Department of Mathematics, University of Hawaii, Honolulu, HI 96822-2273, USA.
Proc Natl Acad Sci U S A. 2010 Apr 6;107(14):6169-74. doi: 10.1073/pnas.1001355107. Epub 2010 Mar 22.
A "mock modular form" is the holomorphic part of a harmonic Maass form f. The nonholomorphic part of f is a period integral of its "shadow," a cusp form g. A direct method for relating the coefficients of shadows and mock modular forms is not known. We solve these problems when the shadow is an integer weight newform. Our solution is p-adic, and it relies on our definition of an algebraic "regularized mock modular form." As an application, we consider the modular solution to the cubic "arithmetic-geometric mean."
“拟模形式”是调和马瑟斯形式 f 的解析部分。f 的非解析部分是其“影子”的周期积分，即一个尖形式 g。我们不知道将影子和拟模形式的系数联系起来的直接方法。当影子是整数权新形式时，我们解决了这些问题。我们的解决方案是 p-adic 的，它依赖于我们对代数“正则化拟模形式”的定义。作为应用，我们考虑三次“算术-几何平均”的模形式解。
