In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:

If $\phi_n$'s are orthonormal Hecke-Maass eigenforms in some arithmetic congruence (compact/noncompact) manifold $M=\Gamma\backslash G$, i.e. normalized eigenforms of Laplacian and Hecke operators in corresponding manifold, then for $I\subset\mathbb{N}$, $$\left(\int_M\prod_{n\in I}\phi_ndvol_M\right)^2=C\frac{L\left(\frac12,\bigotimes\limits_{n\in I}\phi_n\right)}{\prod\limits_{n\in I}L(1,sym^2\phi_n)},$$ for some constant $C$.


1) Is the above (or some modified version) true?

2) If not, why is this only true for THREE eigenforms?

Thanks. Any help or reference is highly appreciated.


1 Answer 1


The reason it works for three (but not other numbers) is the uniqueness of trilinear functionals. Let $G = \mathrm{GL}_2(F)$, $F$ a local field and let $\pi_i$ ($1\leq i\leq 3$) be three irreducible admissible representations. Then there is at most one $G$-invariant functional on $\pi_1 \otimes \pi_2 \otimes \pi_3$ (this is mostly due to D. Prasad). Now let $\pi_i$ instead be global automorphic representations, and $G$ the adelic group. Then one such functional is given by the integral on the LHS of Watson's formula, so by the uniqueness it must factor as a product of local functionals, and this "explains" why the integral is Eulerian.

For the general question of when one should expect such integral representations for L-functions see the papers of Sakellaridis and Sakellaridis–Venkatesh.

  • $\begingroup$ Thank you very much Professor Lior. Can you kindly specify which result of Sakellaridis-Venkatesh you are referring to, as I am being unable to find it? $\endgroup$ Apr 2, 2014 at 20:54
  • 1
    $\begingroup$ References added. $\endgroup$ Apr 3, 2014 at 5:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.