We study the local factor at $p$ of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at $p$ by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field $k$ and any finite set of places $S$ of $k$, one can effectively compute the set of isomorphism classes of hyperelliptic curves over $k$ with good reduction outside $S$. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus $g$ would imply an effective version of Siegel’s theorem for integral points on...

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over $\mathbb{Z}$. We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over ${𝔽}_2$ is smooth is asymptotically $21/64$ as its degree tends to infinity. Much of this paper is an exposition...

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\ge 3$ and $a_p=0$. We generalise the definition of Kobayashi’s plus/minus Selmer groups over $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$ to $p$-adic Lie extensions $K_{\infty }$ of $\mathbb{Q}$ containing $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$, using the theory of $(\varphi ,\Gamma )$-modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case....

We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field ${𝔽}_p$ of $p$ elements, such that the discriminant $D\left(E\right)$ of the quadratic number field containing the endomorphism ring of $E$ over ${𝔽}_p$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

Let $\mathbf{E}$ be an elliptic curve defined over ${\mathbf{F}}_q$, the finite field of $q$ elements. We show that for some constant $\eta \>0$ depending only on $q$, there are infinitely many positive integers $n$ such that the exponent of $\mathbf{E}\left({\mathbf{F}}_{q^n}\right)$, the group of ${\mathbf{F}}_{q^n}$-rational points on $\mathbf{E}$, is at most $q^{n}exp\left(-n^{\eta /loglogn}\right)$. This is an analogue of a result of R. Schoof on the exponent of the group $\mathbf{E}\left({\mathbf{F}}_p\right)$ of ${\mathbf{F}}_p$-rational points, when a fixed elliptic curve $\mathbf{E}$ is defined over $\mathbb{Q}$ and the prime $p$ tends to infinity.

Let $𝕂/k$ be a finite extension of a global field. Such an extension can be generated over $k$ by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.

Let $V$ be an algebraic subvariety of a torus ${𝔾}_m^n\hookrightarrow {\mathbb{P}}^n$ and denote by $V^{*}$ the complement in $V$ of the Zariski closure of the set of torsion points of $V$. By a theorem of Zhang, $V^{*}$ is discrete for the metric induced by the normalized height $\widehat{h}$. We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.