Non-Archimedean convolutions of Hilbert modular forms

Abstract

Now let p be a prime number and S a finite set of primes containing p. In this chapter we consider convolutions of Hilbert modular forms and construct their S-adic analogues; they correspond to certain automorphic forms on the group G = GL₂ × GL₂ over a totally real field F and have the form

$$L(s,f,g) = \mathop \Sigma \limits_n C(n,f)C(n,g)N(n)^{ - s} ,$$

(0.1)

where f, g are Hilbert automorphic forms of “holomorphic type” over F, and C(n, f), C(n, g) are their normalized Fourier coefficients (indexed by integral ideals n of the maximal order O _F ⊂ F).