1549380232-Automorphic_Forms_and_Applications__Sarnak_

280 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L-FUNCTIONS

By Stirling's formula, one has

{ E 00 (z, 1 /2 + i t)dμ1(z) «t,<: (1 + lt1l)-l/2+<: L(7rJ 01ff, 1/ 2 +it).

lxo(l)

For g a primitive cusp form, one uses Watson's formula (4.25),

If xo(l )) g(z)dμ1(z)l

2

_ A(7rJ 01ff 07r 9 , 1/2)

(g,g) - A(sym27r 1 , 1)^2 A(sym^2 7r 9 , 1) ·

The local factors at oo are given by (see [Wats] for instance)

L 00 (sym^2 7r f, s) = Loo(7r f 0 7r f, s)/rR(s),

Loo(1fJ 0 1ff 01fg, s) = IT rR(s ± it1 ± it1 ± itg + Og),

{±}3

if f is a Maass form, and by

Loo(7r f 0 1f f 0 1f 9> s) = IT rR(s+k-1 ± i tg)rR(s+k±itg)rR(s ±itg)rR(s+ 1 ±itg)

±

if f is holomorphic. By Stirling's formula and (2.16), it then follows that

I { g(z)dμ1(z)l^2 «o,g (1 + ltil)-1+"' L(7rJ 0 1ff 0 1fg, 1/2),

Jx 0 ( 1))

I { g(z)dμ1(z)l^2 «c,g k-1+<: L(7rJ 0 1fJ 0 7r 9 ,1/2).
j Xo(l ))
Hence QUE in this case follows from any subconvex bounds for L( 7r 1 07r 1 , 1/2 +it)
and L(7rJ 0 1fJ 0 7r 9 ,1/2) in the t 1 (resp. k 1 ) aspect. Similarly, QUE for other
modular or Shimura curves would follow from an extension of the triple product
identities and from some subconvex bounds for appropriate L-functions.
In the special case of f dihedral, Arithmetic QUE has been proven by Sarnak
(in the holomorphic case) and by Liu/Ye (in the Maass case) [Sa4, LY]. Recall that
a dihedral form, f = f ,p, is a theta series associated -by Hecke and Maass-to a
Grossencharacter 'ljJ on a quadratic field. One has the following factorizations:
L(f, s) = L('l/J, s), L(sym^2 1ff, s) = L('ljJ^2 , s)L(x, s),
L(sym^2 1fJ ® 7r 9 , s) = L(7rJ"' 2 0 7r 9 , s)L(x.7r 9 , s),
where f ,p2 is the theta series corresponding to 'ljJ^2 (and thus has spectral parameter
2it f -resp. weight 2k 1 -), and x is the character corresponding to the quadratic
field. In particular QUE follows in this case from the ScP for Hecke and Rankin-
Selberg L-functions in the spectral aspect (Theorem 4.10).
The above instances of the ScP seem to be hard to prove in general within the
framework of GL 2 analysis. At least one gets some simplifications via the factoriza-
tions
(5.14) L(7rJ 0 1ff, s) = ((s)L(sym^2 7rf, s) ,
L(7r f ® 7r f 0 7f 9, s) = L(7r 9 , s )L(sym^2 7r f ® 7r 9 , s);
in particular the ScP would follow (via the Gelbart/ Jacquet lift) from the more gen-
eral ScP for GL 3 -L-functions and the GL 3 x GLz-Rankin/Selberg L-fonctions. To
achieve this goal, one must probably first develop a good analytic theory of fami-
lies of GL 3 automorphic forms, similar to the GL 2 theory; in particular one needs a
manageable trace type formula. Some analogs of the Petersson/Kuznetsov formula