Abstract
Consider the smooth projective models $C$ of curves $y^2=f(x)$ with $f(x) \in \mathbb{Z}[x]$ monic and separable of degree $2g+1$. We prove that for $g \ge 3$, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to $1$ as $g \to \infty$. Finally, we show that $C(\mathbb{Q})$ can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using $p$-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty’s method that shows that certain computable conditions imply $\#C(\mathbb{Q})=1$; on the other hand, using further $p$-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava–Gross theorems on the average number and equidistribution of nonzero $2$-Selmer group elements, we prove that these conditions are often satisfied for $p=2$.
-
[Faltings1983]
G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern," Invent. Math., vol. 73, iss. 3, pp. 349-366, 1983.
@article{Faltings1983,
author = {Faltings, G.},
journal = {Invent. Math.},
note = {English translation: Finiteness theorems for abelian varieties over number fields, 9--27 in Arithmetic Geometry (Storrs, Conn., 1984), Springer, New York, 1986; erratum in \emph{Invent. Math.} \textbf{75} (1984), 381},
number = {3},
pages = {349--366},
title = {Endlichkeitssätze für abelsche {V}arietäten über {Z}ahlkörpern},
volume = {73},
year = {1983},
doi = {10.1007/BF01388432},
issn = {0020-9910},
} -
[Chabauty1941] C. Chabauty, "Sur les points rationnels des courbes algébriques de genre supérieur à l’unité," C. R. Acad. Sci. Paris, vol. 212, pp. 882-885, 1941.
@article{Chabauty1941,
author = {Chabauty, Claude},
journal = {C. R. Acad. Sci. Paris},
pages = {882--885},
title = {Sur les points rationnels des courbes algébriques de genre supérieur à l'unité},
volume = {212},
year = {1941},
} -
[Coleman1985chabauty]
R. F. Coleman, "Effective Chabauty," Duke Math. J., vol. 52, iss. 3, pp. 765-770, 1985.
@article{Coleman1985chabauty,
author = {Coleman, Robert F.},
journal = {Duke Math. J.},
number = {3},
pages = {765--770},
title = {Effective {C}habauty},
volume = {52},
year = {1985},
doi = {10.1215/S0012-7094-85-05240-8},
issn = {0012-7094},
} -
[Bhargava-Gross-preprint] M. Bhargava and B. Gross, The average size of the $2$-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, 2013.
@misc{Bhargava-Gross-preprint,
author = {Bhargava, Manjul and Gross, Benedict},
title = {The average size of the $2$-{S}elmer group of {J}acobians of hyperelliptic curves having a rational {W}eierstrass point},
year = {2013},
} -
[Reid-thesis] M. Reid, The complete intersection of two or more quadrics, 1972.
@misc{Reid-thesis,
author = {Reid, Miles},
note = {Ph.D. thesis, Trinity College, Cambridge},
title = {The complete intersection of two or more quadrics},
year = {1972},
} -
[Donagi1980]
R. Donagi, "Group law on the intersection of two quadrics," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 7, iss. 2, pp. 217-239, 1980.
@article{Donagi1980,
author = {Donagi, Ron},
journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
number = {2},
pages = {217--239},
title = {Group law on the intersection of two quadrics},
volume = {7},
year = {1980},
url = {http://www.numdam.org/item?id=ASNSP_1980_4_7_2_217_0},
} -
@misc{WangXJ-thesis,
author = {Wang, Xiaoheng},
note = {Ph.D. thesis, Harvard University},
title = {Pencils of quadrics and {J}acobians of hyperelliptic curves},
year = {2013},
url = {http://nrs.harvard.edu/urn-3:HUL.InstRepos:11156784},
} -
[Bhargava-Shankar-preprint1] M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, 2015.
@misc{Bhargava-Shankar-preprint1,
author = {Bhargava, Manjul and Shankar, Arul},
note = {to appear in {\em Annals of Math} {\bf 181}},
title = {Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves},
volume = {181},
year = {2015},
} -
[DeJong2002]
A. J. de Jong, "Counting elliptic surfaces over finite fields," Mosc. Math. J., vol. 2, iss. 2, pp. 281-311, 2002.
@article{DeJong2002,
author = {{de Jong},
A. J.},
journal = {Mosc. Math. J.},
key = {dJ02},
note = {dedicated to Yuri I. Manin on the occasion of his 65th birthday},
number = {2},
pages = {281--311},
title = {Counting elliptic surfaces over finite fields},
volume = {2},
year = {2002},
issn = {1609-3321},
url = {http://www.ams.org/distribution/mmj/vol2-2-2002/dejong.pdf},
} -
[Fouvry1993] É. Fouvry, "Sur le comportement en moyenne du rang des courbes $y^2=x^3+k$," in Séminaire de Théorie des Nombres, Paris, 1990–91, Boston: Birkhäuser, 1993, vol. 108, pp. 61-84.
