论文标题
在Modulo $ p $零模块化表格中，符合Theta系列
On the modulo $p$ zeros of modular forms congruent to theta series
论文作者
论文摘要
对于大于$ 7 $的Prime $ p $，Eisenstein系列重量$ P-1 $具有出色的一致性Properties Modulo $ P $。例如，这意味着其零的$ j $ invariants（已知在间隔$ [0,1728] $中是真实的代数数字），最多是$ p $元素的二次元素，并且是$ p $元素的，并且是一致的Modulo $ p $，对某些the truncest truncest truncunce the truncest thuncunces the the truncest thuncunce the the truncest thuncunce the the truncested超遗传系列。在本文中，我们将重量$ k \ geq 4 $的“ theta模块化形式”介绍为整个模块化组，作为模块化形式，第一个dim $（m_k）$傅立叶系数与某些theta系列相同。我们考虑了Jacobi Theta系列和六边形晶格的Theta系列的这些Theta模块化形式。我们表明，Jacobi Theta系列的theta模块化表格的零j $ invariant是Modulo $ p $，全部在地面上，带有$ p $元素。对于六角形晶格的theta模块化形式，我们表明其零在地面上最多是二次的，带有$ p $元素。此外，我们表明，在这两种情况下，这些零与某些截短的超几何函数的零是一致的。
For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the $j$-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo $p$ all in the ground field with $p$ elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with $p$ elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.
