A curve for which Coleman’s effective Chabauty bound is sharp

Grant, David

doi:10.1090/S0002-9939-1994-1242084-4

A curve for which Coleman’s effective Chabauty bound is sharp
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Proc. Amer. Math. Soc. 122 (1994), 317-319 Request permission

Abstract:

We show that Coleman’s effective Chabauty bound is sharp for the curve $C:{y^2} = x(x - 1)(x - 2)(x - 5)(x - 6)$ defined over $\mathbb {Q}$, by considering its reduction $\bmod \;7$. We also show that the Jacobian of C is absolutely simple.

References

Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French). MR 4484
Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103, DOI 10.1215/S0012-7094-85-05240-8
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
Daniel M. Gordon and David Grant, Computing the Mordell-Weil rank of Jacobians of curves of genus two, Trans. Amer. Math. Soc. 337 (1993), no. 2, 807–824. MR 1094558, DOI 10.1090/S0002-9947-1993-1094558-0

The method of Chabauty-Coleman and the second case of Fermat’s Last Theorem for regular primes

Hecke L-series related with algebraic cycles or with Siegel modular forms