14.3 K-Theory and zeta functions

K-Theory and zeta functions are powerful tools in algebraic geometry and number theory. They provide a bridge between arithmetic and geometric properties of varieties, encoding crucial information about their structure and behavior.

Zeta functions, expressed through Frobenius eigenvalues, connect to K-theory groups. Their special values reveal insights about Picard groups, Brauer groups, and other important algebraic structures. This relationship offers a deep understanding of varieties' arithmetic and geometric nature.

Zeta Function of a Variety

Definition and Connection to K-Theory

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• The zeta function of a variety $X$ over a finite field is a generating function that encodes the number of points of $X$ over finite extensions of the base field
• Defined as a formal power series or an Euler product, with factors corresponding to the Frobenius eigenvalues on the étale cohomology of $X$
• Closely related to the K-theory of $X$, as the Frobenius eigenvalues can be interpreted as the eigenvalues of the Frobenius endomorphism acting on the K-theory groups
• Special values of the zeta function at certain integers are related to the ranks of the K-theory groups and the orders of the torsion subgroups
  • For example, the value at $s=0$ is related to the rank of the Picard group (group of line bundles) and the order of the Brauer group (group of Azumaya algebras)
• The connection between the zeta function and K-theory provides a link between arithmetic and geometric properties of the variety

Expressing the Zeta Function

• Express the zeta function in terms of the Frobenius eigenvalues on the étale cohomology
  • For example, for a smooth projective variety $X$ over $\mathbb{F}_q$, the zeta function is given by: $Z(X,t) = \exp\left(\sum_{n=1}^\infty \frac{|X(\mathbb{F}_{q^n})|}{n}t^n\right) = \prod_{i=0}^{2\dim X} \det(1-tF|H^i_{\acute{e}t}(X,\mathbb{Q}_\ell))^{(-1)^{i+1}}$ where $F$ is the Frobenius endomorphism and $H^i_{\acute{e}t}(X,\mathbb{Q}_\ell)$ is the $\ell$-adic étale cohomology
• Use the Lefschetz fixed-point formula to relate the Frobenius eigenvalues to the trace of the Frobenius endomorphism on the K-theory
• The resulting expression connects the zeta function to the K-theory of the variety

Computing Zeta Functions

Grothendieck-Riemann-Roch Theorem

• The Grothendieck-Riemann-Roch theorem is a powerful tool for computing the zeta function of a variety
• Relates the zeta function to the Chern character of the K-theory and the Todd class of the tangent bundle
• States that the pushforward of the Chern character of a coherent sheaf under the morphism to a point is equal to the integral of the product of the Chern character and the Todd class over the variety
  • For example, for a morphism $f: X \to Y$ and a coherent sheaf $\mathcal{E}$ on $X$, the theorem states: $f_*(\mathrm{ch}(\mathcal{E}) \cdot \mathrm{td}(T_X)) = \mathrm{ch}(f_*\mathcal{E}) \cdot \mathrm{td}(T_Y)$ where $\mathrm{ch}$ is the Chern character, $\mathrm{td}$ is the Todd class, and $T_X$ and $T_Y$ are the tangent bundles of $X$ and $Y$

Applying the Theorem

• To compute the zeta function using the Grothendieck-Riemann-Roch theorem:
  1. Express the zeta function in terms of the Frobenius eigenvalues on the étale cohomology
  2. Use the Lefschetz fixed-point formula to relate the Frobenius eigenvalues to the trace of the Frobenius endomorphism on the K-theory
  3. Apply the Grothendieck-Riemann-Roch theorem to compute the trace of the Frobenius endomorphism in terms of the Chern character and the Todd class
• The resulting formula expresses the zeta function as a product of factors involving the Chern character of the K-theory and the Todd class of the tangent bundle
  • For example, for a smooth projective variety $X$ over $\mathbb{F}_q$, the zeta function can be expressed as: $Z(X,t) = \prod_{i=0}^{\dim X} \det(1-tF|H^i(X,\mathcal{O}_X))^{(-1)^{i+1}}$ where $F$ is the Frobenius endomorphism and $H^i(X,\mathcal{O}_X)$ is the coherent cohomology of the structure sheaf $\mathcal{O}_X$

