Mordell-Weil groups of the Jacobian of the 5-th Fermat curve
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- by Pavlos Tzermias PDF
- Proc. Amer. Math. Soc. 125 (1997), 663-668 Request permission
Abstract:
Let $J_{5}$ denote the Jacobian of the Fermat curve of exponent 5 and let $K=Q(\zeta _{5})$. We compute the groups $J_{5}(K)$, $J_{5}(K^{+})$, $J_{5}(Q)$, where $K^{+}$ is the unique quadratic subfield of $K$. As an application, we present a new proof that there are no $K$-rational points on the 5-th Fermat curve, except the so called “points at infinity".References
- Robert F. Coleman, Torsion points on abelian étale coverings of $\textbf {P}^1-\{0,1,\infty \}$, Trans. Amer. Math. Soc. 311 (1989), no. 1, 185–208. MR 974774, DOI 10.1090/S0002-9947-1989-0974774-5
- D. K. Faddeev, The group of divisor classes on some algebraic curves, Soviet Math. Dokl. 2 (1961), 67–69. MR 0130869
- Ralph Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1980/81), no. 3, 345–359. MR 607375
- Benedict H. Gross and David E. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), no. 3, 201–224. MR 491708, DOI 10.1007/BF01403161
- Serge Lang, Introduction to algebraic and abelian functions, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1972. MR 0327780
- Chong-Hai Lim, The geometry of the Jacobian of the Fermat curve of exponent five, J. Number Theory 41 (1992), no. 1, 102–115. MR 1161149, DOI 10.1016/0022-314X(92)90087-6
- David E. Rohrlich, Points at infinity on the Fermat curves, Invent. Math. 39 (1977), no. 2, 95–127. MR 441978, DOI 10.1007/BF01390104
Additional Information
- Pavlos Tzermias
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- Email: tzermias@math.berkeley.edu, tzermias@crm.es
- Received by editor(s): November 5, 1994
- Received by editor(s) in revised form: September 1, 1995
- Communicated by: William W. Adams
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 663-668
- MSC (1991): Primary 14H25, 14G05; Secondary 11D41
- DOI: https://doi.org/10.1090/S0002-9939-97-03637-X
- MathSciNet review: 1353401