The theory of

**L-functions**has become a very substantial, and still largely conjectural, part of contemporary number theory. In it, broad generalisations of the Riemann Zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.

Table of contents |

2 Conjectural information 3 The example of the Birch and Swinnerton-Dyer conjecture 4 Rise of general theory |

### L-functions

### Conjectural information

- location of zeroes and poles;
- functional equation, with respect to some vertical line Re (
*s*) = constant; - interesting values at integer values.

### The example of the Birch and Swinnerton-Dyer conjecture

One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s. It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of an elliptic curve over the rational numbers: i.e. the number of free generators of its group of rational points.

On the basis of numerical evidence gathered in one of the most important early applications of electronic computing, it appeared that the rank was in all likelihood predicted by the order of the zero of the L-function associated to E by a recipe of Hasse and Weil for its L-series, at a specified point. At that time even the analytic continuation to the point wasn't established.

This then was something like a paradigm example of the nascent theory of L-functions. The details remain highly technical, and the Birch and Swinnerton-Dyer conjecture is now a prize problem in its full generality, though important cases have been proved. In summary:

- initial work concentrated on the case (
*complex multiplication*) where the analytic continuation could be proved; - this connected (via results of Deuring) the theory with earlier work of Hecke on his theory of L-functions constructed with characters more general than Dirichlet's;
- the exact value in the case of rank 0 needed accounting for, and turned out to involve invariants of E studied by Cassels, Tate, Shafarevich and others;
- the exact constant in the functional equation was conjecturally linked to the parity of the rank.

### Rise of general theory

This development preceded Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin's L-functions, which, like Hecke's, were defined several decades earlier.

Gradually it became clearer in what sense the Hasse-Weil construction might be made to work to provide valid L-functions, in the analytic sense: there should be some input from analysis, which meant *automorphic* analysis. The general case now unifies at a conceptual level a number of different research programmes.

Some relevant further links: