On the pullback of an arithmetic theta function

TL;DR

This paper establishes a relationship between arithmetic theta functions on moduli spaces of elliptic curves with CM and abelian varieties with quaternionic multiplication, expressing pullbacks as linear combinations of simpler theta series.

Contribution

It introduces an arithmetic seesaw identity connecting theta functions of different weights and relates them to derivatives of incoherent Eisenstein series, extending previous work.

Findings

Arithmetic degree of pullback expressed as a linear combination of theta functions.

Identifies the arithmetic theta series with derivatives of Eisenstein series.

Generalizes earlier results to broader settings with quaternionic multiplication.

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.