Chabauty and the Mordell-Weil Sieve

These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.

[1]  Nils Bruin.  Chabauty methods and covering techniques applied to generalized Fermat equations , 2002 .

[3]  Michael Stoll,et al.  Deciding Existence of Rational Points on Curves: An Experiment , 2006, Exp. Math..

[4]  T. Browning,et al.  Local Fields , 2008 .

[5]  N. Bruin,et al.  Chabauty methods using elliptic curves , 2003 .

[6]  E. V. Flynn,et al.  A flexible method for applying Chabauty's Theorem , 1997, Compositio Mathematica.

[7]  G. Ballew,et al.  The Arithmetic of Elliptic Curves , 2020, Elliptic Curves.

[8]  J. L. Wetherell.  Bounding the number of rational points on certain curves of high rank , 2001 .

[9]  Bjorn Poonen,et al.  Explicit descent for Jacobians of cyclic coevers of the projective line. , 1997 .

[10]  J. Cassels.  Lectures on elliptic curves , 1991 .

[11]  B. Poonen,et al.  Most odd degree hyperelliptic curves have only one rational point , 2013, 1302.0061.

[12]  J. Silverman.  Advanced Topics in the Arithmetic of Elliptic Curves , 1994 .

[13]  Michael Stoll,et al.  Two-cover descent on hyperelliptic curves , 2008, Math. Comput..

[14]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[15]  E. V. Flynn,et al.  Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Weddle's surface , 1996 .

[16]  M. Mignotte,et al.  Integral points on hyperelliptic curves , 2008, 0801.4459.

[17]  S. Siksek.  Explicit Chabauty over number fields , 2010, 1010.2603.

[18]  M. Stoll,et al.  Partial descent on hyperelliptic curves and the generalized Fermat equation x3+y4+z5=0 , 2011, 1103.1979.

[19]  M. Stoll,et al.  The Mordell-Weil sieve : proving non-existence of rational points on curves , 2009, 0906.1934.

[20]  B. Poonen.  THE METHOD OF CHABAUTY AND COLEMAN WILLIAM MCCALLUM AND , 2017 .

[21]  E. V. Flynn,et al.  Covering collections and a challenge problem of Serre , 2001 .

[22]  Joseph H. Silverman,et al.  Diophantine Geometry: An Introduction , 2000, The Mathematical Gazette.

[23]  Noam D. Elkies,et al.  Trinomials ax and ax with Galois Groups of Order 168 and 8·168 , 2002, ANTS.

[24]  S. Siksek.  Chabauty for symmetric powers of curves , 2009 .