Hasse weil zeta function pdf

The simplest zeta functions from algebraic geometry are the hasse weil zeta functions of elliptic curves over q. The riemann zeta function has many generalizations, notably the hasse weil zeta function. An k be the common zero locus of the polynomials f 1. Hasseweil zeta function of absolutely irreducible sl2representations of the.

Diophantine equations the simplest zeta functions from algebraic geometry are the hasseweil zeta functions of elliptic curves over q. We will say that g has property z if there exists a finite galois extension l of k such that the hasseweil zeta function zetag,k,s of g is an alternating. The local hasse weil zeta function is the following power series. Efficient computation of the hasseweil zeta function. I believe that if the riemann hypothesis is valid then the riemann hypothesis for such a hasseweil zeta function is valid. The hasseweil zeta function of varieties over number fields are conjecturally. This method enables us to handle generic surfaces and threefolds over elds of large characteristic, e. Hasseweil zeta functions of sl2character varieties of. The hasse weil zeta function this is one of the most famous zeta functions, and it played an important role in the development of algebraic geometry in the twentieth century. Topological hochschild homology and the hasseweil zeta. The weil conjectures tell us that the hasse weil zeta function has the form. Sl2character varieties of 3manifolds shinya harada tokyo institute of technology may 21, 2014 supported by the jsps fellowships for young scientists shinya harada hasseweil zeta of 3 manifolds. Qt which turns out to be a rational function dwork that can be algorithmically determined bombieri.

The motivic analogue of the hasseweil zeta function was introduced by kapra. Milne september 14, 2015 abstract the statement of the riemann hypothesis makes sense for all global. Weil stu january 18, 2017 contents 1 introduction 2 2 hasse bound for elliptic curves 2. Odonis questions can then be formulated as follows. Local zeta function hasseweil zeta function hw zeta of. I am looking for references about the hasse weil zeta for arbitrary variety and number field, particularly analytic continuation and functional equation this. The zeta function satis es the following properties. The problem of calculating hasseweil zetafunction for a given. Most research on the hasseweil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. It is implemented in terms of path integrals with the statistics physics interpretation in mind. I am looking for references about the hasseweil zeta for arbitrary variety and number field, particularly analytic continuation and functional equation this. Noncommutative motives in positive characteristic and their. Im thinking of the hasseweil zeta function associated with a certain elliptic curve, which has cm. Topological hochschild homology and the hasseweil zeta function lars hesselholt introduction in this paper, we consider the tate cohomology of the circle group acting on the topological hochschild homology of schemes.

The zeta function zt satis es the functional equation z 1 qt zt qt2g 1. Definition and elementary properties in this lecture we introduce the hasseweil zeta function, and prove some elementary properties. Zeta function of a curve eindhoven university of technology. In class we will see how the riemann zeta function s p 1 n1 1. Odoni, to discuss which we introduce the following terminology and notation. Congruence and hasseweil zeta functions written by jonathan hanke february 14, 2005 abstract many of the most important questions in number theory are connected to questions about \zeta functions associated to various interesting objects. Let g be a linear algebraic group defined over an algebraic number field k. I believe that if the riemann hypothesis is valid then the riemann hypothesis for such a hasse weil zeta function is valid. The hasseweil zeta function let xqbe a projective variety of dimension d, and xza projective model of xq. The relation with riemann zeta function is explained, shedding. Hasseweil zeta functions of closed hyperbolic 3manifolds 3 since we could determine x 0m it also is possible to determine the hasseweil zeta function of psl 2character variety of a closed hyperbolic 3manifold in the sense of heusener and porti as follows. The weiletale topology for number rings 659 and our conjectures concern the scheme zetafunction of a scheme over specz. Definition and elementary properties in this lecture we introduce the hasse weil zeta function, and prove some elementary properties.

In mathematics, the hasse weil zeta function attached to an algebraic variety v defined over an algebraic number field k is one of the two most important types of l function. To have the description of the hasseweil zeta function. Topological hochschild homology and the hasseweil zeta function. This result is again equivalent to the determination of the absolute value of the roots of the local zetafunction of c, and is the analogue of the riemann hypothesis for the function field associated with the curve. Explicit examples of hasseweil zetafunction calculations. We will say that g has property z if there exists a finite galois extension l of k such that the hasse weil zeta function zeta g,k,s of g is an alternating product of artin lfunctions for characters of gallk. The riemann zeta function has many generalizations, notably the hasseweil zeta function. In this case, the periodicity of the zeta function is re. Dynamical zeta functions, nielsen theory alexander felshtyn.

Topological hochschild homology and the hasse weil zeta function lars hesselholt introduction in this paper, we consider the tate cohomology of the circle group acting on the topological hochschild homology of schemes. Diophantine equations the simplest zeta functions from algebraic geometry are the hasse weil zeta functions of elliptic curves over q. Dirichlet species and the hasseweil zeta function in john baez. Is there a riemann hypothesis for the hasseweil zeta. We have zt pt 1 t1 qt where pt y2g i1 1 it 2zt is a polynomial of degree 2gwith reciprocal roots i 2c that are algebraic integers satisfying j ij p q. Im thinking of the hasse weil zeta function associated with a certain elliptic curve, which has cm.

Hasse for g 1 and weil for the general case showed that this function is a rational function of the form pt1. So at least conjecturally one is reduced to the study of these. We show that in the case of a scheme smooth and proper over a nite eld, this cohomology theory naturally gives rise. The weil conjectures tell us that the hasseweil zeta function has the form. The second conjecture asserts that the hasseweil zeta function satis es a functional equation, much like the riemann zeta function. The hasseweil bound reduces to the usual hasse bound when applied to elliptic curves, which have genus g1. Thus, i think that this problem is as difficult as riemann hypothesis. The hasseweil zeta function this is one of the most famous zeta functions, and it played an important role in the development of algebraic geometry in the twentieth century. The aim of the project was to study two questions unpublished of r. Such l functions are called global, in that they are defined as euler products in terms of local zeta functions. The hasseweil zetafunction is then defined as a product over all finite places of q.

It is easier to work in still more generality, starting from any functor. If the scheme is smooth and proper over specz, then the zetafunction of the scheme is the same as the hasseweil zetafunction of the generic. Introduction the purpose of this paper is to serve as the. Effective computations of hasseweil zeta functions mit math. The hasseweil zeta function to state the weil conjectures we will use the hasseweil zeta function. The local hasseweil zeta function is the following power series. Hasse weil zeta functions of closed hyperbolic 3manifolds 3 since we could determine x 0m it also is possible to determine the hasse weil zeta function of psl 2character variety of a closed hyperbolic 3manifold in the sense of heusener and porti as follows. Hasseweil zeta functions of smooth hypersurfaces over finite fields, based on kedlayas. Let m be an orientable closed hyperbolic 3manifold of. Lfunctions in particular dedekind zeta functions, artin lfunctions, hasseweil zeta functions. Most research on the hasse weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.

Motivic versions of the above local zeta functions4 1. The riemann hypothesis over finite fields from weil to the present day james s. They form one of the two major classes of global l functions, the other being the l functions associated. References for general hasseweil zeta function mathoverflow. Automorphic forms representations and l functions ed borel. Hasseweil zeta functions of sl2character varieties of 3. Explicit examples of hasse weil zetafunction calculations for curves. The course will also develop selected topics of the theory of etale cohomology. Zeta functions in algebraic geometry mircea mustat.