| Let V (lambda) denote a local system of weight lambda on X = A2,n( C ), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with fixed n-level structure. The inner cohomology of X with coefficients in V (lambda), H3! (X, V (lambda)), has a Hodge filtration of weight 3. Each term of this Hodge filtration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3! (X, V (lambda)) denoted by H3Ends (X, V (lambda)).;First of all we show that there is a non-zero subspace of H3Ends (X, V (lambda)) denoted by Vtheta ( K), where K is an open compact subgroup of GSp(4, A ), such that elements of Vtheta( K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) x GL(2)/ Gm . This means that there is a non-zero part of H3Ends (X, V (lambda)) which is endoscopic.;Secondly, we consider the local theta correspondence and find an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4, F) over a non-archimedean local field F. Therefore, we can present an explicit way for finding a basis for Vtheta(K) for a fixed level structure K..;There is a part of the Hodge structure that only contributes in H3,0! (X, V (lambda)) ⊕ H3,0! (X, V (lambda)). This part is endoscopic and coming from the Yoshida lift from O(4).;Finally, in the case X = A2, if eendo(A2, V (lambda)) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge filtration), then we have eendoA2, Vl =-sl1+l2+4S l1-l2+2 Ll2+1, 1 where lambda = (lambda1, lambda2) and lambda is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). |
