TY - JOUR

T1 - Maximal abelian extension of X(p) unramified outside cusps

AU - Yamazaki, Takao

AU - Yang, Yifan

N1 - Funding Information:
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.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - Let p be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of X(p) over Q is given by the Shimura covering X2(p) → X(p) , that is, a unique subcovering of X1(p) → X(p) of degree Np: = (p- 1) / gcd (p- 1 , 12). In this short paper, we show that a geometrically maximal abelian covering X2′(p)→X0(p) of X(p) over Q unramified outside cusps is cyclic of degree 2 Np. The main ingredient for the construction of X2′(p) is the generalized Dedekind eta functions.

AB - Let p be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of X(p) over Q is given by the Shimura covering X2(p) → X(p) , that is, a unique subcovering of X1(p) → X(p) of degree Np: = (p- 1) / gcd (p- 1 , 12). In this short paper, we show that a geometrically maximal abelian covering X2′(p)→X0(p) of X(p) over Q unramified outside cusps is cyclic of degree 2 Np. The main ingredient for the construction of X2′(p) is the generalized Dedekind eta functions.

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U2 - 10.1007/s00229-019-01136-7

DO - 10.1007/s00229-019-01136-7

M3 - Article

AN - SCOPUS:85069725943

VL - 162

SP - 441

EP - 455

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3-4

ER -