UNIVERSITA` DI PISA DOTTORATO IN MATEMATICA XXVIII CICLO Francesca Malagoli Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures TESI DI DOTTORATO Relatore: Umberto Zannier arXiv:1704.02640v2 [math.NT] 7 Jun 2017 ANNO ACCADEMICO 2015/2016 Acknowledgements I would like to express my sincere gratitude to my advisor, Professor Umberto Zan- nier, for providing me with the opportunity to work on such an interesting subject and for the help offered by means of worthy remarks and insights. I also would like to thank Professors Marmi and Panti for their help and attention, for their precious suggestions and for the time that they kindly devoted to it. I am grateful to the referees for their precise and valuable remarks. I finally thank Professors Mirella Manaresi and Rita Pardini for their constant help and support. Contents Introduction 7 1 Continued fractions of formal Laurent series 13 1.1 General continued fraction formalism . 13 1.2 Continued fractions of formal Laurent series . 17 1.2.1 Formal Laurent series . 18 1.2.2 Regular continued fraction expansions of formal Laurent series . 21 1.3 Some operations with continued fractions . 26 1.3.1 Continued fractions with some constant partial quotients . 27 1.3.2 Folding Lemma . 28 1.3.3 Unary operations . 29 1.3.4 Multiplication of a continued fraction by a polynomial . 31 1.3.5 A polynomial analogue of Serret's Theorem . 35 2 Quadratic irrationalities in function fields 39 2.1 Periodicity and quasi-periodicity . 43 2.2 Algebro-geometric approach . 51 2.2.1 Reminder of basic results on curves . 51 2.2.2 Quadratic continued fractions and hyperelliptic curves . 53 2.2.2.1 Reduction of abelian varieties and Pellianity . 57 3 Zaremba's and McMullen's conjectures in the real case 61 3.1 Zaremba's Conjecture . 61 3.2 McMullen's Conjecture . 64 3.3 The work of Bourgain and Kontorovich . 65 4 Polynomial analogue of Zaremba's Conjecture 71 4.1 Likelihood of Zaremba's Conjecture . 72 4.1.1 General remarks on the \orthogonal multiplicity" . 74 4.2 Results on Zaremba's Conjecture . 75 4.2.1 Finite fields . 79 4.2.2 K = F2 ................................. 83 5 5 Polynomial analogue of McMullen's conjecture 85 5.1 K = Fp: a connection between Zaremba's and McMullen's Conjectures . 89 5.2 K an uncountable field . 91 5.3 K = Q: reduction of Laurent series . 92 5.3.1 Reduction of a formal Laurent series modulo a prime . 93 5.3.2 Proof of McMullen's strong Conjecture for K = Q ......... 98 5.4 K = Q: generalized Jacobians . 103 5.4.1 Construction of generalized Jacobians . 103 5.4.1.1 Generalized Jacobian associated to a modulus . 105 5.4.1.2 Pullback of generalized Jacobians . 107 5.4.2 McMullen's strong Conjecture over number fields . 114 A Hyperbolic plane and continued fractions 117 A.1 Elements of hyperbolic geometry . 117 A.2 Cutting sequences and continued fractions . 119 A.3 Modular surface . 122 6 Introduction The classical theory of real continued fractions, whose origins probably date back to the antiquity, was already laid down by Euler in the 18th century and then further developed, among others, by Lagrange and Galois. As it is well known, there are close connections between the arithmetic behaviour of algebraic number fields and that of the algebraic function fields in one variable, this analogy being stronger in the case of function fields over finite fields. For instance, in the 19th century Abel [1] and Chebyshev [16] started to developed a theory of polynomial continued fractions. More precisely, the ring of polynomials K[T ] over a given field K can play the role of the ring of integers Z and thus its field of fractions K(T ) will correspond to the field of rational numbers Q. Then, the role of R will be played by the completion of K(T ) with respect to the valuation ord associated to the degree, that is, −1 −1 by L = K((T )), the field of formal Laurent series in T . Every formal Laurent series P i α = ciT can be written (uniquely) as a regular simple continued fraction, that is, as i≤n 1 α = a0 + ; 1 a1 + 1 a + 2 . .. where a0 2K[T ] and where the ai are polynomials of positive degree. As in the real case, the ai are called the partial quotients of α and we will write α = [a0; a1;::: ]: Almost all the classical notions and results can be translated in the function fields setting; in particular in this thesis we will consider the polynomial analogues of Zaremba's and McMullen's Conjectures on continued fractions with bounded partial quotients. We will focus especially on the second one, proving it when K is an infinite algebraic extension of a finite field, when K is uncountable, when K = Q and when K is a number field; we will also see that, if K is a finite field, then the polynomial analogue of Zaremba's Conjecture over K would imply the polynomial analogue of McMullen's. Of course, according to the base field K, we will have to use different methods. 