志村曲線在西格爾模三維流型上的映射

Quaternionic loci in Siegel’s modular threefold

林易萱(Yi-Hsuan Lin)

指導教授 ： 楊一帆

國立交通大學/理學院/應用數學系暨研究所/博士(2016年)

摘要

在這篇論文中，我們首先計算在西格爾模三維流型上有幾個不同的志村曲線的映射。接著我們用特定志村曲線的Hauptmodul來參數化表示這些虧格為０的像。

關鍵字

西格爾模三維流型 ； 志村曲線

並列摘要

Let O be a maximal order in an indefinite quaternion algebra of discriminant D over Q and Q_D be the set of points in Siegel’s modular threefold A_2 whose corresponding abelian surfaces have quaternionic multiplication by O. In this thesis, we first determine the number of irreducibe components in Q_D and then for each irreducible component of genus 0, we will find a parameterization in terms of the Hauptmodul of a certain Shimura curve associated to O.

並列關鍵字

QM ； abelian surface ； Shimura curve ； Siegel's modular threefold

參考文獻

[2] Srinath Baba and H°akan Granath. Genus 2 curves with quaternionic multiplication. Canad. J. Math., 60(4):734–757, 2008.

[7] Richard E. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math., 132(3):491–562, 1998.

[8] Richard E. Borcherds. Reflection groups of Lorentzian lattices. Duke Math. J., 104(2):319–366, 2000.

[9] Jan H. Bruinier. Borcherds products on O(2, l) and Chern classes of Heegner divisors, volume 1780 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.

[11] M. Eichler. U¨ ber die Einheiten der Divisions algebren. Math. Ann., 114(1):635–654, 1937.