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arXiv · 1106.4732

On the pullback of an arithmetic theta function

Stephen Kudla·Tonghai Yang
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Abstract

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.

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Stephen Kudla, Tonghai Yang. 2011-06-23. On the pullback of an arithmetic theta function. https://arxiv.org/abs/1106.4732

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