Disparity in Selmer  ranks of quadratic twists of elliptic curves

Zev Klagsbrun; Barry Mazur; Karl Rubin

doi:https://doi.org/10.4007/annals.2013.178.1.5

Abstract

We study the parity of $2$-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve $E$ over an arbitrary number field $K$. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even $2$-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve $E$ such that as $K$ varies, these fractions are dense in $[0, 1]$. More generally, our results also apply to $p$-Selmer ranks of twists of $2$-dimensional self-dual $\mathbf{F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

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[cassels] $Go to document$ J. W. S. Cassels, "Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer," J. Reine Angew. Math., vol. 217, pp. 180-199, 1965.

Show bibtex

@article {cassels, MRKEY = {0179169}, AUTHOR = {Cassels, J. W. S.}, TITLE = {Arithmetic on curves of genus 1. {VIII}. {O}n conjectures of {B}irch and {S}winnerton-{D}yer}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {217}, YEAR = {1965}, PAGES = {180--199}, ISSN = {0075-4102}, MRCLASS = {14.48 (14.49)}, MRNUMBER = {0179169}, MRREVIEWER = {T. Ono}, ZBLNUMBER = {0241.14017}, DOI = {10.1515/crll.1965.217.180}, }
[cremona] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge: Cambridge Univ. Press, 1992.

Show bibtex

@book {cremona, MRKEY = {1201151}, AUTHOR = {Cremona, J. E.}, TITLE = {Algorithms for Modular Elliptic Curves}, PUBLISHER = {Cambridge Univ. Press}, ADDRESS = {Cambridge}, YEAR = {1992}, PAGES = {vi+343}, ISBN = {0-521-41813-5}, MRCLASS = {11G40 (11G05 11Y16 11Y35)}, MRNUMBER = {1201151}, MRREVIEWER = {Philippe Satg{é}}, ZBLNUMBER = {0758.14042}, }
[cfoss] $Go to document$ J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon, and M. Stoll, "Explicit $n$-descent on elliptic curves. I. Algebra," J. Reine Angew. Math., vol. 615, pp. 121-155, 2008.

Show bibtex

@article {cfoss, MRKEY = {2384334}, AUTHOR = {Cremona, J. E. and Fisher, T. A. and O'Neil, C. and Simon, D. and Stoll, M.}, TITLE = {Explicit {$n$}-descent on elliptic curves. {I}. {A}lgebra}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {615}, YEAR = {2008}, PAGES = {121--155}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11G05 (11R34 11Y50 14G25)}, MRNUMBER = {2384334}, MRREVIEWER = {Imin Chen}, DOI = {10.1515/CRELLE.2008.012}, ZBLNUMBER = {1242.11039}, }
[dokchitser] $Go to document$ T. Dokchitser and V. Dokchitser, "Elliptic curves with all quadratic twists of positive rank," Acta Arith., vol. 137, iss. 2, pp. 193-197, 2009.

Show bibtex

@article {dokchitser, MRKEY = {2491537}, AUTHOR = {Dokchitser, Tim and Dokchitser, Vladimir}, TITLE = {Elliptic curves with all quadratic twists of positive rank}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {137}, YEAR = {2009}, NUMBER = {2}, PAGES = {193--197}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11G05 (11G40)}, MRNUMBER = {2491537}, MRREVIEWER = {Conjeeveram S. Rajan}, DOI = {10.4064/aa137-2-7}, ZBLNUMBER = {05529145}, }
[dokdok] $Go to document$ T. Dokchitser and V. Dokchitser, "Regulator constants and the parity conjecture," Invent. Math., vol. 178, iss. 1, pp. 23-71, 2009.

Show bibtex

@article {dokdok, MRKEY = {2534092}, AUTHOR = {Dokchitser, Tim and Dokchitser, Vladimir}, TITLE = {Regulator constants and the parity conjecture}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {178}, YEAR = {2009}, NUMBER = {1}, PAGES = {23--71}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11G05 (11G10 11G40 19A22)}, MRNUMBER = {2534092}, MRREVIEWER = {Joseph H. Silverman}, DOI = {10.1007/s00222-009-0193-7}, ZBLNUMBER = {1219.11083}, }
[goldfeld] $Go to document$ D. Goldfeld, "Conjectures on elliptic curves over quadratic fields," in Number Theory, Carbondale 1979, New York: Springer-Verlag, 1979, vol. 751, pp. 108-118.

