Singular moduli for real quadratic fields: A rigid analytic approach
Singular moduli for real quadratic fields: A rigid analytic approach
复制标题
实二次场的奇异模量：严格的分析方法
DOI：
10.1215/00127094-2020-0035
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发表时间：
2020
影响因子：
2.5
通讯作者：
Jan Vonk
中科院分区：
文献类型：
--
作者：
H. Darmon;Jan Vonk
刚性亚纯上圈是一类离散群Γ：= SL 2（Z[1/p]）的第一上同调，其值在p进上半平面Hp：= P1（Cp）− P1（Qp）上的非零刚性亚纯函数的乘法群中。这样的类可以在Hp中的真实的二次无理数处求值，这被称为“RM点”。刚性亚纯上环可以被设想为Borcherds奇异theta lis的真实的二次对应：它们的零点和极点包含在RM点的r-轨道的nite并中，并且它们的RM值被推测位于真实的二次elds的环类elds中。这些RM值与SL 2（Z）\H上的模函数的CM值有着惊人的相似之处：特别是它们似乎就像Gross和Zagier描述的经典奇异模的模的因子一样。一个快速算法计算刚性亚纯上循环高p-adic精度导致令人信服的数值证据的代数性和因式分解的奇异模的真实的二次ELD。
A rigid meromorphic cocycle is a class in the rst cohomology of the discrete group Γ := SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid meromorphic functions on the p-adic upper half plane Hp := P1(Cp) − P1(Qp). Such a class can be evaluated at the real quadratic irrationalities in Hp, which are referred to as “RM points”. Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lis: their zeroes and poles are contained in a nite union of Γ-orbits of RM points, and their RM values are conjectured to lie in ring class elds of real quadratic elds. ese RM values enjoy striking parallels with the CM values of modular functions on SL2(Z)\H: in particular they seem to factor just like the dierences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p-adic accuracy leads to convincing numerical evidence for the algebraicity and factorisation of the resulting singular moduli for real quadratic elds.