TY - JOUR
T1 - The modularity of elliptic curves over all but finitely many totally real fields of degree 5
AU - Ishitsuka, Yasuhiro
AU - Ito, Tetsushi
AU - Yoshikawa, Sho
N1 - Funding Information:
The work of Y.I. was supported by JSPS KAKENHI Grant Number 21K13773. The work of T.I. was supported by JSPS KAKENHI Grant Number 21H00973. The work of Y.I. and T.I. was supported by JSPS KAKENHI Grant Number 21K18577. The work of S.Y. was supported by JSPS KAKENHI Grant Number 19K14514. The authors of this paper would like to thank Masao Oi and Masataka Chida for invaluable discussions on modular forms and the Mordell–Weil groups. The authors would also like to thank the referees for reading the first draft of this paper carefully and giving helpful comments and suggestions. The results of this paper cannot be obtained without use of the L-functions and modular forms database (LMFDB) [31]. The authors would like to thank all the people working for this wonderful database.
Funding Information:
The work of Y.I. was supported by JSPS KAKENHI Grant Number 21K13773. The work of T.I. was supported by JSPS KAKENHI Grant Number 21H00973. The work of Y.I. and T.I. was supported by JSPS KAKENHI Grant Number 21K18577. The work of S.Y. was supported by JSPS KAKENHI Grant Number 19K14514. The authors of this paper would like to thank Masao Oi and Masataka Chida for invaluable discussions on modular forms and the Mordell–Weil groups. The authors would also like to thank the referees for reading the first draft of this paper carefully and giving helpful comments and suggestions. The results of this paper cannot be obtained without use of the L-functions and modular forms database (LMFDB) []. The authors would like to thank all the people working for this wonderful database.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/12
Y1 - 2022/12
N2 - We study the finiteness of low degree points on certain modular curves and their Atkin–Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree 5. On the way, we prove a criterion for the finiteness of rational points of degree 5 on a curve of large genus over a number field using the results of Abramovich–Harris and Faltings on subvarieties of Jacobians.
AB - We study the finiteness of low degree points on certain modular curves and their Atkin–Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree 5. On the way, we prove a criterion for the finiteness of rational points of degree 5 on a curve of large genus over a number field using the results of Abramovich–Harris and Faltings on subvarieties of Jacobians.
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U2 - 10.1007/s40993-022-00383-0
DO - 10.1007/s40993-022-00383-0
M3 - Article
AN - SCOPUS:85139876807
VL - 8
JO - Research in Number Theory
JF - Research in Number Theory
SN - 2363-9555
IS - 4
M1 - 82
ER -