Maximal abelian extension of $X_0(p)$ unramified outside cusps

Takao Yamazaki; Yifan Yang

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arXiv · 1901.06564

Maximal abelian extension of $X_0(p)$ unramified outside cusps

Takao Yamazaki·Yifan Yang

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Abstract

Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb Q$ is given by the Shimura covering $X_2(p) \to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ of degree $N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \to X_0(p)$ of $X_0(p)$ over $\mathbb Q$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2'(p)$ is the generalized Dedelind eta functions.

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Takao Yamazaki, Yifan Yang. 2019-01-19. Maximal abelian extension of $X_0(p)$ unramified outside cusps. https://arxiv.org/abs/1901.06564

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