UncoveringaNew L-function Andrew R. Booker n March of this year, my student, Ce Bian, plane, with the exception of a simple pole at s = 1. announced the computation of some “degree Moreover, it satisfies a “functional equation”: If 1 3 transcendental L-functions” at a workshop −s=2 γ(s) = ÐR(s) := π Ð (s=2) and Λ(s) = γ(s)ζ(s) at the American Institute of Mathematics (AIM). This article aims to explain some of the then I (1) (s) = (1 − s). motivation behind the workshop and why Bian’s Λ Λ computations are striking. I begin with a brief By manipulating the ζ-function using the tools of background on L-functions and their applications; complex analysis (some of which were discovered for a more thorough introduction, see the survey in the process), one can deduce the famous Prime article by Iwaniec and Sarnak [2]. Number Theorem, that there are asymptotically x ≤ ! 1 about log x primes p x as x . L-functions and the Selberg Class Other L-functions reveal more subtle properties. There are many objects that go by the name of For instance, inserting a multiplicative character L-function, and it is difficult to pin down exactly χ : (Z=qZ)× ! C in the definition of the ζ-function, what one is. In one of his last published papers we get the so-called Dirichlet L-functions, [5], the late Fields medalist A. Selberg tried an Y 1 L(s; χ) = : axiomatic approach, basically by writing down the 1 − χ(p)p−s common properties of the known examples. This p prime resulted in what is generally known as the “Selberg (Here we extend the definition of χ by setting class”. Before discussing the list of axioms, it is χ(p) = 0 when p divides q.) It turns out that these helpful to consider a few concrete examples. also continue to entire functions and satisfy a The simplest and most familiar example of an functional equation similar to (1): If ( L-function is the Riemann ζ-function, R(s) if χ(−1) = 1; γ(s; χ) = Ð and 1 X Y 1 ÐR(s + 1) if χ(−1) = −1 ζ(s) = n−s = ; − −s (s; χ) = γ(s; χ)L(s; χ) n=1 p prime 1 p Λ then which encodes information about the prime num- 1=2−s bers. Using either formula above, one can see that (2) Λ(s; χ) = χq Λ(1 − s; χ), ζ(s) is an analytic function of complex numbers s where χ is a certain constant of absolute value 1 with <(s) > 1. However, as discovered by Riemann, and χ is the conjugate character. The Dirichlet ζ has an analytic continuation to the entire complex L-functions encode information about primes in arithmetic progressions, which is revealed by ma- Andrew R. Booker is senior lecturer of mathematics at the nipulations similar to those for the ζ-function; in University of Bristol. His email address is andrew.booker particular, Dirichlet’s theorem says that the primes @bristol.ac.uk. distribute themselves evenly among the invertible 1Computing arithmetic spectra, March 9–14, 2008. residue classes modulo q. 1088 Notices of the AMS Volume 55, Number 9 Another example arises from arithmetic ge- • L(s) should continue to an entire function, ometry. Given an elliptic curve E of the form with the exception of at most finitely many y 2 = x3 + ax + b with a; b 2 Q, one can consider poles on the line <(s) = 1. Moreover, the the number of solutions, #E(Fp), to its defin- analytically continued function should sat- ing equation reduced modulo prime numbers p. isfy a functional equation of the following Based on the fact that about half of the numbers form: If r mod p are squares, a simple heuristic analysis Y suggests that there are roughly p such solutions, γ(s) = ÐR(s + µj ) and j= and indeed a theorem of Hasse implies the sharp 1 p (s) = γ(s)L(s), bound jp − #E(Fp)j < 2 p. Thus, it is natural to Λ p−#E(Fp) = p for certain complex constants µj , then consider the normalized quantity λ(p) p . (As in the case of the Dirichlet L-functions, there 1=2−s Λ(s) = N Λ(1 − s),¯ are finitely many primes for which the definition where N is a positive integer and is a needs some adjustment, such as those dividing the constant of absolute value 1. (Note that denominators of a and b. The conductor of E is an the above examples are all of this form; in integer N whose prime factorization contains all of particular, (s; χ) = (s;¯ χ).) these exceptional primes.) We can then associate Λ Λ an L-function, called a Hasse-Weil L-function in this It is also expected that all of the functions in the Selberg