Arithmetic zeta function

In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most fundamental objects of number theory.

Definition

The arithmetic zeta function ${\zeta _{X}(s)}$ is defined by an Euler product analogous to the Riemann zeta function:

${\zeta _{X}(s)}=\prod _{x}({1-N(x)^{-s}})^{-1}$

where the product is taken over all closed points $x$ of the scheme $X$. Equivalently, the product is over all points whose residue field is finite. The cardinality of this field is denoted N(x).

Examples

For example, if $X$ is a finite field with $q$ elements, then ${\zeta _{X}(s)}={\frac {1}{1-q^{-s}}}$. If $X$ is the spectrum of the ring of integers, then ${\zeta _{X}(s)}$ is the Riemann zeta function. More generally, if $X$ is the spectrum of the ring of integers of an algebraic number field, then ${\zeta _{X}(s)}$ is the Dedekind zeta function.

The zeta function of affine and projective spaces over a scheme X are given by

$\zeta _{\mathbf {A} ^{n}(X)}(s)=\zeta _{X}(s-n)$

$\zeta _{\mathbf {P} ^{n}(X)}(s)=\prod _{i=0}^{n}\zeta _{X}(s-i)$.

Main conjectures

Let $X$ be a regular irreducible equidimensional scheme proper over integers (characteristic zero case) or over a finite field (positive characteristic). Then

1. The arithmetic zeta function ${\zeta _{X}(s)}$ has a meromorphic continuation to the complex plane and satisfies a functional equation with respect to $s\to n-s$ where $n$ is the absolute dimension of $X$.

2 (generalized Riemann Hypothesis). The zeros of ${\zeta _{X}(s)}$ inside the critical strip $Re(s)\in [0,n]$ lie on the vertical lines $Re(s)=1/2,3/2,\dots ,$ and the poles of ${\zeta _{X}(s)}$ inside the critical strip $Re(s)\in [0,n]$ lie on the vertical lines $Re(s)=0,1,2,\dots$

3 (special values). The order of the zero or pole and the residue of ${\zeta _{X}(s)}$ at integer points inside the critical strip can be described by formulas involving important arithmetic invariants of $X$.

What is proved

Conjecture 1 is proved when $n=1$ and some few special cases when $n>1$ in characteristic zero and for all $n$ in positive characteristic.

Conjecture 2 is proved (Emil Artin, Helmut Hasse, André Weil, Alexander Grothendieck, Pierre Deligne) in positive characteristic for all $n$. It is not proved for any schemes in characteristic zero. The Riemann hypothesis is a partial case of Conjecture 2.

Conjecture 3 is known when $n=1$ and is known in few very special cases when $n>1$. The Birch and Swinnerton-Dyer conjecture is a partial case of Conjecture 3.

Methods and theories

The arithmetic zeta function of a regular connected equidimensional arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor. Hence, results on L-functions imply corresponding results for the arithmetic zeta functions. However, there is still very little amount of proven results about the L-factors of arithmetic schemes in characteristic zero and dimensions 2 and higher. Ivan Fesenko initiated a theory which studies the arithmetic zeta functions directly, without working with their L-factors. It is a higher dimensional generalisation of Tate's thesis, i.e. it uses higher adele groups, higher zeta integral and objects which come from higher class field theory.

References

• François Bruhat (1963). Lectures on some aspects of p-adic analysis. Tata Institute of Fundamental Research.
• Fesenko, Ivan (2010), "Analysis on arithmetic schemes. II", Journal of K-theory, 5: 437–557
• Jean-Pierre Serre (1965). Zeta and L-functions. Harper and Row. {{cite book}}: Unknown parameter |book= ignored (help)
• Serre, Jean-Pierre (1969/70), "Facteurs locaux des fonctions zeta des varietés algébriques (définitions et conjectures)", Séminaire Delange-Pisot-Poitou, 19 {{citation}}: Check date values in: |year= (help)CS1 maint: year (link)