7 Real numbers with bounded partial quotients appear in many fields of mathematics and computer science, for instance in Diophantine approximation, fractal geometry, tran- scendental number theory, ergodic theory, numerical analysis, pseudo-random number generation, dynamical systems and formal language theory (for a survey, see [59]). While studying numerical integration, pseudo-random number generation and quasi- Monte Carlo methods, Zaremba in [68] (page 76) conjectured that for every positive integer d ≥ 2 there exists an integer b < d, relatively prime to d, such that all of the partial quotients in the continued fraction of b=d are less than or equal to 5. More generally, we can consider the following Conjecture: Conjecture 1 (Zaremba). There exists an absolute constant z such that for every integer d ≥ 2 there exists b, relatively prime to d, with the partial quotients of b=d bounded by z. p In [62], Wilson proved that any real quadratic field Q( d) contains infinitely many purely periodic continued fractions whose partial quotients are bounded by a constant md depending only on d (for example, we can take m5 = 2). McMullen [34] explained these phenomena in terms of closed geodesics on the modular surface and conjectured the existence of an absolute constant m, independent from d: Conjecture 2 (McMullen). There exists an absolutep constant m such that for every positive squarefree integer d the real quadratic field Q( d) contains infinitely many purely periodic continued fractions whose partial quotients are bounded by m. Actually, McMullen conjectured that md = 2 should be sufficient for every d. Zaremba's and McMullen's Conjectures have been deeply studied by Bourgain and Kontorovich, who showed that they are special cases of a much more general local-global Conjecture and proved a density-one version of Zaremba's Conjecture (Theorem 1.2 in [13]). However, both Conjectures are still open. In the polynomial setting, the property of having bounded partial quotients corre- sponds to the property of having partial quotients of bounded degree. When K is a finite field, Laurent series whose continued fraction expansions have bounded partial quotients appear in stream cipher theory, as they are directly linked to the study of lin- ear complexity properties of sequences and pseudorandom number generation. Indeed, Niederreiter ([42], Theorem 2) proved that the linear complexity profile of a sequence is as close as possible to the expected behaviour of random sequences if and only if all the partial quotients of the corresponding Laurent series are linear. We will study the following analogue of Zaremba's Conjecture for polynomial con- tinued fractions over a field K: Conjecture Z (Polynomial analogue of Conjecture 1). There exists a constant zK such that for every non-constant polynomial f 2K[T ] there exists g 2K[T ], relatively prime to f, such that all the partial quotients of f=g have degree at most zK. Actually, it is believed that it is enough to take zK = 1 for any field K 6= F2 and zF2 = 2. This conjecture has already been studied many authors, including Blackburn [9], Friesen [19], Lauder [32] and Niederreiter [41]: 8 Theorem 3 (Blackburn [9], Friesen [19]). Conjecture Z holds with zK = 1 whenever K is an infinite field. If K = Fq is a finite field and deg f < q=2, then there exists a polynomial g, relatively prime to f, such that all the partial quotients of f=g are linear. Analogously, we will study a polynomial version of McMullen's Conjecture. Let K be a field with characteristic different from 2. Then for every polynomial D 2 K[T ] of even degree, which is not a square in K[T ] and whose leading coefficient is a square in K, the square root of D is well defined as a formal Laurent series. We can then consider the following conjecture: Conjecture M (Polynomial analogue of Conjecture 2). Let K be a field with char K 6= 2. Then there exists a constant mK such that for every polynomial D 2 K[T ] satisfying 1 the previousp conditions there exist infinitely many pairwise non-equivalent elements of K(T; D) n K(T ) whose partial quotients have degree at most mK. It is believed that for every admissible field K it is enough to take mK = 1. We will see that this is true in all the cases where the Conjecture has been proved. We will also consider the following strengthening of Conjecture M (with mK = 1): Conjecture 4.