Average equidistribution of Heegner points associated to the 3-part of the class group of imaginary quadratic fields

Details Average equidistribution of Heegner points associated to the 3-part of the class group of imaginary quadratic fields Average equidistribution of Heegner points associated to the 3-part of the class group of imaginary quadratic fields

2010-05-10

arxiv.org/abs/1005.1458v2

Mathematics

Number Theory

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Scientific paper

We prove that the Heegner points attached to the 3-part of the class group of an imaginary quadratic field $\Q(\sqrt{-d})$ equidistribute in $\mathcal{F} = SL_2(\zed)\backslash \uH$ on average over $d$ as $d \to \infty$. As a result, we obtain a proof of the Davenport-Heilbronn theorem on the mean size of the 3-part of the class group without first passing through cubic fields. We also prove a uniform vertical density of Heegner points associated to the $k$-part of the class group high in the cusp of $\mathcal{F}$, for any odd $k$. This leads to a conjectural negative secondary main term in the mean size of the $k$-part of the class group, refining the prediction of the Cohen-Lenstra heuristic.

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