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Computing the Zeta Functions of Two Classes of Singular Curves By Computing the Zeta Functions of Two Classes of Singular Curves by Robert M. Burko A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2014 by Robert M. Burko Abstract Computing the Zeta Functions of Two Classes of Singular Curves Robert M. Burko Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2014 Motivated by applications to cryptography, for over a decade mathematicians have suc- cessfully used p-adic cohomological methods to compute the zeta functions of various classes of varieties defined over finite fields of order q = pa in an amount of time polyno- mial in a, assuming the characteristic p is fixed. In all instances, the varieties considered had smooth representations in either affine or projective space. In this thesis, two non-smooth situations are introduced: the case of superelliptic curves with singular points that are rational over the field of definition, and the case of nodal projective plane curves. In each case, we present a polynomial-time algorithm which computes the zeta function the curve, and provide the results of an implementation in MAGMA. The case of singular superelliptic curves extends a method of Gaudry and G¨urel,and the case of nodal projective curves extends a method of Kedlaya, Abbott, and Roe. Assuming the curve has geometric genus g, and that the characteristic p is fixed, the running time of the first algorithm is O(a3+"g5+") and the running time of the second is O(a7+"g10+"). Both methods involve computing the matrix of the Frobenius automorphism on the cohomology groups of Monsky and Washnitzer up to a certain amount of p-adic accuracy. Estimates on the amount of accuracy needed are drawn from the theory of crystalline cohomology introduced by Grothendieck and developed by Berthelot. ii Dedication For my parents, Tom and Risa, my brother Jeremy, my sister Rachel, and my fianc´ee Rebecca. Acknowledgements I am greatly appreciative of the many people who have supported me throughout the process of writing this thesis. Without their encouragement and generosity, it is clear to me that this work would not have at all been possible. I would first like to thank my thesis advisor, V. Kumar Murty, who kept me guided as I wandered the seas of mathematical investigation. I am forever grateful, not only for his time and dedication in meeting with me and ensuring that my work moves forward, but in his unwavering confidence in my ability to conduct my own research. I would like to thank all the members of the GANITA seminar group at the Univer- sity of Toronto, who patiently listened to all the various stages in the development of this thesis and offered countless hours in useful discussion and mutual support. I am privileged to have had such a cohesive group filled with warmth and friendship. Included in this group are Catalina Anghel, Aaron Chow, Payman Eskandari, William George, Nataliya Laptyeva, Meng Fai Lim, Mariam Mourtada, and Ying Zong. In particular I would like to single out Ying, an exceptional mathematician who donated so much of his time to me in explaining various theories and entertaining my mathematical curiosities. The staff members of the mathematics department at the University of Toronto cre- ate an outstanding model of how a department should be run, and only in their absence do we realize the amount they keep us on course each day. I would like to thank in particular Ida Bulat and Jemima Merisca for their care, dedication, and endless personal iii emails which helped me navigate my way through graduate studies. I would surely have continued struggling to find a thesis topic if not for the guidance of Alan Lauder, who not only gave useful insight and discussion via email, but also helped fund two separate visits to the University of Oxford and introduced me to his number theory group. I would like to extend thank yous to Jan Tuitman who welcomed me into the Oxford community and carefully read and scrutinized early drafts of this thesis, as well as George M. Walker for our conversations and for welcoming me into his home in Bristol. I would also like to thank Keith Gillow for allowing me extended usage of the servers at Oxford to run MAGMA programs. Lastly, I would like to thank my dear friends and my family members, immediate and extended, old and new, for keeping my spirits lifted, and for their endless patience and nurturing. iv Contents 1 Introduction 1 1.0.1 Point Counting and the Zeta Function . 