A 𝑝-adic approach to rational points on curves

Poonen, Bjorn

doi:10.1090/bull/1707

A $p$-adic approach to rational points on curves
HTML articles powered by AMS MathViewer

by Bjorn Poonen HTML | PDF

Bull. Amer. Math. Soc. 58 (2021), 45-56 Request permission

Abstract:

In 1922 Mordell conjectured the striking statement that, for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983 and again by a different method by Vojta in 1991. But neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.

References

Benjamin Bakker and Jacob Tsimerman, The Ax-Schanuel conjecture for variations of Hodge structures, Invent. Math. 217 (2019), no. 1, 77–94. MR 3958791, DOI 10.1007/s00222-019-00863-8
Jennifer Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman, and Jan Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13, Ann. of Math. (2) 189 (2019), no. 3, 885–944. MR 3961086, DOI 10.4007/annals.2019.189.3.6
Enrico Bombieri, The Mordell conjecture revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615–640. MR 1093712
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
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
E. Victor Flynn and Joseph L. Wetherell, Covering collections and a challenge problem of Serre, Acta Arith. 98 (2001), no. 2, 197–205. MR 1831612, DOI 10.4064/aa98-2-9
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
Minhyong Kim, The motivic fundamental group of $\mathbf P^1\sbs \{0,1,\infty \}$ and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR 2181717, DOI 10.1007/s00222-004-0433-9
Minhyong Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133. MR 2512779, DOI 10.2977/prims/1234361156
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
Brian Lawrence and Akshay Venkatesh, Diophantine problems and $p$-adic period mappings, Invent. Math. 221 (2020), no. 3, 893–999. MR 4132959, DOI 10.1007/s00222-020-00966-7
L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Cambridge Phil. Soc. 21 (1922), 179–192.
A. N. Paršin, Quelques conjectures de finitude en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 467–471. MR 0427323
Bjorn Poonen, Computing rational points on curves, Number theory for the millennium, III (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 149–172. MR 1956273
Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathematical Society, Providence, RI, 2017. MR 3729254, DOI 10.1090/gsm/186
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Paul Vojta, Siegel’s theorem in the compact case, Ann. of Math. (2) 133 (1991), no. 3, 509–548. MR 1109352, DOI 10.2307/2944318