{"title":"论与θ级数全等的模数形式的模数p零点","authors":"Berend Ringeling","doi":"10.1016/j.jnt.2024.03.019","DOIUrl":null,"url":null,"abstract":"<div><p>For a prime <em>p</em> larger than 7, the Eisenstein series of weight <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> has some remarkable congruence properties modulo <em>p</em>. Those imply, for example, that the <em>j</em>-invariants of its zeros (which are known to be real algebraic numbers in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1728</mn><mo>]</mo></math></span>), are at most quadratic over the field with <em>p</em> elements and are congruent modulo <em>p</em> to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span> for the full modular group as the modular forms for which the first <span><math><mi>dim</mi><mo></mo><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> 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 <em>j</em>-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo <em>p</em> all in the ground field with <em>p</em> elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with <em>p</em> elements. 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Those imply, for example, that the <em>j</em>-invariants of its zeros (which are known to be real algebraic numbers in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1728</mn><mo>]</mo></math></span>), are at most quadratic over the field with <em>p</em> elements and are congruent modulo <em>p</em> to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span> for the full modular group as the modular forms for which the first <span><math><mi>dim</mi><mo></mo><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> 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. 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引用次数: 0
摘要
对于大于 7 的素数 p,权重为 p-1 的爱森斯坦数列具有一些显著的同调性质,例如,这些性质意味着其零点(已知为区间 [0,1728] 中的实代数数)的 j 不变式在具有 p 元素的域上最多是二次,并且与某个截断超几何数列的零点同调。在本文中,我们引入了权重 k≥4 的全模态群的 "θ模态",即第一个 dim(Mk) 傅里叶系数与某些θ级数相同的模态。我们考虑了雅可比θ级数和六方格的θ级数的这些θ模形式。我们证明,雅可比θ级数的θ模形式零点的 j 不变性都是在具有 p 元素的基域中模数为 p 的。对于六边形网格的θ模形式,我们证明其零点在有 p 个元素的基域中最多是二次。此外,我们还证明了这两种情况下的这些零点都与某些截断超几何函数的零点相等。
On the modulo p zeros of modular forms congruent to theta series
For a prime p larger than 7, the Eisenstein series of weight 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 ), 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 for the full modular group as the modular forms for which the first 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.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
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