@incollection{Fouvry1993, address = {Boston},
author = {Fouvry, {É}.},
booktitle = {Séminaire de {T}héorie des {N}ombres, {P}aris, 1990--91},
pages = {61--84},
publisher = {Birkhäuser},
series = {Progr. Math.},
title = {Sur le comportement en moyenne du rang des courbes {$y\sp 2=x\sp 3+k$}},
volume = {108},
year = {1993},
} -
[Poonen2013-bourbaki] B. Poonen, "Average rank of elliptic curves [after Manjul Bhargava and Arul Shankar]," in Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058, Paris: Soc. Math. France, 2013, vol. 352, p. exp. no. 1049, viii, 187-204.
@incollection{Poonen2013-bourbaki, address = {Paris},
author = {Poonen, Bjorn},
booktitle = {S{é}minaire Bourbaki. Vol. 2011/2012. Expos{é}s 1043--1058},
pages = {Exp. No. 1049, viii, 187--204},
publisher = {Soc. Math. France},
series = {Astérisque},
title = {Average rank of elliptic curves [after {M}anjul {B}hargava and {A}rul {S}hankar]},
volume = {352},
year = {2013},
isbn = {978-2-85629-371-3},
issn = {0303-1179},
} -
[McCallum-Poonen2012] W. McCallum and B. Poonen, "The method of Chabauty and Coleman," in Explicit Methods in Number Theory, Paris: Soc. Math. France, 2012, vol. 36, pp. 99-117.
@incollection{McCallum-Poonen2012, address = {Paris},
author = {McCallum, William and Poonen, Bjorn},
booktitle = {Explicit Methods in Number Theory},
pages = {99--117},
publisher = {Soc. Math. France},
series = {Panor. Synthèses},
title = {The method of {C}habauty and {C}oleman},
volume = {36},
year = {2012},
} -
[Stoll2006-chabauty]
M. Stoll, "Independence of rational points on twists of a given curve," Compos. Math., vol. 142, iss. 5, pp. 1201-1214, 2006.
@article{Stoll2006-chabauty,
author = {Stoll, Michael},
journal = {Compos. Math.},
number = {5},
pages = {1201--1214},
title = {Independence of rational points on twists of a given curve},
volume = {142},
year = {2006},
doi = {10.1112/S0010437X06002168},
issn = {0010-437X},
} -
[Skolem1934] . T. Skolem, "Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen," in 8. Skand. Mat.-Kongr., Stockholm, , 1934, pp. 163-188.
@incollection{Skolem1934,
author = {Skolem, {\relax Th}},
booktitle = {8. Skand. Mat.-Kongr., Stockholm},
note = {(German)},
pages = {163--188},
title = {Ein Verfahren zur {B}ehandlung gewisser exponentialer {G}leichungen und diophantischer {G}leichungen},
year = {1934},
} -
[McCallum1994]
W. G. McCallum, "On the method of Coleman and Chabauty," Math. Ann., vol. 299, iss. 3, pp. 565-596, 1994.
@article{McCallum1994,
author = {McCallum, William G.},
journal = {Math. Ann.},
number = {3},
pages = {565--596},
title = {On the method of {C}oleman and {C}habauty},
volume = {299},
year = {1994},
doi = {10.1007/BF01459799},
issn = {0025-5831},
} -
[Lorenzini-Tucker2002]
D. Lorenzini and T. J. Tucker, "Thue equations and the method of Chabauty-Coleman," Invent. Math., vol. 148, iss. 1, pp. 47-77, 2002.
@article{Lorenzini-Tucker2002,
author = {Lorenzini, Dino and Tucker, Thomas J.},
journal = {Invent. Math.},
number = {1},
pages = {47--77},
title = {Thue equations and the method of {C}habauty-{C}oleman},
volume = {148},
year = {2002},
doi = {10.1007/s002220100186},
issn = {0020-9910},
} -
[Katz-Zureick-Brown]
E. Katz and D. Zureick-Brown, "The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions," Compos. Math., vol. 149, iss. 11, pp. 1818-1838, 2013.
@article{Katz-Zureick-Brown,
author = {Katz, Eric and Zureick-Brown, David},
journal = {Compos. Math.},
number = {11},
pages = {1818--1838},
title = {The {C}habauty-{C}oleman bound at a prime of bad reduction and {C}lifford bounds for geometric rank functions},
volume = {149},
year = {2013},
doi = {10.1112/S0010437X13007410},
issn = {0010-437X},
} -
[Saito1988]
T. Saito, "Conductor, discriminant, and the Noether formula of arithmetic surfaces," Duke Math. J., vol. 57, iss. 1, pp. 151-173, 1988.
