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay–Navarro conjecture.

We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over $\mathbb{Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso and Zannier and Amoroso and Dvornicich, as well as properties of the Arakelov–Zhang pairing.