class satisfy an analogue of the case, given by the product Riemann hypothesis, i.e., all zeros of the completed Y 1 function (s) should have real part 1 . It would L(s; E) = ; Λ 2 − −s + −2s p prime 1 λ(p)p χN (p)p seem sensible to include that as an axiom, but for the fact that not a single instance of the Riemann where χN (p) = 0 if p divides N and 1 otherwise. hypothesis is yet known to be true! Again from the definition and Hasse’s bound we see that L(s; E) is analytic for <(s) > 1. However, the Langlands’ Philosophy Shimura-Taniyama-Weil conjecture, now a theorem Another common feature of the examples above is of Wiles et al., implies that L(s; E) continues to an that they are all generating functions for sequences entire function and satisfies a functional equation: that occur naturally in number theory. In order If to extract the information that they contain, one needs to know the nice analytic properties (i.e., γ(s; E) = R(s) R(s + 1) and Ð Ð analytic continuation and functional equation) that Λ(s; E) = γ(s; E)L(s; E) the generating functions possess. However, as is then apparent from the example of elliptic curves, those properties can be difficult to establish. Fortunately, = 1=2−s Λ(s; E) E N Λ(s; E), we have a source of L-functions with good analytic for a certain E 2 {±1g. As is now well known, the properties, known as automorphic forms or modu- analytic properties of these L-functions lie at the lar forms. The problem is thus reduced to showing heart of Wiles’ proof of Fermat’s Last Theorem. that an L-function of arithmetic interest is equal So what do these examples have in common? to one arising from an automorphic form; this is One difficulty with Selberg’s approach is that it is in fact what Wiles et al. proved in their resolution not obvious which properties should be considered of the Shimura-Taniyama-Weil conjecture. intrinsic and which not, and there is no general The study of automorphic forms is a discipline agreement on that point. A few things are clear, in its own right, called the Langlands program, however: after R. P. Langlands, who was arguably the first to understand the scope for applying them to num- • An L-function should be given by an Euler ber theory. In particular, Langlands predicted the product. In all known cases, it takes the existence of certain “functorial transfers” between form different types of automorphic forms, which may Y 1 be viewed as a (largely conjectural) set of rules L(s) = ; −s governing the expected equalities of L-functions. p prime fp(p ) where fp is a polynomial of a fixed degree Maass Forms r ≥ 1, with the exception of at most finite- After accounting for all of these equalities, it turns ly many primes p, for which the degree out that there are far more automorphic forms, can be smaller than usual. Moreover, the each with an associated L-function, than there product should converge absolutely for s are L-functions that can be properly interpreted with <(s) > 1, and thus define an analytic as generating functions. Thus, there are many function in that region. so-called “transcendental L-functions”, which are October 2008 Notices of the AMS 1089 associated to automorphic forms but not obviously With a few exceptions arising from instances connected to number theory. The most classical of the Langlands functoriality conjectures, no ex- examples are known as Maass forms, after H. Maass, plicit examples of Maass forms are known. In fact, who was the first to construct them. These are the data associated to a typical Maass form are functions f on the hyperbolic upper half plane believed to be transcendental, which explains the H = fz = x + iy 2 C : y > 0g, equipped with terminology above; thus, in a sense one can never 2 2 2 = dx +dy know a typical form exactly, and all constructions the Riemannian metric ds y2 , which are modular, in the sense that of them are approximate and numerical. (This is in contrast to the more classical holomorphic az + b (3) f = f (z) modular forms, which have algebraic data and can cz + d be described by explicit formulas.) a b for all matrices c d in a discrete subgroup Ð of SL(2; R), the group of orientation-preserving Applications isometries of H. That begs the question why number theorists The prototypical case is Ð = SL(2; Z), for which should be interested in these functions at all.