We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain “special” subvarieties, not unlike those in Manin conjectures, dominate. also prove new general results hypersurfaces. Our work continues line inquiry initiated by Hooley.

We prove, as an analogy of a conjecture of Artin, that if Y −→ X is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides those of Y in Z[T ]. Then, we give some interpretations of this result in terms of semi-abelian varieties.

We provide explicit equations for moduli spaces of Drinfeld shtukas over the projective line with $\Gamma(N)$, $\Gamma_1(N)$ and $\Gamma_0(N)$ level structures, where $N$ is an effective divisor on $\mathbb{P}^1$. If degree high enough, these are relative surfaces. study some invariants space structure several $4$ divisors

Examples of projective homogeneous varieties over the field of Laurent series over p-adic fields which admit zero cycles of degree one and which do not have rational points are constructed. Let k be a field and X a quasi-projective variety over k. Let Z0(X) denote the group of zero cycles on X and deg : Z0(X) → Z the degree homomorphism which associates to a closed point x of X, the degree [k(x...

Definition 1. Let k be a field. An algebraic variety over k is a k-scheme X such that there exists a covering by a finite number of affine open subschemes Xi which are affine varieties over k, i.e. each Xi is the affine scheme associated to a finitely generated algebra over k. A projective variety over k is a projective scheme over k, i.e. a k-scheme isomorphic to Proj k[T0, . . . , Tn]/I for a...

In arithmetic geometry, cohomology groups are not vector spaces as in classical algebraic geometry but rather euclidean lattices. As a consequence, to understand these groups we need to evaluate not only their rank, but also their successive minima, which are fundamental invariants in the geometry of numbers. The goal of this article is to perform this task for line bundles on projective curves...

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order p whose p-torsion cohomology can be killed by finite covers. As an application we derive the section conjecture for the real points of a large class of varieties defined over the field of real numbers and the nat...

Introduction Let K be a number field, ax the ring of integers of K. Suppose we are given a projective curve 2, and a morphism 4 : Z 4 Pf(C) (where Pf(C) is the projective line) such that 4 and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z) "/K(Pt(C))) (where K(Z) A is...