Properties of Zeta Functions

Analytic Properties

• Meromorphic continuation: The zeta function, initially defined as a formal power series, can be analytically continued to a meromorphic function on the complex plane
• Functional equation: The zeta function satisfies a functional equation relating its values at $s$ and $1-s$, up to a factor involving the Euler characteristic and the dimensions of the cohomology groups
  • For example, for a smooth projective variety $X$ over $\mathbb{F}_q$, the functional equation is given by: $Z(X,q^{-s}) = \pm q^{\chi(X)s/2}Z(X,q^{s-1})$ where $\chi(X)$ is the Euler characteristic of $X$
• Poles and zeros: The poles and zeros of the zeta function are related to the eigenvalues of the Frobenius endomorphism on the étale cohomology, and their locations and orders provide information about the arithmetic and geometric properties of the variety

Algebraic Properties

• Rationality: The zeta function is a rational function in the variable $t=q^{-s}$, where $q$ is the cardinality of the base field
  • The degree of the numerator and denominator are related to the Betti numbers of the variety
  • For example, for a smooth projective curve $C$ over $\mathbb{F}_q$, the zeta function has the form: $Z(C,t) = \frac{P(t)}{(1-t)(1-qt)}$ where $P(t)$ is a polynomial of degree $2g$, with $g$ being the genus of the curve
• Special values: The special values of the zeta function at certain integers (e.g., $s=0,1$) have interpretations in terms of the K-theory and cohomology of the variety, such as the order of the Brauer group or the class number

Special Values of Zeta Functions

Interpretations in K-Theory and Cohomology

• The value of the zeta function at $s=0$ is related to the rank of the Picard group (the group of line bundles) and the order of the Brauer group (the group of Azumaya algebras)
  • These groups are related to the K-theory of the variety
  • For example, for a smooth projective variety $X$ over $\mathbb{F}_q$, we have: $Z(X,0) = \frac{|\mathrm{Pic}(X)|}{|\mathrm{Br}(X)|}$ where $\mathrm{Pic}(X)$ is the Picard group and $\mathrm{Br}(X)$ is the Brauer group
• The value of the zeta function at $s=1$ is related to the class number of the variety (the degree of the Picard variety) and the order of the Shafarevich-Tate group (a group measuring the failure of the Hasse principle)
  • These quantities are related to the étale cohomology of the variety
  • For example, for an abelian variety $A$ over a global field $K$, the Birch and Swinnerton-Dyer conjecture predicts: $\lim_{s \to 1} (s-1)^{-r} L(A,s) = \frac{|\Sha(A/K)| \cdot \mathrm{Reg}(A/K) \cdot \prod_v c_v}{|\mathrm{Tors}(A/K)|^2}$ where $r$ is the rank of the Mordell-Weil group, $\Sha(A/K)$ is the Shafarevich-Tate group, $\mathrm{Reg}(A/K)$ is the regulator, $c_v$ are the Tamagawa numbers, and $\mathrm{Tors}(A/K)$ is the torsion subgroup of the Mordell-Weil group

Research and Conjectures

• The residue of the zeta function at $s=1$ is equal to the product of the Euler characteristic of the variety and a factor involving the Tamagawa numbers (local densities) of the variety over the local fields
  • This residue formula is a manifestation of the Birch and Swinnerton-Dyer conjecture
• The special values of the derivatives of the zeta function at $s=0$ are related to the determinants of the cohomology groups and the regulators of the variety, which are defined using the K-theory and the Chow groups of the variety
• Studying the special values of zeta functions and their interpretations in terms of K-theory and cohomology is an active area of research, with connections to various conjectures and problems in arithmetic geometry, such as the Tate conjecture and the Birch and Swinnerton-Dyer conjecture