1 1.0.2 Weil Cohomology . 3 1.1 Algorithmic Approaches . 4 1.1.1 `-Adic Methods . 4 1.1.2 p-Adic Methods . 5 1.1.3 Deformation and Fibration Methods . 6 1.1.4 Approaches to Singular Varieties . 7 1.2 This Thesis . 8 1.3 Applications and Future Work . 8 1.3.1 Cryptography . 9 1.3.2 Support for Dimca's Conjecture . 10 1.3.3 Potential Generalizations and Improvements . 11 2 Cohomology Theories 13 2.1 Preliminaries . 13 2.1.1 p-Adic numbers and Witt vectors . 13 2.1.2 Useful Properties of Etale´ Maps . 14 2.2 Algebraic de Rham Cohomology . 15 2.2.1 K¨ahlerDifferentials . 15 2.2.2 de Rham Cohomology for Schemes . 15 2.2.3 de Rham Cohomology with Logarithmic Singularities . 16 2.3 p-Adic Cohomology Theories . 22 2.3.1 Monsky-Washnitzer Cohomology . 23 2.3.2 Rigid Cohomology and Crystalline Cohomology . 24 2.3.3 Comparisons Theorems Between p-Adic and de Rham Cohomology 25 2.4 Exact Sequences . 26 v 3 Superelliptic Curves 28 3.1 Basic Properties . 28 3.1.1 The Genus . 29 3.1.2 The Zeta Function . 33 1 0 − 3.1.3 The Vector Space HMW(Ck=K) ................... 34 3.1.4 Some Useful Order-preserving Functions . 37 3.2 Computing a Basis for Cohomology . 41 3.2.1 The Reduction Process . 42 3.2.2 Two Lemmas . 46 3.3 The Matrix of Frobenius . 54 3.4 Working Within a Crystalline Basis . 55 3.5 p-Adic Precision Analysis . 62 4 Nodal Plane Curves 64 4.1 Cohomology of the Affine Complement of a Hypersurface in Pn . 64 4.2 Basic Properties of Nodal Plane Curves . 70 4.2.1 Computing a Lift . 70 4.2.2 The Zeta Function of a Nodal Curve . 75 4.3 A Crystalline Lattice of the Affine Complement . 78 4.4 The Matrix of Frobenius . 88 4.5 p-Adic Precision Analysis . 90 5 Algorithms and Complexity Estimates 93 5.1 Superelliptic Curve . 93 5.1.1 Algorithm . 93 5.1.2 Complexity Analysis . 95 5.2 Nodal Plane Curve . 95 5.2.1 Algorithm . 95 5.2.2 Complexity Analysis . 96 6 Experiments 98 6.1 Examples of Superelliptic Curves . 98 6.2 Examples of Nodal Plane Curves . 100 vi Chapter 1 Introduction 1.0.1 Point Counting and the Zeta Function Let p be a prime, let Fp be a finite field with p elements, and let X be an algebraic variety defined over Fp. For instance, X might be the simultaneous solution of the system of polynomial equations 8 > f1(x1; :::; xn) ≡ 0 mod p > <> f2(x1; :::; xn) ≡ 0 mod p . > . > : fm(x1; :::; xn) ≡ 0 mod p where the variables x1; :::; xn lie in Fp. If we restrict the variables to take values in a finite extension of Fp, then there are only a finite set of possibilities for each variable, and so this system has a finite number of solutions. An interesting question in number theory, dating back at least of far as Gauss' Disquisitiones Arithmeticae [30], asks for the number of solutions to such systems. An almost equivalent but slightly more involved question is to calculate what is known as the zeta function of X. In general, we let X be an algebraic variety defined over a a finite field with q = p elements, and let #X(Fqk ) denote the number of solutions its defining equation has over the finite field Fqk , the so-called \Fqk -rational points" of X. One defines the zeta function, a formal power series associated to X, by 1 ! X T k Z(X; T ) = exp #X( k ) 2 [[T ]]: Fq k Q k=1 The zeta function has many interesting properties. For instance, from Galois theory one can determine that the coefficients of its expanded power series are integers. More 1 Chapter 1. Introduction 2 astonishingly, in 1960 it was proven by Dwork that the zeta function is rational, that is, a quotient of two polynomials with integer coefficients [22]. Dwork's proof is analytic in nature − he shows that the zeta function is meromorphic, both as a function on the complex plane and on the completion of the algebraic closure of the field of rational p-adic numbers. The following theorem was conjectured by Weil in 1948 and proven by him in the case of curves, but not proven in full generality until 1974 by Deligne [15]. Theorem 1.0.1. Let X be a smooth projective variety of dimension n defined over Fq. Then 1. Z(X; T ) is a rational function of T , and can be written P (T )P (T ) ··· P (T ) Z(X; T ) = 1 3 2n−1 P0(T )P1(T ) ··· P2n(T ) with Pi(T ) 2 1 + T Z[T ]. Moreover, the polynomials Pi(T ) satisfy the following properties: n i) P0(T ) = 1 − T and P2n(T ) = 1 − q T . n ii) The map x ! q =x sends the roots of Pi(T ) to the roots of P2n−i(T ), pre- serving multiplicities.
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