Computing the zeta function of a curve 54 exercises 55 chapter 4. Pdf motivic zeta functions for degenerations of abelian. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. Dedekind zeta function of a number eld epstein zeta function of a quadratic form goss zeta function of a function eld padic zeta function from algebraic geometry. In fact, for every z e %, define a lattice d, in the twodimensional complex vector space c2 by d o. We proved in hn11 that this is indeed the case when x is an abelian variety. Download introduction to abelian varieties ebook free in pdf and epub format. Basic properties for these nonabelian zeta functions, such as meromorphic extensions, rationality and functional equations, are established.
The order of the pole of the zeta function zx,tat tq. Then the complex torus c2dz has a structure of abelian variety. On the periods of abelian varieties 5 that the zeta function of kis the dirichlet series given by the sum over all nonzero ideals a of the ring o of integers of k ks x na s. Two points z and y of s are equivalent with respect to rb if and only. Pdf introduction to abelian varieties download ebook for free. Two riemann forms h 1 and h 2 are called equivalent if there are positive integers n and m such that nh 1 mh 2. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense.
Weil conjectures for abelian varieties over nite elds kwun chung abstract this is an expository paper on zeta functions of abelian varieties over nite elds. The study of abelian manifolds forms a natural generalization of the theory of elliptic functions, that is, of doubly periodic functions of one complex variable. An introduction to height functions what are height functions let xk be a variety over a number. On the zeta function of a fibre variety whose fibres are abelian varieties. Abelian varieties and curves of genus g over a finite field f q are bound together in their study since the jacobian of the curve c is an abelian variety jacc of dimension g over f q. Every abelian variety of dimension 2is the jacobian variety of a curve of genus 2, and every curve of genus 2has an equation of the form y2z4 df0x6 cf1x5zcc f6z6. Let zbe a smooth algebraic variety over f q, where q pa, for pa prime. The hasseweil zeta function of varieties over number fields are conjecturally. More precisely, we prove that for every tamely rami.
Generalized nonabelian reciprocity laws 3 than in q. Evaluating zeta functions of abelian number fields at. The motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to calabiyau varieties and to compute them explicitly on a model with strict normal. The weil zeta function of a variety over a finite field. The main purpose of this paper is to ll in more details to the proofs provided in milne. This should serve the community well as there appears to be a distinct lack of numerical data regarding zeta values at negative integers, despite there being great overall interest in these numbers. The riemann hypothesis for varieties over finite fields. Motivic zeta functions of abelian varieties, and the.
Zeta function of the category of finite abelian groups. Recall that the zeta function of kis the dirichlet series given by the sum over all nonzero ideals a of the ring o of integers of k ks x na s where the norm naof an ideal is equal to its index in the. More precisely, we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at chai. On the zeta function of a fibre variety whose fibres are. An abelian variety is a complete connected group variety over some. Recall that the zeta function of an algebraic variety xover a. These integers are encoded in the hasseweil zeta function of v as described below. An abelian variety is simple if it has no nontrivial abelian subvarieties.
An abelian variety is a commutative algebraic group a on an irreducible projective variety. This zeta function is expressed in terms of the riemann zeta function. Congruences between abelian padic zeta functions mahesh kakde this article is a reproduction of lectures in the workshop based on section 6 of kak10 with a slight change in the notation to make it consistent with previous articles in the volume. Thus, the zeta function ze,t can be computed efficiently by schoofs work. An elliptic curve is a onedimensional abelian variety. We will end by discussing the results of larsen and lunts on kapranov zeta functions of algebraic surfaces. The langweil estimate and the zeta function of an arithmetic scheme. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The basic example is the wellknown factorization of the.
Every abelian variety is isogenous to the product of powers of nonisogenous simple abelian varieties. Motivic zeta functions of abelian varieties, and the monodromy conjecture. Here x p x f p is the bre of xover p, and xp s is the usual zeta function of the. Thus, partial zeta functions are generalizations of moment zeta functions. The basic example is the wellknown factorization of the riemann zeta function. Motivic zeta functions of abelian varieties, and the monodromy.
