Ogg 和 Ligozat 函数的继续属性 (Further properties of a function of Ogg and Ligozat)

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function w_N(g) = w_N(f) - 1/w_N(f) on the modular curve X_0(N), where f is a certain modular unit on the Nebentypus cover X_\chi(N) introduced by Ogg and Ligozat for N prime congruent to 1 (mod 4) and w_N is the Fricke involution. These correspond to levels N = 5, 13, where the genus of X_0(N) is zero. In this paper we produce some analogs of these identities for each w_N(g) such that X_0(N) has genus 1, 2, and also for each h = g + w_N(g) such that the Atkin-Lehner quotient X_0+(N) has genus 1, 2. We also found that if n is the degree of the field of definition F of the non-trivial zeros of the latter, then the degree of the normal closure of F over Q is the n-th solution of Singmaster's Problem.

翻译：Ramanujan(N) 的某些特性可以用模块曲线X_0(N) 上的 w_N(g) = w_N(f) - 1/w_N(f) / 1/w_N(f) 上的合理函数简洁表示。在模块曲线X_0(N) 上, f 是 Ogg 和 Ligozat 引入的 Nebendypus 封面 X ⁇ chi(N) 上的一个特定模块单位, 由 Ogg 和 Ligozat 引入的 N 质同为 1 (modd4), w_N 是 Fricke 。这些特性相当于 N= 5, 13 13, X_ 0(N) 的genus为零。在本文中, 我们为每个 w_ N(g) 生成了这些特性的一些类比喻, 例如 X_ 0(N) 有 1 genus, 2, 并且对于每个 h = g + w w_ w_ (N) + (N) 。因此, Atkinnerneral plast master.

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Ogg is a free, open container format maintained by the http://Xiph.Org Foundation. The creators of the Ogg format state that it is unrestricted by software patents and is designed to provide for efficientstreaming and manipulation of high quality digital multimedia.