1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

One chooses V a smooth non-negative function compactly supported on

[1/2, 1 ], and N? 1, and one evaluates the sum

Ev(! x g,N) := 'L>.1(n).A 9 (n)V(~).

n;;,l

By the inverse Mellin transform,

1 j L(f x g, s) • s

Ev(! x g,N) = 2 7ri ((qq')( 2 s) V(s)N ds,

(3)

where L(f x g, s) is defined as in (2.14). Shifting the contour to ales= 1 /2, we hit
a pole at s = 1 only if f = g; hence for f -j. g one has
1 j L(f x g, s) •
Ev(! x g,N) = 2 7ri ((qq')( 2 s) V(s)ds,
(1/2)

while for f = g one has

L(g x g, s) • 8 1 J L(g x g, s ) •

Ev(g x g, N) = ress=l (Cq')( 2 s) V (s)N + 2 7ri (Cq')( 2 s) V(s)ds.

(1/2)

Now, it follows, from (2 .14) and (2.16), that

L(g x g,s). 8 • L(g x g,s) i-c.

Ress=l (Cq')( 2 s) V(s)N = V(l )Nress=l (Cq')( 2 s) »c q V(l )N.

For simplicity, assume that q and q' are squarefree; in this case the conductor of

L(f ® g, s) equals [q, q' J^2 and the conductor of L(g ® g, s ) equals q'^2 ; hence, it

follows from the convexity bounds that

(5 .1) Ev(! x g , N) = Oc,v((qq'y N1;2([q, q'J1f2)

an d" LtV ( g x g , N) = v· ( 1 )N ress-1 L(g ( ')( x g, ) s) +^0 c v ( q 'cN^1 ;2 q i1/2).

• ( q 2s '
  Now if we take N = N(f,g) we have
  Ev(! x g, N) = Ev(g x g, N),
  and it follows from (5.1) that N(f,g) «c (qq^1 )£[q,q^1 ]. In particular the above
  method retrieves (essentially) the Riemann/ Roch bound. More importantly any
  subconvex bound for L(f ® g, s) and L(g ® g, s) in the level aspects would improve
  the Riemann/ Roch bound; and ultimately, GLH would give N(f,g) «c (qq')c for
  any c: > 0.
  Remark 5.1. When g is fixed (so that q' = 0(1)), the Subconvexity Problem for
  L(f ® g) has been solved in [KMV2] and yields
  N(f,g) «q' ql-1/41.

Remark 5.2. Even without subconvexity, the approach via L-functions is inter-
esting for distinguishing modular forms by means of their first Hecke eigenvalues
when Riemann/ Roch is not available, as is the case with Maass forms; moreover,
it can be extended to automorphic forms on G Ld, for which there is no underlying
Shimura variety.