Siegel's modular forms and the arithmetic of quadratic forms

Abstract

Recently many authors utilized indefinite quadratic forms and Weil's representations to construct automorphic forms with Euler products on metaplectic groups or on orthogonal groups. For example, Shintani [27] employed indefinite quadratic forms of signature (2, 1) and associated Weil's representations to construct automorphic forms on the metaplectic group of genus 1, answering to a question raised by Shimura [26]. His results were modulated and generalized by Niwa [17], Gelbart [-8], Oda [18] and Rallis [20]. Howe [10] has given conjectures in most general cases. In this paper, we shall be concerned with definite quadratic forms of four variables and associated Weil's representations to construct Siegel's modular forms of genus 2. More precisely, we shall construct a Siegel modular form of genus 2, which is an eigenfunction of Hecke operators, from a pair of automorphic forms on a definite quaternion algebra over Q or from an automorphic form on a totally definite quaternion algebra over a real quadratic field. To avoid excessive overlaps with the text, we shall explain our result in the simplest case. Let D be a definite quaternion algebra over Q of discriminant d 2 and R be a maximal order of D. Let D~ denote the adelization of D • For a rational prime p, we put R p = R | p and let H denote the Hamilton quaternion algebra. We set K = I I R~ x H • Let q01 and ~02 be complex valued functions on p D~ which are left D • and right K-invariant. For every rational prime p.~d, we can define the action of Hecke operator T'(p) on qh (i= 1, 2) in the same manner as in Well [33]. We assume that T'(p)~0i=2~(p)qh, 2/(p)EC(i=I, 2) if n p,t/d. Let D~ = U D • y~K be a double coset decomposition of D~, where H is i= l the class number of D. We can assume that the reduced norm of Yi is 1 (1 N i < H). For l<i<H, we put ei=lD• For l < i < H and I<j<-H, we p define a lattice L~j in D by Lij=DnYi(1-lRv)yf 1. For x~D, we put N(x)=xx* p