Goss zeta function

Goss zeta function
Name	Goss zeta function
Introduced	1970s–1980s
Introduced by	David Goss
Field	Number theory, Algebraic geometry
Related	Riemann zeta function, Dedekind zeta function, Drinfeld module, L-series

Goss zeta function The Goss zeta function is a zeta function for global function fields introduced by David Goss that plays a role analogous to the Riemann zeta function and the Dedekind zeta function in the arithmetic of function fields over finite fields, connecting ideas from André Weil, Emil Artin, John Tate, Vladimir Drinfeld and Gerd Faltings. It encodes arithmetic of rings of functions on curves over Finite fields and interacts with structures studied by Carlitz, Leonard Carlitz, Serre, Grothendieck, Alexander Grothendieck and Jean-Pierre Serre.

Definition and Basic Properties

In the function field setting one fixes a global function field K (finite extension of F_q(T)), a ring A of functions regular outside a chosen place, and a notion of positive divisors as in work of André Weil, Friedrich Karl Schmidt and Hasse, then David Goss defined the zeta function ζ_A(s) as a sum over nonzero ideals or as an Euler product over primes modeled on constructions by Emil Artin, Helmut Hasse and Erich Hecke, paralleling classical constructions by Bernhard Riemann, Dedekind and Ernst Kummer. Basic properties include multiplicativity, convergence in a half-plane analogous to regions in the Riemann zeta function theory, and compatibility with degree maps studied by John Tate and Pierre Deligne. The definition uses completions at infinite places akin to constructions in Alexander Grothendieck's theory of schemes and draws on function field techniques developed by Max Deuring, I. Reiner, and Gerald J. Janusz.

Analogy with Riemann and Dedekind Zeta Functions

The analogy between Goss's construction and the Riemann zeta function or the Dedekind zeta function manifests through Euler products, factorization by primes studied by Hecke and Ernst Zermelo, and through spectral interpretations reminiscent of approaches by Atle Selberg, Harish-Chandra, and Peter Sarnak. The role of the archimedean place in number fields is replaced by the ``infinite'' place of a function field as treated by Emil Artin and John Tate, while class field theoretic consequences echo results of Emil Artin, Helmut Hasse and modern formulations by Milne and Serre.

Analytic Continuation and Functional Equation

Goss established analytic continuation results and functional equations in the framework of non-Archimedean analysis influenced by the work of Klaus Schmidt and Igor Shafarevich; these mirror functional identities for the Riemann zeta function and for Hecke L-series studied by Erich Hecke and Atle Selberg. The analytic continuation uses techniques from p-adic analysis and from the theory of Tate's thesis by John Tate, while functional equations relate values at s and 1 − s or analogous shifts, reflecting dualities akin to those in conjectures of Bernhard Riemann and the functional equations proven by André Weil and Pierre Deligne.

Special Values and Arithmetic Applications

Special values of the Goss zeta function at integers and at analogues of negative integers yield algebraic information reminiscent of results by Bernhard Riemann for ζ(−n), by Kummer for cyclotomic values, and by Dirichlet and Hecke for L-values; they connect to class number formulas and regulators in the function field context as in the work of Emil Artin, Helmut Hasse and John Tate. Applications include explicit class number relations, analogues of the Birch and Swinnerton-Dyer conjecture for function field abelian varieties investigated by Vladimir Drinfeld, Gerd Faltings and Matthew Baker, and interactions with special value formulas developed by Taelman, David Goss, and Richard Pink.

Relation to Drinfeld Modules and Taelman's L-values

The Goss zeta function is intimately linked to the theory of Drinfeld modules introduced by Vladimir Drinfeld and to Taelman's L-values for motives over function fields; Taelman's theorems draw connections between special values of L-series and class modules paralleling Dirichlet's class number formula, building on the arithmetic of Drinfeld's shtuka and on ideas of Grothendieck and Serre. These relations involve moduli studied by Drinfeld, cohomological techniques in the style of Pierre Deligne, and explicit constructions by Taelman that echo classical work of Emil Artin and John Tate.

Examples and Explicit Computations

Concrete computations appear in the simplest A = F_q[T] case studied originally by Leonard Carlitz and extended by David Goss, where explicit formulas mimic those of Carlitz module theory and classical computations by Gauss, Jacobi, and Kummer in cyclotomic settings. Further explicit examples involve elliptic curves over Finite fields treated by André Weil and by Jean-Pierre Serre, Drinfeld modules analyzed by Vladimir Drinfeld and Goss, and computational approaches influenced by algorithmic work of Victor Shoup, Henri Cohen, and Andrew Granville.

Category:Zeta functions