Hasse–Weil zeta function

Hasse–Weil zeta function
Name	Hasse–Weil zeta function
Field	Number theory, Arithmetic geometry, Algebraic geometry
Introduced	1920s–1960s
Key people	Emil Artin, Helmut Hasse, André Weil, John Tate, Alexander Grothendieck, Pierre Deligne, Benedict Gross, Barry Mazur

Hasse–Weil zeta function The Hasse–Weil zeta function is an analytic invariant attached to an algebraic variety defined over a global field, encoding data about reductions of the variety at places of the field and governing deep links between arithmetic, geometry, and analysis. It generalizes the Riemann zeta function and Dedekind zeta functions and connects to conjectures of Bernhard Riemann, André Weil, John Tate, Peter Scholze, and others concerning special values, functional equations, and analytic continuation. The function lies at the crossroads of work by Helmut Hasse, Alexander Grothendieck, Pierre Deligne, Goro Shimura, and contributors to the Langlands program such as Robert Langlands and Pierre Cartier.

Definition and Basic Properties

For a smooth projective variety X over a global field K (for example Q, a number field like a number field such as Kurt Gödel's birthplace city Bonn is unrelated), the Hasse–Weil zeta function is formally defined as an Euler product over nonarchimedean primes v of K, with local factors built from the action of Frobenius at v on the étale cohomology groups of X. Early formulations trace to counting points over finite fields as in work of Helmut Hasse and Weil, and the modern cohomological description uses the formalism developed by Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne. Basic expected properties include meromorphic continuation, a functional equation relating s to 1−s via a global epsilon factor influenced by conductors appearing in the work of John Tate and local factors studied by Igor Shafarevich and Jean-Loup Waldspurger.

Examples and Special Cases

Classical examples include the zeta function of projective space and curves: for projective line over Q the zeta reduces to the Dedekind zeta function of Q and for elliptic curves over Q one obtains L-series studied in the modularity theorem proved by Andrew Wiles, Richard Taylor, Christophe Breuil, Brian Conrad, and Fred Diamond. For varieties with complex multiplication the Hasse–Weil zeta function decomposes into Hecke L-series related to work of Hecke, Goro Shimura, and Yuri Manin. Examples from Shimura varieties tie into the theories developed by James Arthur, Michael Harris, and Richard Taylor, while zeta functions of K3 surfaces, Calabi–Yau varieties, and higher-dimensional abelian varieties connect to investigations by Pierre Deligne, Barry Mazur, and Kazuya Kato.

Relation to L-functions and Automorphic Forms

Conjecturally, the Hasse–Weil zeta function factors into automorphic L-functions predicted by the Langlands program formulated by Robert Langlands and refined by contributors like James Arthur, Michael Harris, and Richard Taylor. The modularity results for elliptic curves over Q and potential modularity for higher-dimensional motives relate to work of Andrew Wiles, Richard Taylor, Clozel, Harris, and Sheldon Katz; instances use the local Langlands correspondence established by Pierre Deligne, Guy Henniart, and Colin Bushnell. Compatibility of local factors with epsilon factors and conductors draws on results by Jean-Pierre Serre, John Tate, and Ken Ribet.

Analytic and Arithmetic Conjectures (Riemann Hypothesis, Birch–Swinnerton-Dyer)

The analogue of the Riemann hypothesis for varieties over finite fields was proved by Pierre Deligne using Grothendieck's étale cohomology; the global Riemann hypothesis for Hasse–Weil zeta functions remains open and is entwined with the Langlands correspondence conjectures of Robert Langlands and analytic properties studied by Harish-Chandra and Atle Selberg. The Birch–Swinnerton-Dyer conjecture for elliptic curves over Q predicts that the order of vanishing of the associated Hasse–Weil L-function at s=1 equals the Mordell–Weil rank, a problem advanced by Bryan Birch, Peter Swinnerton-Dyer, and major progress by Andrew Wiles, Benedict Gross, Don Zagier, and Kolyvagin. Special value formulas and regulators link to the Beilinson conjectures and Bloch–Kato conjectures formulated by Alexander Beilinson, Kazuya Kato, and Florian Popescu.

Methods of Construction and Cohomological Interpretation

Construction uses the Grothendieck–Lefschetz trace formula, étale cohomology developed by Alexander Grothendieck and Jean-Pierre Serre, and the formalism of weights and monodromy studied by Pierre Deligne and Nicholas Katz. Motives, as envisioned by Alexander Grothendieck and developed in conjectural form by Yves André and Uwe Jannsen, provide a conceptual framework in which Hasse–Weil zeta functions correspond to motivic L-functions. Tools include crystalline cohomology introduced by Pierre Berthelot, p-adic Hodge theory advanced by Jean-Marc Fontaine and Gerd Faltings, and arithmetic geometry techniques used by Gerd Faltings in finiteness theorems and by Faltings, Raynaud, and Serre in descent methods.

Known Results and Open Problems

Proven cases include Weil's conjectures for varieties over finite fields resolved by Alexander Grothendieck and Pierre Deligne, modularity of elliptic curves over Q by Andrew Wiles and Richard Taylor, and numerous instances of potential automorphy by Michael Harris, Richard Taylor, and collaborators. Major open problems include the general analytic continuation and functional equation for Hasse–Weil zeta functions in arbitrary dimension anticipated by the Langlands program, the global Riemann hypothesis in the spirit of Bernhard Riemann, and deep special-value conjectures such as Birch–Swinnerton-Dyer and Bloch–Kato posed by Alexander Beilinson, Kazuya Kato, and Florian Popescu. Progress continues through advances in p-adic methods by Peter Scholze, categorical and geometric Langlands work influenced by Edward Frenkel, and explicit arithmetic geometry computations by Bjorn Poonen and William Stein.

Category:Zeta functions