Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and 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). ii

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and 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). ii