Dec 22, 2010 the motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to calabiyau varieties and to compute them explicitly on a model with strict normal. On the zeta function of a fibre variety whose fibres are abelian. More generally, when x is an abelian variety of a fixed dimension embedded in a. Our object of study is a structure a, 8 formed by an abelian variety a of dimension n defined over an algebraic number field k. Recall that a complex abelian variety is a proper irreducible algebraic variety over c with an algebraic group. Cohomology of the kummer sequence for an abelian variety 72 4. Abelian varieties and curves of genus g over a finite field f q are bound together in their study since the jacobian of the curve c is an abelian variety jacc. Local zetafunction of a characteristic p variety hasseweil lfunction of a variety motivic zeta function automorphic lfunctions of. In this case one expects to have a product formula in terms of local factors. Zeta functions in algebraic geometry mircea mustat. The zeta function associated with the category of finite abelian groups is defined and studied. To get an lfunction for a itself, one takes a suitable euler product of such local functions.
Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by. An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods. The compact complex torus x of the form vl is isomorphic to the cpoints of a complex abelian variety if and only if the pair v. We prove for abelian varieties a global form of denef and loeser. Lk is a finite separable field ex tension, a is an abelian variety over l, and a, is the abelian variety over k obtained from a by restriction of scalars. This is a survey on motivic zeta functions associated to abelian varieties and calabiyau varieties over a discretely valued field.
Motivic complexes and special values of zeta functions. Abelian varieties, theta functions and the fourier transform alexander polishchuk. For abelian varieties such as a p, there is a definition of local zetafunction available. Weil conjectures for abelian varieties over nite elds. Motivic zeta functions for degenerations of abelian varieties.
The analytic proof involves the computation of the logarithmic derivative of the zeta function of kat the point s 0 in two different ways. Frobenius on an associated abelian variety, and we develop the necessary machinery of abelian varieties. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. The third topic explored by shimura is the various algebraic relations among the periods of abelian integrals. Aug 12, 2017 the zeta function associated with the category of finite abelian groups is defined and studied. Over the complex numbers, a polarised abelian variety can also be defined as an abelian variety a together with a choice of a riemann form h. The exposition is selfcontained, and required a preliminary knowledge of only the ele.
Read introduction to abelian varieties online, read in mobile or kindle. R whose value hp measures the arithmetic complexity of the point p. In general, it is not possible to write down explicit 1the case gd2is something of an exception to this statement. For x a smooth proper variety over a finite field fq, its zeta function is. L is a discrete free abelian subgroup of v whose rank is equal to 2dim cv.
Main theorems throughout the paper, we let f denote a totally real algebraic number field of degree n, k a totally. The prime number theorem, hardys theorem on the zeros of. By a global zeta function we mean an object that takes into account all primes. Abelian varieties with complex multiplication and modular. In addition, jacc is related to the divisor class group of the curve pic 0 c. It can be shown that an abelian variety ak is projective as kvariety. Local zeta function of a characteristic p variety hasseweil l function of a variety motivic zeta function automorphic lfunctions of cusp forms on glm. A choice of an equivalence class of riemann forms on a is called a polarisation of a. Frobenius on an associated abelian variety, and we develop the. Models of an abelian variety with complex multiplication. One notes that an algebraic group on an irreducible projective variety is always commutative. The zeta function of an abelian variety over an algebraic number field always refers to the onedimensional part see 12, 0. Addendum to values of zeta functions of varieties over. Addendum to values of zeta functions of varieties over finite fields by j.
This method enables us to handle generic surfaces and threefolds over elds of large characteristic, e. If ris any ring, we can then use vr to mean the solutions to the equations which. Introduction let k be a cmfield of degree 2n, q, a cmtype of k. Then the number of primes p xsuch that d 1p 1 satis es the asymptotic formula 3 x1 m1 m. This series converges in the halfplane where the real part of sis greater than one. Pdf introduction to abelian varieties download ebook for. Models of an abelian variety with complex multiplication over. The weil conjectures tell us that the hasseweil zeta function has the form. The partial zeta function measures the distribution of rational points of xindependently along the. Every abelian variety of dimension 2is the jacobian variety of a curve of genus 2, and every curve of genus 2has an. The hasseweil zeta function let xqbe a projective variety of dimension d, and xza projective model of xq. Determinant formulas for zeta functions for real abelian function fields shiomi, daisuke, proceedings of the japan academy, series a, mathematical sciences, 2011 zeta functions of generalized permutations with application to their factorization formulas koyama, shinya and nakajima, sachiko, proceedings of the japan academy, series a.