@article{Saito1988,
author = {Saito, Takeshi},
journal = {Duke Math. J.},
number = {1},
pages = {151--173},
title = {Conductor, discriminant, and the {N}oether formula of arithmetic surfaces},
volume = {57},
year = {1988},
doi = {10.1215/S0012-7094-88-05706-7},
issn = {0012-7094},
} -
[Liu1994]
Q. Liu, "Conducteur et discriminant minimal de courbes de genre $2$," Compositio Math., vol. 94, iss. 1, pp. 51-79, 1994.
@article{Liu1994,
author = {Liu, Qing},
journal = {Compositio Math.},
number = {1},
pages = {51--79},
title = {Conducteur et discriminant minimal de courbes de genre {$2$}},
volume = {94},
year = {1994},
issn = {0010-437X},
url = {http://www.numdam.org/item?id=CM_1994__94_1_51_0},
} -
[Liu1996]
Q. Liu, "Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète," Trans. Amer. Math. Soc., vol. 348, iss. 11, pp. 4577-4610, 1996.
@article{Liu1996,
author = {Liu, Qing},
journal = {Trans. Amer. Math. Soc.},
number = {11},
pages = {4577--4610},
title = {Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète},
volume = {348},
year = {1996},
doi = {10.1090/S0002-9947-96-01684-4},
issn = {0002-9947},
} -
[Bhargava-most-preprint] M. Bhargava, Most hyperelliptic curves over $\mathbb{Q}$ have no rational points, 2013.
@misc{Bhargava-most-preprint,
author = {Bhargava, Manjul},
title = {Most hyperelliptic curves over {$\mathbb{Q}$} have no rational points},
year = {2013},
} -
[Shankar-Wang-preprint] A. Shankar and X. Wang, Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves, 2014.
@misc{Shankar-Wang-preprint,
author = {Shankar, Arul and Wang, Xiaoheng},
title = {Average size of the 2-{S}elmer group of {J}acobians of monic even hyperelliptic curves},
year = {2014},
} -
[Lichtenbaum1968]
S. Lichtenbaum, "Curves over discrete valuation rings," Amer. J. Math., vol. 90, pp. 380-405, 1968.
@article{Lichtenbaum1968,
author = {Lichtenbaum, Stephen},
journal = {Amer. J. Math.},
pages = {380--405},
title = {Curves over discrete valuation rings},
volume = {90},
year = {1968},
doi = {10.2307/2373535},
issn = {0002-9327},
} -
[Koblitz1984]
N. Koblitz, $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions, Second ed., New York: Springer-Verlag, 1984, vol. 58.
@book{Koblitz1984, address = {New York},
author = {Koblitz, Neal},
edition = {Second},
pages = {xii+150},
publisher = {Springer-Verlag},
series = {Grad. Texts in Math.},
title = {{$p$}-adic Numbers, {$p$}-adic Analysis, and Zeta-Functions},
volume = {58},
year = {1984},
doi = {10.1007/978-1-4612-1112-9},
isbn = {0-387-96017-1},
} -
[Chiodo-Eisenbud-Farkas-Schreyer2013]
A. Chiodo, D. Eisenbud, G. Farkas, and F. Schreyer, "Syzygies of torsion bundles and the geometry of the level $\ell$ modular variety over $\overline{\mathcal{M}}_g$," Invent. Math., vol. 194, iss. 1, pp. 73-118, 2013.
@article{Chiodo-Eisenbud-Farkas-Schreyer2013,
author = {Chiodo, Alessandro and Eisenbud, David and Farkas, Gavril and Schreyer, Frank-Olaf},
journal = {Invent. Math.},
note = {Theorem~2.3 is misstated in the journal article, but a correct statement appears in the version at \url {http://arxiv.org/abs/1205.0661v4}},
number = {1},
pages = {73--118},
title = {Syzygies of torsion bundles and the geometry of the level {$\ell$} modular variety over {$\overline{\mathcal{M}}\sb g$}},
volume = {194},
year = {2013},
doi = {10.1007/s00222-012-0441-0},
issn = {0020-9910},
} -
[ACampo1979]
N. A’Campo, "Tresses, monodromie et le groupe symplectique," Comment. Math. Helv., vol. 54, iss. 2, pp. 318-327, 1979.
@article{ACampo1979,
author = {A'Campo, Norbert},
journal = {Comment. Math. Helv.},
number = {2},
pages = {318--327},
title = {Tresses, monodromie et le groupe symplectique},
volume = {54},
year = {1979},
doi = {10.1007/BF02566275},
issn = {0010-2571},
} -
[Kneser1965c]
M. Kneser, "Starke Approximation in algebraischen Gruppen. I," J. Reine Angew. Math., vol. 218, pp. 190-203, 1965.
@article{Kneser1965c,
author = {Kneser, Martin},
journal = {J. Reine Angew. Math.},
pages = {190--203},
title = {Starke {A}pproximation in algebraischen {G}ruppen. {I}},
volume = {218},
year = {1965},
doi = {10.1515/crll.1965.218.190},
issn = {0075-4102},
} -
[Bosch-Lutkebohmert-Raynaud1990]
S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990, vol. 21.
