论文标题
$ p $ - 亚种奇异模量的Quaternionic构造
A quaternionic construction of $p$-adic singular moduli
论文作者
论文摘要
Darmon和Vonk引入了刚性的Meromorormormorphic Cocycles，作为一种猜想的$ p $ - addic-adic扩展，将单数模量理论延伸至真正的二次基础场。它们是$ \ mathrm {sl} _2（\ mathbb {z} [1/p]）$的某些同胞类别，可以在实际二次不合理性上进行评估，因此所获得的值猜测为基础场的代数扩展。在本文中，我们介绍了类似的共同体casses结构，其中$ \ mathrm {sl} _2（\ mathbb {z} [1/p]）$被不确定的Quaternion代数中的订单代替，而不是完全实物数字$ f $。这些Quaternionic的共同体学类可以在几乎完全复杂的扩展名中的元素上进行评估，$ f $的$ k $ $ f $，我们猜想相应的值位于$ k $的代数扩展中。我们还报告了该代数猜想的大量数值证据。
Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions $K$ of $F$, and we conjecture that the corresponding values lie in algebraic extensions of $K$. We also report on extensive numerical evidence for this algebraicity conjecture.
