Congruences for Certain Theta Series
Congruences for Certain Theta Series
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某些 Theta 级数的同余式
DOI：
10.1006/jnth.1998.2234
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发表时间：
1998
影响因子：
0.7
通讯作者：
P. Tiep
中科院分区：
文献类型：
--
作者：
N. Dummigan;P. Tiep
一个偶数、整数、正定格的θ级数是一个生成函数，它记录了格的每个壳层中的点数。上半平面上的这个函数，以这样一种特殊的方式出现，满足成为特定类型的模形式的非常强的条件。如果格是自对偶的，则它具有第一级，并且它的权重是格的秩的一半（我们假设是偶数）。对于某些秩相对较小的格，包括E8格和Leech格，有一个相当明确的公式来计算θ级数的系数，但一般来说它们是相当神秘的。在本文中，我们研究他们的可分性。在每一种情况下，我们考虑，格具有高度的对称性，我们使用轨道来证明，所有的系数的θ系列是整除的一些数字N，这就是说，θ系列是全等1（模N）作为一个q-展开。现在模形式的同余理论表明，模形式的权严格限制了它的q-展开式可以同余为1的模N的可能性。模形式的例子，满足所有的同余，其重量允许是爱森斯坦系列。具有相同性质的格包括E8格、Leech格、
The theta series of an even, integral, positive-definite lattice is a generating function which records the number of points in each shell of the lattice. This function on the upper half-plane, arising in such a special manner, satisfies the very strong condition of being a modular form of a specific type. It has level one if the lattice is self-dual, and its weight is half the rank of the lattice (which we assume to be even). For certain lattices of relatively small rank, including the E8 lattice and the Leech lattice, there is a reasonably explicit formula for the coefficients of the theta series, but in general they are fairly mysterious. In this paper we investigate their divisibility. In each case we consider, the lattice possesses a high degree of symmetry and we use orbits to prove that all the coefficients of the theta series are divisible by some number N, which is to say that the theta series is congruent to 1 (mod N) as a q-expansion. Now the theory of congruences of modular forms shows that the weight of a modular form severely restricts the possible N modulo which its q-expansion may be congruent to 1. Examples of modular forms satisfying all the congruences which their weights allow are the Eisenstein series. Lattices whose theta series enjoy the same property include the E8 lattice, the Leech