| Summary: | 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.
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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).
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