@book{Bosch-Lutkebohmert-Raynaud1990, address = {New York},
author = {Bosch, Siegfried and L{ü}tkebohmert, Werner and Raynaud, Michel},
pages = {x+325},
publisher = {Springer-Verlag},
series = {Ergeb. Math. Grenzgeb.},
title = {Néron Models},
volume = {21},
year = {1990},
doi = {10.1007/978-3-642-51438-8},
isbn = {3-540-50587-3},
} -
[Poonen-Rains2012-selmer]
B. Poonen and E. Rains, "Random maximal isotropic subspaces and Selmer groups," J. Amer. Math. Soc., vol. 25, iss. 1, pp. 245-269, 2012.
@article{Poonen-Rains2012-selmer,
author = {Poonen, Bjorn and Rains, Eric},
journal = {J. Amer. Math. Soc.},
number = {1},
pages = {245--269},
title = {Random maximal isotropic subspaces and {S}elmer groups},
volume = {25},
year = {2012},
doi = {10.1090/S0894-0347-2011-00710-8},
issn = {0894-0347},
} -
[Poonen-Voloch2004]
B. Poonen and J. F. Voloch, "Random Diophantine equations," in Arithmetic of Higher-Dimensional Algebraic Varieties, Boston: Birkhäuser, 2004, vol. 226, pp. 175-184.
@incollection{Poonen-Voloch2004, address = {Boston},
author = {Poonen, Bjorn and Voloch, Jos{é} Felipe},
booktitle = {Arithmetic of Higher-Dimensional Algebraic Varieties},
note = {with appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz},
pages = {175--184},
publisher = {Birkhäuser},
series = {Progr. Math.},
title = {Random {D}iophantine equations},
volume = {226},
year = {2004},
doi = {10.1007/978-0-8176-8170-8_11},
} -
[Poonen2006-heuristic]
B. Poonen, "Heuristics for the Brauer-Manin obstruction for curves," Experiment. Math., vol. 15, iss. 4, pp. 415-420, 2006.
@article{Poonen2006-heuristic,
author = {Poonen, Bjorn},
journal = {Experiment. Math.},
number = {4},
pages = {415--420},
title = {Heuristics for the {B}rauer-{M}anin obstruction for curves},
volume = {15},
year = {2006},
doi = {10.1080/10586458.2006.10128974},
issn = {1058-6458},
} -
[Granville2007]
A. Granville, "Rational and integral points on quadratic twists of a given hyperelliptic curve," Int. Math. Res. Not., vol. 2007, iss. 8, p. I, 2007.
@article{Granville2007,
author = {Granville, Andrew},
journal = {Int. Math. Res. Not.},
number = {8},
pages = {Art. ID 027, 24},
title = {Rational and integral points on quadratic twists of a given hyperelliptic curve},
volume = {2007},
year = {2007},
doi = {10.1093/imrn/rnm027},
issn = {1073-7928},
} -
[Stoll2009-preprint] M. Stoll, On the average number of rational points on curves of genus $2$, 2009.
@misc{Stoll2009-preprint,
author = {Stoll, Michael},
title = {On the average number of rational points on curves of genus $2$},
year = {2009},
} -
[Poonen2001-torsion]
B. Poonen, "Computing torsion points on curves," Experiment. Math., vol. 10, iss. 3, pp. 449-465, 2001.
@article{Poonen2001-torsion,
author = {Poonen, Bjorn},
journal = {Experiment. Math.},
number = {3},
pages = {449--465},
title = {Computing torsion points on curves},
volume = {10},
year = {2001},
doi = {10.1080/10586458.2001.10504462},
issn = {1058-6458},
} -
[Stoll2001]
M. Stoll, "Implementing 2-descent for Jacobians of hyperelliptic curves," Acta Arith., vol. 98, iss. 3, pp. 245-277, 2001.
@article{Stoll2001,
author = {Stoll, Michael},
journal = {Acta Arith.},
number = {3},
pages = {245--277},
title = {Implementing 2-descent for {J}acobians of hyperelliptic curves},
volume = {98},
year = {2001},
doi = {10.4064/aa98-3-4},
issn = {0065-1036},
}