On the pullback of an arithmetic theta function

Abstract

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series \({\widehat{\phi}_{\mathcal C}(\tau)}\) defined using arithmetic 0-cycles on the moduli space \({\mathcal C}\) of elliptic curves with CM by the ring of integers \({O_{\kappa}}\) of an imaginary quadratic field. The second such series \({\widehat{\phi}_{\mathcal M}(\tau)}\) has weight 3/2 and takes values in the arithmetic Chow group \({\widehat{{\rm CH}}^1(\mathcal{M})}\) of the arithmetic surface associated to an indefinite quaternion algebra \({B/\mathbb{Q}}\). For an embedding \({O_\kappa \rightarrow O_B}\), a maximal order in B, and a two sided O B -ideal Λ, there is a morphism \({j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}\) and a pullback \({j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}\). Our main result is an expression for the pullback \({j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}\) as a linear combination of products of \({\widehat{\phi}_{\mathcal C}(\tau)}\)’s and classical weight \({\frac{1}{2}}\) theta series.