Show bibtex

@incollection {goldfeld, MRKEY = {0564926}, AUTHOR = {Goldfeld, Dorian}, TITLE = {Conjectures on elliptic curves over quadratic fields}, BOOKTITLE = {Number Theory, {C}arbondale 1979}, VENUE={{P}roc. {S}outhern {I}llinois {C}onf., {S}outhern {I}llinois {U}niv., {C}arbondale, {I}ll., 1979}, SERIES = {Lecture Notes in Math.}, VOLUME = {751}, PAGES = {108--118}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1979}, MRCLASS = {12A70 (14K07)}, MRNUMBER = {0564926}, MRREVIEWER = {Kenneth Kramer}, ZBLNUMBER = {0417.14031}, DOI = {10.1007/BFb0062705}, }
[heath-brown] $Go to document$ D. R. Heath-Brown, "The size of Selmer groups for the congruent number problem. II," Invent. Math., vol. 118, iss. 2, pp. 331-370, 1994.

Show bibtex

@article {heath-brown, MRKEY = {1292115}, AUTHOR = {Heath-Brown, D. R.}, TITLE = {The size of {S}elmer groups for the congruent number problem. {II}}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {118}, YEAR = {1994}, NUMBER = {2}, PAGES = {331--370}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11G40 (11G05)}, MRNUMBER = {1292115}, MRREVIEWER = {Fernando Q. Gouv{ê}a}, DOI = {10.1007/BF01231536}, ZBLNUMBER = {0815.11032}, NOTE={(with an appendix by P. Monsky)}, }
[howard] B. Howard, "The Heegner point Kolyvagin system," Compos. Math., vol. 140, iss. 6, pp. 1439-1472, 2004.

Show bibtex

@article {howard, MRKEY = {2098397}, AUTHOR = {Howard, Benjamin}, TITLE = {The {H}eegner point {K}olyvagin system}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {140}, YEAR = {2004}, NUMBER = {6}, PAGES = {1439--1472}, ISSN = {0010-437X}, MRCLASS = {11G05 (11R23)}, MRNUMBER = {2098397}, MRREVIEWER = {Henri Darmon}, ZBLNUMBER = {1139.11316}, }
[kane] D. M. Kane, On the ranks of the $2$-Selmer groups of twists of a given elliptic curve.

Show bibtex

@misc{kane, author={Kane, D. M.}, TITLE={On the ranks of the $2$-{S}elmer groups of twists of a given elliptic curve}, NOTE={\emph{Algebra Number Theory}, to appear}, ARXIV={1009.1365}, }
[kmr2] Z. Klagsbrun, B. Mazur, and K. Rubin, Distribution of Selmer ranks of quadratic twists of elliptic curves.

Show bibtex

@misc{kmr2, author={Klagsbrun, Z. and Mazur, B. and Rubin, K.}, TITLE={Distribution of {S}elmer ranks of quadratic twists of elliptic curves}, NOTE={to appear}, }
[kramer] $Go to document$ K. Kramer, "Arithmetic of elliptic curves upon quadratic extension," Trans. Amer. Math. Soc., vol. 264, iss. 1, pp. 121-135, 1981.

Show bibtex

@article {kramer, MRKEY = {0597871}, AUTHOR = {Kramer, Kenneth}, TITLE = {Arithmetic of elliptic curves upon quadratic extension}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {264}, YEAR = {1981}, NUMBER = {1}, PAGES = {121--135}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {14G25 (10B10 14K07)}, MRNUMBER = {0597871}, MRREVIEWER = {Andrew Bremner}, DOI = {10.2307/1998414}, ZBLNUMBER = {0471.14020}, }
[MRvisibility] $Go to document$ B. Mazur and K. Rubin, "Finding large Selmer rank via an arithmetic theory of local constants," Ann. of Math., vol. 166, iss. 2, pp. 579-612, 2007.

Show bibtex

@article {MRvisibility, MRKEY = {2373150}, AUTHOR = {Mazur, B. and Rubin, K.}, TITLE = {Finding large {S}elmer rank via an arithmetic theory of local constants}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {166}, YEAR = {2007}, NUMBER = {2}, PAGES = {579--612}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11G05 (11G40 11R23)}, MRNUMBER = {2373150}, MRREVIEWER = {Benjamin V. Howard}, DOI = {10.4007/annals.2007.166.579}, ZBLNUMBER = {1219.11084}, }
[MRstablerank] $Go to document$ B. Mazur and K. Rubin, "Ranks of twists of elliptic curves and Hilbert’s tenth problem," Invent. Math., vol. 181, iss. 3, pp. 541-575, 2010.

