We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over 𝔽q[x], 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

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

The Asai (or twisted tensor) L-function of a Bianchi modular form Ψ is the L-function attached to the tensor induction to ℚ of its associated Galois representation. When Ψ is ordinary at p we construct a p-adic analogue of this L-function: that is, a p-adic measure on ℤp× that interpolates the critical values of the Asai L-function twisted by Dirichlet characters of p-power conductor. The construction

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 ℙ1 ∖{0,μN,∞}. 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

Soit K un corps valué de hauteur 1 et d’inégales caractéristiques (0,p), 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 AK-modules, avec AK une K-algèbre de « périodes abstraites p-adiques » qui admet une description explicite par générateurs et relations. Nous démontrons des théorèmes de comparaison

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

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 qβ, with β > 1∕2 − 1∕(2(n + 1)). This is the first Burgess bound that applies in all dimensions to

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 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