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

  • Takao YamazakiEmail author
  • Yifan Yang
Article

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 \(X_2(p) \rightarrow X_0(p)\), that is, a unique subcovering of \(X_1(p) \rightarrow X_0(p)\) of degree \(N_p := (p-1)/\gcd (p-1, 12)\). In this short paper, we show that a geometrically maximal abelian covering \(X_2'(p) \rightarrow X_0(p)\) 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 Dedekind eta functions.

Mathematics Subject Classification

11G18 (11F03 · 11G45 · 14G35 ) 

Notes

Acknowledgements

The first author would like to thank Masataka Chida and Fu-Tsun Wei for fruitful discussion. He is partially supported by JSPS KAKENHI Grant (18K03232). The second author was partially supported by Grant 106-2115-M-002-009-MY3 of the Ministry of Science and Technology, Republic of China (Taiwan). The authors would like to thank the anonymous referee for the detailed comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversityAobaJapan
  2. 2.Department of MathematicsNational Taiwan University and National Center for Theoretical SciencesTaipeiTaiwan

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