Motivic zeta function

Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety X is the formal power series

Z(X,t)=\sum_{n=0}^\infty [X^{(n)}]t^n

Here X(n) is the n-th symmetric power of X, i.e., the quotient of Xn by the action of the symmetric group Sn, and [X(n)] is the class of X(n) in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to Z(X,t), one obtains the local zeta function of X.

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to Z(X,t), one obtains 1 / (1 − t)χ(X).

Motivic measures

A motivic measure is a map μ from the set of finite type schemes over a field k to a commutative ring A, satisfying the three properties

\mu(X)\, depends only on the isomorphism class of X,
\mu(X)=\mu(Z)+\mu(X\setminus Z) if Z is a closed subscheme of X,
\mu(X_1\times X_2)=\mu(X_1)\mu(X_2).

For example if k is a finite field and A={\Bbb Z} is the ring of integers, then \mu(X)=\#(X(k)) defines a motivic measure, the counting measure.

If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers.

The zeta function with respect to a motivic measure μ is the formal power series in A[[t]] given by

Z_\mu(X,t)=\sum_{n=0}^\infty\mu(X^{(n)})t^n.

There is a universal motivic measure. It takes values in the K-ring of varieties, A = K(V), which is the ring generated by the symbols [X], for all varieties X, subject to the relations

[X']=[X]\, if X' and X are isomorphic,
[X]=[Z]+[X\setminus Z] if Z is a closed subvariety of X,
[X_1\times X_2]=[X_1]\cdot[X_2].

The universal motivic measure gives rise to the motivic zeta function.

Examples

Let \Bbb L=[{\Bbb A}^1] denote the class of the affine line.

Z({\Bbb A}^n,t)=\frac{1}{1-{\Bbb L}^n t}
Z({\Bbb P}^n,t)=\prod_{i=0}^n\frac{1}{1-{\Bbb L}^i t}

If X is a smooth projective irreducible curve of genus g admitting a line bundle of degree 1, and the motivic measure takes values in a field in which {\Bbb L} is invertible, then

Z(X,t)=\frac{P(t)}{(1-t)(1-{\Bbb L}t)}\,,

where P(t) is a polynomial of degree 2g. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If S is a smooth surface over an algebraically closed field of characteristic 0, then the generating function for the motives of the Hilbert schemes of S can be expressed in terms of the motivic zeta function by Göttsche's Formula

\sum_{n=0}^\infty[S^{[n]}]t^n=\prod_{m=1}^\infty Z(S,{\Bbb L}^{m-1}t^m)

Here S[n] is the Hilbert scheme of length n subschemes of S. For the affine plane this formula gives

\sum_{n=0}^\infty[({\Bbb A}^2)^{[n]}]t^n=\prod_{m=1}^\infty \frac{1}{1-{\Bbb L}^{m+1}t^m}

This is essentially the partition function.


Wikimedia Foundation. 2010.

Поможем решить контрольную работу

Look at other dictionaries:

  • Motivic L-function — In mathematics, motivic L functions are a generalization of Hasse–Weil L functions to general motives over global fields. The local L factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v… …   Wikipedia

  • Dedekind zeta function — In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function which is obtained by specializing to the case where K is the rational numbers Q. In particular,… …   Wikipedia

  • Riemann zeta function — ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… …   Wikipedia

  • Dirichlet L-function — In mathematics, a Dirichlet L series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole… …   Wikipedia

  • Дзета-функции — Эта страница информационный список. См. также основную статью: Дзета функция Римана В математике дзета функция обычно это функция родственная или аналогичная дзета функции Римана …   Википедия

  • Riemann hypothesis — The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011 …   Wikipedia

  • Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… …   Wikipedia

  • List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

  • Class number formula — In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function Contents 1 General statement of the class number formula 2 Galois extensions of the rationals 3 A …   Wikipedia

  • Birch and Swinnerton-Dyer conjecture — Millennium Prize Problems P versus NP problem Hodge conjecture Poincaré conjecture Riemann hypo …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”