Theta-functions and Hilbert modular forms

Abstract

The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

On a relation between ̃𝑆𝐿₂ cusp forms and cusp forms on tube domains associated to orthogonal groups

“…In [8] and [10] a correspondence is set up between SL2 cusp forms and O(Q) cusp forms (see also [2], [4]). Essentially starting with a Schwartz function <p, which transforms according to a finite dimensional representation of a maximal compact subgroup of SL2 X O(Q), we form the 0 series T^{G, g) with G 6E SL2 and g e O(Q) (where L is a lattice in R* so that £ is a direct summand of L).…”

“…T¿D(G,g)= 2 ßX(g)Gn(G) (1)(2)(3)(4) n<-l where Ai(e? )= 2 ^(s-'a) \ex" with Xn the subset of all lattice points X in {X G R*|ö(Ar, X) = n} n L satisfying Q(\, v) = 0 (where v is a given nonzero isotropic vector in L).…”

“…Although ß* is not an automorphic form relative to O(Q), it is invariant under the discrete group TL n 0(Q)lv], where 0(Q) [v] is the parabolic subgroup of O(Q) stabilizing the line determined by the vector v. The important point is that the validity of (1)(2)(3)(4) depends on the "cuspidal" behavior of we(g~')<P£, in the Weil representation. That is, if 7re(g~')<pD is a cusp form in the Weil representation (which means that 7re(g"')<pD satisfies the first and second Cusp Vanishing Theorems in [II] relative to the parabolic 0(Q)lc]) then the formula in (1)(2)(3)(4) holds. Thus in §2 we have set up a rather elaborate technical machinery to deduce (1)(2)(3)(4).…”

“…That is, if 7re(g~')<pD is a cusp form in the Weil representation (which means that 7re(g"')<pD satisfies the first and second Cusp Vanishing Theorems in [II] relative to the parabolic 0(Q)lc]) then the formula in (1)(2)(3)(4) holds. Thus in §2 we have set up a rather elaborate technical machinery to deduce (1)(2)(3)(4). We have adopted this point of view in order to prove a more general formula than in [14] and [9] and to show the dependence of the formula on the cuspidal properties of the Weil representation.…”

“…Using (1)(2)(3)(4) we deduce that (when Q has signature (k -2, 2)) <Tl(,g)\f()>= 2 af(\n\)ß;(g)\n\1-*.…”