Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for the distribution of the zeros near s = 1 2 . In this note we carry this out after introducing some basic invariants associated to a family.
Definition of families and ConjecturesThe zoo of automorphic cusp forms π on G = GL n over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse-Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)'s in families in which the π's have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises.For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter. Other examples are the GL 2 subconvexity results which are proved by deforming the given form in a family (see [60] and [85] for accounts). In the analogous function field setting the notion of a family of zeta functions is well defined, coming from the notion of a family of varieties defined over a base. Here too the power of deforming in a family in order to understand individual members is amply demonstrated in the work of Deligne [30]. In the number field setting there is no formal definition of a family F of L-functions.Our aim is to give a working definition for the formation of a family which will correspond to parametrized subsets of A(G), the set of isobaric automorphic representations on G(A). As far as we can tell these include almost all families that have been studied. For the most part our families can be investigated using the trace formula, monodromy groups in arithmetic geometry and the geometry of numbers, and these lead to a determination of the distribution of the zeros near s = 1 2 of members of the family. For the high zeros of a given Λ(s, π), it was shown in [98] that the local scaled spacing statistics follows the universal GUE
