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Computing the Zeta Function of Two Classes of Singular Curves By Computing the Zeta Function of Two Classes of Singular Curves by Robert Burko A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright (year of graduation) by Robert Burko 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 . 20 2.3.1 Monsky-Washnitzer Cohomology . 20 2.3.2 Rigid Cohomology and Crystalline Cohomology . 21 2.3.3 Comparisons Theorems Between p-Adic and de Rham Cohomology 22 2.4 Exact Sequences . 23 ii 3 Superelliptic Curves 25 3.1 Basic Properties . 25 3.1.1 The Genus . 26 3.1.2 The Zeta Function . 30 1 0 − 3.1.3 The Vector Space HMW(CK =K) . 31 3.1.4 Some Useful Order-preserving Functions . 34 3.2 Computing a Basis for Cohomology . 38 3.2.1 The Reduction Process . 39 3.2.2 Two Lemmas . 43 3.3 The Matrix of Frobenius . 51 3.4 Working Within a Crystalline Basis . 52 3.5 p-Adic Precision Analysis . 59 4 Nodal Curves in P2 62 4.1 Cohomology of the Affine Complement of a Hypersurface in Pn . 62 4.2 Basic Properties of Nodal Curves . 68 4.2.1 Computing a Lift . 68 4.2.2 The Zeta Function of a Nodal Curve . 74 4.3 A Crystalline Lattice of the Affine Complement . 77 4.4 The Matrix of Frobenius . 85 4.5 p-Adic Precision Analysis . 88 5 Algorithms and Complexity Estimates 90 5.1 Superelliptic Curve . 90 5.1.1 Algorithm . 90 5.1.2 Complexity Analysis . 92 5.2 Nodal Plane Curve . 92 5.2.1 Algorithm . 92 5.2.2 Complexity Analysis . 93 6 Experiments 95 6.1 Examples of Superelliptic Curves . 95 6.2 Examples of Nodal Plane Curves . 97 iii 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 [1], 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 [2]. 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 [3]. Theorem 1.0.1. Let X by 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. iii) If one writes Y Pi(T ) = (1 − αijT ) j i=2 then αij is an algebraic integer with complex absolute value q . P i 2. Let E = i(−1) deg(Pi). There is a functional equation 1 Z X; = ±qnE=2tEZ(X; T ): qnT 3. If R is the integer ring of a number field, p is a prime ideal in R lying over p, and X is the reduction modulo p of a smooth scheme Y over R, then deg(Pi) is the i-th Betti number of Y ×R C as a topological space Note that the condition for this theorem is that X is smooth and projective, whereas Dwork's proof of the rationality of the zeta function is valid for any variety defined over Fq. Chapter 1. Introduction 3 1.0.2 Weil Cohomology The proof of Theorem (1.0.1) requires finding a \good" cohomology theory for X, known as a Weil cohomology. Such a theory is not unique for X and can be defined in the following way. k Definition 1.0.2. Let = Fq be a finite field of characteristic p, and let K be a field of characteristic 0. A Weil cohomology is a contravariant functor from smooth proper varieties X over k to graded algebras H•(X), where each Hi(X) is a finite dimensional K-vector space satisfying the following properties: 1. If n = dim(X), then Hi(X) = 0 for i2 = f0; 1; :::; 2ng. i i 2. (Lefschetz fixed point theorem) There are \Frobenius maps" Fi : H (X) ! H (X), i 2 f0; :::; 2ng such that 2n X i k i #X(Fqk ) = (−1) Tr(Fi jH (X)) i=0 3. (Poincar´eduality) The vector space H2n(X) is one-dimensional, and the map Hi(X) × H2n−i(X)(n) ! H2n(X)(n) is a perfect pairing for i 2 f0; 1; :::; 2ng, and equivariant under the Frobenius maps i i −n (here H (X)(n) denotes the vector space H (X) with Fi replaced by q Fi). In actuality, Weil cohomologies possess more structure than this, including a cycle class map, K¨unneth formula, and a Lefschetz hyperplane theorem, as well as Frobenius compatibility maps, but for the purposes of this thesis we are interested in the above properties. When X is not smooth or proper, it is possible that X ! Hi(X) is still a well-defined functor, and satisfies some of the above properties. For instance, the rigid cohomology i groups Hrig(X) are defined for any Fq-scheme X, and are finite dimensional and satisfy the Lefschetz fixed point theorem when X is smooth and affine. If X is a singular, projective hypersurface, then Poincar´eduality still holds for 0 ≤ i ≤ dim(X) − dim(Y ), where Y is the singular locus of X. From the following lemma, to ensure rationality of the zeta function it is sufficient to have finite dimensionality of each Hi(X) combined with the Lefschetz fixed point formula. Chapter 1. Introduction 4 Lemma 1.0.3. Let V be a finite dimensional vector space over a field K, and let I denote the identity on V . Then for any endomorphism F of V , there is an identity in the power series ring K[[T ]] 1 ! X T k exp Tr(F jV ) = det(I − TF jV )−1: k k=1 Proof. A simple calculation, see for instance [4, Appx C, Lemma 4.1]. i It follows immediately that if one defines Pi(T ) = det(I − TFijH (X)), then the zeta function of X can be written 1 2n ! X X T k Z(X; T ) = exp (−1)iTr(F kjHi(X)) i k k=1 i=0 i 2n 1 !!(−1) Y X T k = exp Tr(F kjHi(X)) i k i=0 k=1 2n Y (−1)i+1 = Pi(T ) : i=0 1.1 Algorithmic Approaches 1 For a variety X defined over Fq, the problem of computing the sequence f#X(Fqk )gk=1 is considered to be a hard problem. The brute force approach of inserting values into the defining equation of X has running time polynomial in q (i.e.
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