  In this paper, we consider the relation between the simplest types of
arithmetic theta series, those associated to the cycles on the moduli space
$\Cal C$ of elliptic curves with CM by the ring of integers $\OK$ in an
imaginary quadratic field $\kay$, on the one hand, and those associated to
cycles on the arithmetic surface $\M$ parametrizing 2-dimensional abelian
varieties with an action of the maximal order $O_B$ in an indefinite quaternion
algebra $B$ over $\Q$, on the other. We show that the arithmetic degree of the
pullback to $Cal C$ of the arithmetic theta function of weight 3/2 valued in
$\hat CH^1(\M)$ can be expressed as a linear combination of arithmetic theta
functions of weight 1 for $\Cal C$ and unary theta series. This identity can be
viewed as an arithmetic seesaw identity. In addition, we show that the
arithmetic theta series of weight 1 coincide with the central derivative of
certain incoherent Eisenstein series for SL(2)/Q, generalizing earlier joint
work with M. Rapoport for the case of a prime discriminant.
