a directory of all known L-functions

[This page is under construction! Contributions are strongly encouraged.]

"It's a whole beautiful subject and the Riemann zeta function is just the first one of these, but it's just the tip of the iceberg. They are just the most amazing objects, these L-functions - the fact that they exist, and have these incredible properties are tied up with all these arithmetical things - and it's just a beautiful subject. Discovering these things is like discovering a gemstone or something. You're amazed that this thing exists, has these properties and can do this."

B. Conrey, Dr. Riemann's Zeros (Atlantic, 2002), p.166

J. Baez, J. Baez, This Week's Finds in Mathematical Physics week 217 includes very helpful discussion of the Riemann Hypothesis, Extended Riemann Hypothesis, Grand Riemann Hypothesis, Weil Conjectures, Langlands Programme, the functional equations of zeta and L-functions, modularity of theta functions, etc. week 218 follows this up, framing certain issues concerning bewildering array of zeta functions and L-functions in terms of category theory.

D.W. Farmer, "Modeling families of L-functions" (preprint 11/05)

[abstract:] "We discuss the idea of a 'family of L-functions' and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model and the Interaction Model."

abelian L-functions

J.-F. Burnol, "An adelic causality problem related to abelian L-functions", Journal of Number Theory 87 no.2 (2001) 253-269.

L-functions of algebraic varieties

Artin L-functions

M.R. Murty, "Selberg's conjectures and Artin L-functions", Bulletin of the AMS (New Series) 31 (1994) 1-14.

"In his thesis (published in 1920) the German mathematician E. Artin had developed the arithmetic theory of ‘function fields over a finite field’, in particular the field of functions on a curve over a finite field, and noted the many similarities with the theory of number fields developed by Dirichlet, Dedekind, Kronecker and Hilbert. The analogies between function fields and number fields had been known since Dedekind’s time (at least in characteristic zero), but Artin’s work was perhaps the first to take the base field to have positive characteristic as opposed to subfields of the complex numbers. This required an entirely algebraic development of the subject since the transcendental techniques derived from working over the field of complex numbers are inapplicable in this new context. Artin also (later) developed a quite general theory of L-functions which, once again by purely algebraic means, defined functions akin to the zeta function for general number fields and for function fields. Artin may thus be seen to have been working to ‘geometrize or at least "algebraicize") number theory’ while Weil was trying to ‘arithmetize geometry’, and Weil has remarked on the excitement with which he and his colleagues in those days awaited new numbers of the journals which regularly contained Artin’s work. These results showed that a ‘Riemann hypothesis’ could be formulated for the L-functions arising in function fields. In an astonishing piece of work (announced in 1940 but only fully written up in 194?) Weil was able to use geometric techniques to prove the ‘Riemann hypothesis’ for the function field of any curve over a finite field. n the course of this effort he convinced himself that a reformulation of the foundations of algebraic geometry was imperative.

His approach to the ‘Riemann hypothesis for finite fields’ was to use the theory of ‘correspondences’ on an algebraic variety which had been developed by Severi and the ‘Italian school’ of algebraic geometers. correspondences on a curve give rise to the transformations of the associated abelian variety (the Jacobian of a curve) into itself. eil showed that these could be described by matrices with l-adic enties l being a prime number different from the charateristic of the base field) and that simple algebraic properties of these matrices were sufficient to prove the results. It is only fitting that, years later, when systematic theories generalizing Weil’s approach were developed, they were entitled ‘Weil cohomology theories’."

D. Reed, Figures of Thought (Routledge, New York, 1995)

Artin L-functions of graph coverings

A. Terras and H. Stark, Artin L-Functions of Graph Coverings, in Contemporary Math., Vol. 290, Dynamical, Spectral, and Arithmetic Zeta Functions - Edited by Michel L. Lapidus, and Machiel van Frankenhuysen (AMS, 2001) 181-195

Artin-Hecke L-functions

A. Weil, "Sur les formules explicites de la théorie des nombres", Izv. Mat. Nauk (ser. Mat.) 36 (1972)

L-functions of automorphic cusp forms

cuspidal L-functions

L-functions of degree 1

Brian Conrey's L-functions page

L-functions of degree 2

Brian Conrey's L-functions page

Dirichlet L-functions

Wikipedia entry on Dirichlet character and L-series

Z. Rudnick, "Zeta functions in arithmetic and their spectral statistics"

L-functions of elliptic curves

Charles Daney's notes, from an accessible explanation of Wiles' proof of Fermat's Last Theorem

global L-functions

global nth symmetric power L-functions

L-functions of holomorphic modular forms

local L-functions

Maass L-function

T. Meurman, "On the order of the Maass L-function on the critical line", Number Theory Vol. 1, Budapest Colloq. Math. Soc., Janos Bolyai 51 (1990)

"Not all L-functions are directly associated to arithmetic or geometric objects. The simplest example of L-functions not of arithmetic/geometric nature are those arising from Maass waveforms for a Riemann surface X uniformized by an arithmetic subgroup... of PGL(2,R). They are pull-backs f(z), to the universal covering space of X, of simultaneous eigenfunctions for the action of the hyperbolic Laplacian and of the Hecke operators on X."  [E. Bombieri]

L-function attached to the modular discriminant

Z. Rudnick, "Zeta functions in arithmetic and their spectral statistics"

L-function attached to eigenfunctions of the Laplacian on the modular domain

Z. Rudnick, "Zeta functions in arithmetic and their spectral statistics"

L-functions of modular forms

Charles Daney's notes, from an accessible explanation of Wiles' proof of Fermat's

motivic L-functions

C. Deninger, "Motivic L-functions and regularized determinants", Proc. Symp. Pure Math. 55 (1) (1994) 707-743.

C. Deninger, "Motivic L-functions and regularized determinants II", from F. Catanese (editor) Arithmetic Geometry, Symp. Math. 37 (Cortand, 1994)

p-adic L-functions

J. Coates and W. Sinnott, "On p-adic L-functions over real quadratic fields", Inventiones Mathematicae 25 (1974) 252-279.

J. Diamond, "On the values of p-adic L-functions at positive integers", Acta Arith. 35 (1979) 223-237.

B. Ferrero and R. Greenberg, "On the behaviour of p-adic L-functions at s = 0", Inventiones Mathematicae 50 (1978) 91-102.

N.M. Katz, "p-adic L-functions via moduli of elliptic curves", Proc. Symp. in Pure Math. 29 (1975) 479-506.

N.M. Katz, "p-adic L-functions for CM fields", Inventiones Mathematicae 49 (1978) 199-297.

N. Koblitz, "A new proof of certain formulas for p-adic L-functions", Duke Mathematical Journal 46 (1979) 455-468.

principal L-functions

quadratic L-functions

K. Soundararajan, "Nonvanishing of quadratic L-functions at s = 1/2", Annals of Mathematics 77 (1963) 387-405.

Rankin-Selberg L-functions

E. Kowalski, P. Michel, J. Vanderkam, "Rankin-Selberg L-functions in the level aspect" (preprint, 2000)

Rankin triple L-functions

T. Watson, "Central value of the Rankin triple L-function for unramified Maass cuspforms" (?)

L-functions from Selberg class S

Brian Conrey's L-functions page

standard L-functions

symmetric power L-functions of GL2

Kim and Shahidi, "Symmetric cube L-functions for GL2 are entire", Annals of Mathematics 150 (1999) 645-662.

triple product L-functions

M. Harris and S. Kudla, "The central value of a triple product L-function", Annals of Mathematics 133 (1991) 605-672.

twisted/elliptic L-functions


archive      tutorial      mystery      new      search      home      contact