The exposition is selfcontained, and required a preliminary knowledge of only the ele. Zeta functions in algebraic geometry mircea mustat. By a global zeta function we mean an object that takes into account all primes. Download introduction to abelian varieties ebook free in pdf and epub format. The main purpose of this paper is to ll in more details to the proofs provided in milne. In addition, jacc is related to the divisor class group of the curve pic 0 c. 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. This series converges in the halfplane where the real part of sis greater than one. 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. The order of the pole of the zeta function zx,tat tq. 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. Cohomology of the kummer sequence for an abelian variety 72 4.
Evaluating zeta functions of abelian number fields at. Motivic zeta functions for degenerations of abelian. In this case one expects to have a product formula in terms of local factors. The zeta function of an abelian variety over an algebraic number field always refers to the onedimensional part see 12, 0. Pdf introduction to abelian varieties download ebook for free. Every abelian variety is isogenous to the product of powers of nonisogenous simple abelian varieties. Then the number of primes p xsuch that d 1p 1 satis es the asymptotic formula 3 x1 m1 m. The weil conjectures tell us that the hasseweil zeta function has the form. In general, it is not possible to write down explicit 1the case gd2is something of an exception to this statement. An abelian variety is a commutative algebraic group a on an irreducible projective variety. 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. Local zeta function of a characteristic p variety hasseweil l function of a variety motivic zeta function automorphic lfunctions of cusp forms on glm. The langweil estimate and the zeta function of an arithmetic scheme. 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 basic example is the wellknown factorization of the. These integers are encoded in the hasseweil zeta function of v as described below. We proved in hn11 that this is indeed the case when x is an abelian variety. The hasseweil zeta function let xqbe a projective variety of dimension d, and xza projective model of xq.
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. 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. The prime number theorem, hardys theorem on the zeros of. Pdf introduction to abelian varieties download ebook for.
This zeta function is expressed in terms of the riemann zeta function. Pdf motivic zeta functions for degenerations of abelian. Abelian varieties, theta functions and the fourier transform alexander polishchuk. Addendum to values of zeta functions of varieties over finite fields by j. Main theorems throughout the paper, we let f denote a totally real algebraic number field of degree n, k a totally. 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.
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. Then the complex torus c2dz has a structure of abelian variety. Frobenius on an associated abelian variety, and we develop the necessary machinery of abelian varieties. It can be shown that an abelian variety ak is projective as kvariety. Read introduction to abelian varieties online, read in mobile or kindle. Zeta function of the category of finite abelian groups. Models of an abelian variety with complex multiplication. 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. For abelian varieties such as a p, there is a definition of local zetafunction available. Generalized nonabelian reciprocity laws 3 than in q. Recall that the zeta function of an algebraic variety xover a. Addendum to values of zeta functions of varieties over.
One notes that an algebraic group on an irreducible projective variety is always commutative. 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. R whose value hp measures the arithmetic complexity of the point p. To get an lfunction for a itself, one takes a suitable euler product of such local functions. A choice of an equivalence class of riemann forms on a is called a polarisation of a. L is a discrete free abelian subgroup of v whose rank is equal to 2dim cv. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense. This method enables us to handle generic surfaces and threefolds over elds of large characteristic, e.
The zeta function associated with the category of finite abelian groups is defined and studied. Models of an abelian variety with complex multiplication over. 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. Abelian varieties with complex multiplication and modular. Motivic complexes and special values of zeta functions. The basic example is the wellknown factorization of the riemann zeta function. If ris any ring, we can then use vr to mean the solutions to the equations which. In fact, for every z e %, define a lattice d, in the twodimensional complex vector space c2 by d o. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. An elliptic curve is a onedimensional abelian variety. On the zeta function of a fibre variety whose fibres are. Introduction let k be a cmfield of degree 2n, q, a cmtype of k.
This is a survey on motivic zeta functions associated to abelian varieties and calabiyau varieties over a discretely valued field. The zeta function is an important function in mathematics. On the zeta function of a fibre variety whose fibres are abelian. For x a smooth proper variety over a finite field fq, its zeta function is.
Motivic zeta functions of abelian varieties, and the monodromy conjecture. Computing the zeta function of a curve 54 exercises 55 chapter 4. Frobenius on an associated abelian variety, and we develop the. The third topic explored by shimura is the various algebraic relations among the periods of abelian integrals.
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. 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. 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. Weil conjectures for abelian varieties over nite elds. Aug 12, 2017 the zeta function associated with the category of finite abelian groups is defined and studied. An abelian variety is a complete connected group variety over some. 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. The analytic proof involves the computation of the logarithmic derivative of the zeta function of kat the point s 0 in two different ways. The hasseweil zeta function of varieties over number fields are conjecturally.
Motivic zeta functions of abelian varieties, and the monodromy. The riemann hypothesis for varieties over finite fields. Two points z and y of s are equivalent with respect to rb if and only. We prove for abelian varieties a global form of denef and loeser. Recall that a complex abelian variety is a proper irreducible algebraic variety over c with an algebraic group. An introduction to height functions what are height functions let xk be a variety over a number. Here x p x f p is the bre of xover p, and xp s is the usual zeta function of the. Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by. More generally, when x is an abelian variety of a fixed dimension embedded in a. 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. We will end by discussing the results of larsen and lunts on kapranov zeta functions of algebraic surfaces. Every abelian variety of dimension 2is the jacobian variety of a curve of genus 2, and every curve of genus 2has an. 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. Thus, the zeta function ze,t can be computed efficiently by schoofs work.
The weil zeta function of a variety over a finite field. Basic properties for these nonabelian zeta functions, such as meromorphic extensions, rationality and functional equations, are established. Let zbe a smooth algebraic variety over f q, where q pa, for pa prime. More precisely, we prove that for every tamely rami. Local zetafunction of a characteristic p variety hasseweil lfunction of a variety motivic zeta function automorphic lfunctions of. An abelian variety is simple if it has no nontrivial abelian subvarieties. The partial zeta function measures the distribution of rational points of xindependently along the. Motivic zeta functions for degenerations of abelian varieties. 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. Motivic zeta functions of abelian varieties, and the. 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. 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 varieties.