1549380232-Automorphic_Forms_and_Applications__Sarnak_

Lecture 5. Some applications of subconvexity

In this last section, we describe several applications of subconvex bounds in

arithmetic and related fields. Although analytic by nature, the (sub)convexity

bound may have deep geometric or arithmetic meaning and applications.

5.1. Subconvexity vs. Riemann/Roch

We consider again the question stated in section 3.3.1, of distinguishing two mod-

ular forms by their first Hecke eigenvalues:

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

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

To fix ideas we consider f E SHq) and g E S~(q') two primitive holomorphic

forms of weight two. Then f ( z)dz and g(z)dz define two holomorphic differentials

on the modular curve X 0 ([q, q']), and picking q = exp(27riz ) the uniformizer at oo,

one has

f(z)dz -g(z)dz = L(>..1(n) - >.. 9 (n))qn-^1 dq = L (>.. 1 (n) ->.. 9 (n))qn-^1 dq

n;;:,l n ;;;,N(f,g)

so that N (!, g) - 1 is the order of vanishing of this differential at oo. By the Rie-

mann/Roch Theorem it follows that

N(f,g) :( degDxoq,q' + 1=2genus(Xo[q,q'])-1

_ - [q, q'J I1v1qq' (^1 + i) (l (l)) [ 'J1+e

6

+ 0 «e q, q

for any c: > 0. Note that at this point we have not used the fact that f and g are
primitive.
On can also approach this question via Rankin-Selberg L-functions (see [GH]
for instance). The basic idea is that the modular forms f and g can be distinguished
by the different analytic properties of L(f 0 g , s) and L(g 0 g , s): the latter has a
pole at s = 1 and the former has no poles. The method amounts to making this
difference explicit at the level of Hecke eigenvalues.

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