Questions related to primitive points on elliptic curves and statistics for biquadratic curves over finite fields

Abstract

Algebraic curves represent a very wide _eld of research in mathematics, and there are many possibilities about the point of view one can adopt to approach this topic. In this thesis we want to study some properties of two particular families of curves: elliptic curves and biquadratic curves. For this reason, we divided the thesis in two parts, one for each of these topics. The _rst part begins with an introduction to the theory of elliptic curves. This is itself an incredibly rich area of research, because of the beautiful property of elliptic curves to have a group structure on the set of points with coordinates on a _xed _eld. We mainly focus on reductions modulo a prime number p of an elliptic curve E de_ned over Q. This is an interesting thing in itself, because, if p does not divide the discriminant of the curve, we have a group homomorphism. Moreover, if we _x E and let the prime p vary in the set of prime numbers, we can ask very deep and interesting questions about the reduction modulo p. Most of these questions are about the density of primes for which a certain property holds in the reduction mod p of the elliptic curve (we denote the reduction of E as _E mod p. One of these questions, called the Lang-Trotter conjecture for Primitive Points (cf. [LT77]), comes from a classical analytic number theoretical question about reductions of rational numbers, namely the Artin primitive root conjecture (cf. [Hoo67]). The main tool to attack this kind of problems is the Chebotarev Density Theorem, of which we write an e_ective version of Serre. At the end of the _rst chapter we give details about the paper of 1977 in which Lang and Trotter gave the statement of their conjecture. In the second chapter we study a strange, but not so rare, situation when a point on an elliptic curve is \very far" from being primitive. A point P on an elliptic curve is primitive modulo p if P mod p generates the group _E Fpq. The aim of the Lang-Trotter conjecture is to give a density of the primes for which a _xed P in EpQq is primitive modulo p. Going in some sense in the opposite direction, we give the de_nition of a never-primitive point as a point P that is primitive modulo p only for a _nite set of primes p. We give some necessary conditions for the presence of these points on a given elliptic curve and we give a non trivial example where all points on an elliptic curve of positive rank are never-primitive. These conditions involve a precise structure of the Galois representations at a _xed prime p, both for curves with trivial and non trivial torsion in the group of rational points. In the _nal part of the chapter, we give a splitting condition on a polynomial depending on E, P and a prime p, to assure the fact that P is a never-primitive point. In the third chapter we consider a problem that is weaker than the one in the Lang-Trotter conjecture. It is essentially the content of the submitted paper [Mel15]. Given an elliptic curve E over Q of rank at least r ¡ 0, and a free subgroup 􀀀 of EpQq of rank exactly r, we want to _nd the density of primes (of good reduction for E) for which _E pFpq{_􀀀 mod p is a cyclic group. For the proof of this result, we use Chebotarev Density Theorem on some special extensions of Q, that are the equivalent of Kummer extensions, in the setting of elliptic curves. The splitting of a rational prime p in such extensions says something very precise about the structure of _E pFpq{_􀀀 mod p, from which we can deduce a good asymptotic formula for the density. The second part is an extract of a joint work (cf. [LMM15]) with Elisa Lorenzo Garcia, postdoctoral researcher at the University of Leiden, and Piermarco Milione, graduate student at the University of Barcelona. The big area of research in which this work can be placed is Arithmetic Statistics, or more precisely Statistics about Curves over Finite Fields. The aim of our work is to extend, to families of biquadratic curves, some statistics that were already done for other families (i.e. hyperelliptic curves, cyclic trigonal curves and others). The interest is in the fact that we consider curves that are non-cyclic covers of the projective line, since this can be the starting point for future studies for general abelian covers. The fourth chapter is dedicated to the introduction of notions and methods we need to do statistics on families of curves over _nite _elds. We start with the de_nition of zeta functions over function _elds (i.e. _nite extensions of Fqptq, with Fq _nite _eld) starting with the example when the function _eld is Fqptq. We see that, as in the classical case of Riemann zeta function, we can write the zeta function of Fqptq as an Euler product, in which the role of primes is played by irreducible polynomials in Fqrts. There is a whole dictionary between number _elds and function _elds, of which we give also a brief description. Then, we see that there is an equivalence of categories between smooth curves over Fq and _nite extensions of Fqptq, that allows one to de_ne a zeta function attached to a curve. The most important properties of zeta functions of curves over _nite _elds are expressed by Weil Theorem. Thanks to these properties we can do statistics about the number of points of curves in a given family looking at traces of matrices called Frobenius classes. We conclude the chapter with the exposition of some questions one can ask about families of curves, that are the questions we will answer for biquadratic curves. In the _fth chapter, we _rst de_ne a biquadratic curve and we describe the family of bi- quadratic curves of _xed genus. The main result is about a subfamily, that algebraic geometers call connected component of the coarse muduli space of biquadratic curves of genus g. We use some methods developed, among others, in [KR09] and [BDFL10] respectively for hyperelliptic curves and cyclic l-covers of the projective line. We prove that the number of points of curves in such a subfamily can be written, for the limit of g that tends to in_nity, as a sum of q 􀀀 1 independent and identically distributed random variables. Nowadays, we are trying to prove a version of this theorem for r-quadratic curves, namely curves whose function _eld has Galois group pZ{2Zqr over Fqptq. We have a good description of the family, and we found an interesting generalization in this case (cf. [LMM15, Theorem 6.6]). In the sixth and last chapter, we want to compute the average of the number of points on biquadratic curves in a certain family. A biquadratic curve has an a_ne model that is given by a system of two equations y2 i _ hiptq for i _ 1; 2 and we can see that, up to a certain change of generators of the extension, one of the two polynomials has even degree. So we have to prove a result that Rudnick gave for hyperelliptic curves de_ned by polynomial of odd degree (cf. [Rud10]), in the case of polynomials of even degree. More precisely, we compute the average of (powers of) traces of the Frobenius classes in the family of hyperelliptic curves with a_ne model given by y2 _ hptq, with hptq polynomial of degree 2g 􀀀 2 (g is the genus of the curves), at the limit g Ñ 8. Even if we use the methods of Rudnick, some intermediate results are quite di_erent, and so this is an interesting result in itself. The next step is the application of this result to a certain family of biquadratic curves,di_erent from the one in Chapter 4, that is chosen to let the results for hyperellitpic curves be applicable. We do not arrive to a _nal version of an analogous theorem for biquadratic curves, but we strongly believe that if we can give an estimate for a certain sum of characters, we can _nally have the desired result.