1549380232-Automorphic_Forms_and_Applications__Sarnak_

232 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

Question. Given f and g two distinct primitive holomorphic forms, what is the small-

est possible n = N(f, g)for which >..1(n) =f. >.. 9 (n)?

Serre [Ser2] observed that under GRH, one has N(f,g) « logA(Q 1 Q 9 ) for

some absolute constant A(= 4), and we will discuss some weaker unconditional
bounds for N(f, g) in the last lecture. The following result of Duke and Kowalski
[DK] is the analog of Thm. 3.6:
Theorem 3. 7. Let sg (:::;; Q) be the set of primitive holomorphic forms with trivial
nebentypus, weight 2 and level q :::;; Q. Then there exists an absolute constant B > 0
such thatfor any A> 1, the number of pairs (f,g) E sg(:::;; Q)^2 such that N(f,g) ~
log(Q 1 Q 9 )A is bounded by
[Sf(:::; Q)[3/2+B/A_

In particular, if A is sufficiently large, the probability that two forms are not distin-
guished by their first log(Q 1 Q 9 )A Hecke eigenvalues goes to 0 as Q ___, +oo.

The proof of this statement is similar to the previous one; one uses Theorem

3.5 for the automorphic representations 'fff, f E sg(:::;; Q). However, the fact that

one does not have (unconditionally) a good lower bound for

L [>..1(P)l

2

nEN

n~N

creates some difficulty. The latter is solved with the help of the identity

AJ(P)^2 - AJ(P^2 ) = 1,

which shows that at least one of the two quantities >.. 1 (p )^2 or >.. 1 (p^2 ) is large. Using

this idea requires a further modification of Linnik's original method, for which we

refer to [DK], and in particular, another application of Thm 3.5, but this time for

the family of symmetric squares { sym^2 7r f, f E Sg(:::;; Q)}.

3.3.2. Duke's bound for [Sf (q, x)I

Another recent striking application of the large sieve inequalities is to the problem

of estimating the size of the set of primitive holomorphic forms of weight one (of

some given nebentypus). There is no Selberg trace nor Petersson's type formula on

the space generated by these forms, hence apparently no manageable formula for

its dimension. However, an application of the trace formula to the whole space of

automorphic forms of weight 1 yields a generic bound (Weyl's law),

(3.12) [Sf(q, x)[ « vol(Xo(q)).

With a more careful choice of the test function this bound can be improved by a
factor log q (Sarnak) and this seems to be the best one can do without exploiting
further the specific (arithmetic) nature of the forms of weight one.
On the other hand, Deligne and Serre [DS] attached to each f E Sf (q, x)
an irreducible 2-dimensional Galois representation PJ : Gal(Q/Q) 1-+ GL 2 (C),
unramified outside the primes dividing q and satisfying for any prime p Jq,
tr(p1(Frobp)) = >..1(P), det(PJ(Frobp)) = x(p).
In particular (since the image of p 1 is finite), f satisfies the Ramanujan-Petersson
conjecture: [>..1(P)[ :::;; 2. The form f can then be further classified according to
the image of PJ via the projection GL 2 (C) 1-+ PGL 2 (C): the latter is isomorphic