1549380232-Automorphic_Forms_and_Applications__Sarnak_

Lecture 5. Converse theorems

Let us return first to Hecke. Recall that to a modular form

00

f(T) = L ane2"inr

n-l

for say SL 2 (Z) Hecke attached an L function L(s, f) and they were related via the
Mellin transform

A(s, f) = (27r)-sr(s)L(s, f) = fo

00

f(iy)y^8 dxy

and derived the functional equation for L(s, f) from the modular transformation
law for f(T) under the transformation T ~ -l/T. In his fundamental paper [33]
he inverted this process by taking a Dirichlet series

00

D(s) = "'an

L ns

n=l

and assuming that it converged in a half plane, had an entire continuation to a
function of finite order, and satisfied the same functional equation as the L-function
of a modular form of weight k, then he could actually reconstruct a modular form
from D(s) by Mellin inversion

and obtain the modular transformation law for j(T) under T ~ -l/T from the

functional equation for D(s) under s ~ k - s. This is Hecke's Converse Theorem.
In this Lecture we will present some analogues of Hecke's theorem in the context
of L-functions for GLn. Surprisingly, the technique is exactly the same as Hecke's,
i.e., inverting the integral representation. This was first done in the context of
automorphic representation for GL 2 by Jacquet and Langlands [39] and then ex-
tended and significantly strengthened for GL 3 by Jacquet, Piatetski-Shapiro, and
Shalika [40]. For a more extensive bibliography and history, see [13].
This section is taken mainly from our survey [13]. Further details can be found
in [9, 12].

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