Show bibtex

@article {MRstablerank, MRKEY = {2660452}, AUTHOR = {Mazur, B. and Rubin, K.}, TITLE = {Ranks of twists of elliptic curves and {H}ilbert's tenth problem}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {181}, YEAR = {2010}, NUMBER = {3}, PAGES = {541--575}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11G05 (11G40)}, MRNUMBER = {2660452}, MRREVIEWER = {Pete L. Clark}, DOI = {10.1007/s00222-010-0252-0}, ZBLNUMBER = {1227.11075}, }
[mrs] $Go to document$ B. Mazur, K. Rubin, and A. Silverberg, "Twisting commutative algebraic groups," J. Algebra, vol. 314, iss. 1, pp. 419-438, 2007.

Show bibtex

@article {mrs, MRKEY = {2331769}, AUTHOR = {Mazur, B. and Rubin, K. and Silverberg, A.}, TITLE = {Twisting commutative algebraic groups}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {314}, YEAR = {2007}, NUMBER = {1}, PAGES = {419--438}, ISSN = {0021-8693}, CODEN = {JALGA4}, MRCLASS = {14L15}, MRNUMBER = {2331769}, MRREVIEWER = {Andy R. Magid}, DOI = {10.1016/j.jalgebra.2007.02.052}, ZBLNUMBER = {1128.14034}, }
[milne] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, 1986, vol. 1.

Show bibtex

@book {milne, MRKEY = {0881804}, AUTHOR = {Milne, J. S.}, TITLE = {Arithmetic Duality Theorems}, SERIES = {Perspect. Math.}, VOLUME = {1}, PUBLISHER = {Academic Press}, ADDRESS = {Boston, MA}, YEAR = {1986}, PAGES = {x+421}, ISBN = {0-12-498040-6}, MRCLASS = {14F20 (11G99 11R34 12G10)}, MRNUMBER = {0881804}, MRREVIEWER = {Gerd Faltings}, ZBLNUMBER = {0613.14019}, }
[monsky] $Go to document$ P. Monsky, "Generalizing the Birch-Stephens theorem. I. Modular curves," Math. Z., vol. 221, iss. 3, pp. 415-420, 1996.

Show bibtex

@article {monsky, MRKEY = {1381589}, AUTHOR = {Monsky, P.}, TITLE = {Generalizing the {B}irch-{S}tephens theorem. {I}. {M}odular curves}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {221}, YEAR = {1996}, NUMBER = {3}, PAGES = {415--420}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {11G40 (11G05)}, MRNUMBER = {1381589}, MRREVIEWER = {M. Ram Murty}, DOI = {10.1007/PL00004518}, ZBLNUMBER={0853.11048}, }
[poonenrains] $Go to document$ B. Poonen and E. Rains, "Random maximal isotropic subspaces and Selmer groups," J. Amer. Math. Soc., vol. 25, iss. 1, pp. 245-269, 2012.

Show bibtex

@article {poonenrains, MRKEY = {2833483}, AUTHOR = {Poonen, Bjorn and Rains, Eric}, TITLE = {Random maximal isotropic subspaces and {S}elmer groups}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {25}, YEAR = {2012}, NUMBER = {1}, PAGES = {245--269}, ISSN = {0894-0347}, MRCLASS = {11G10 (11G05 11G30 14G25 14K15)}, MRNUMBER = {2833483}, MRREVIEWER = {Philipp Habegger}, DOI = {10.1090/S0894-0347-2011-00710-8}, ZBLNUMBER = {06005475}, }
[rohrlich] $Go to document$ D. E. Rohrlich, "Galois theory, elliptic curves, and root numbers," Compositio Math., vol. 100, iss. 3, pp. 311-349, 1996.

Show bibtex

@article {rohrlich, MRKEY = {1387669}, AUTHOR = {Rohrlich, David E.}, TITLE = {Galois theory, elliptic curves, and root numbers}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {100}, YEAR = {1996}, NUMBER = {3}, PAGES = {311--349}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {11G05 (11F80 11G07 11G40 11R32)}, MRNUMBER = {1387669}, MRREVIEWER = {Kenneth Kramer}, URL = {http://www.numdam.org/item?id=CM_1996__100_3_311_0}, ZBLNUMBER = {0860.11033}, }
[rubinES] K. Rubin, Euler Systems, Princeton, NJ: Princeton Univ. Press, 2000, vol. 147.

Show bibtex

@book {rubinES, MRKEY = {1749177}, AUTHOR = {Rubin, Karl}, TITLE = {Euler Systems}, SERIES = {Ann. of Math. Studies}, VOLUME = {147}, NOTE = {Hermann Weyl Lectures. The Institute for Advanced Study}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, NJ}, YEAR = {2000}, PAGES = {xii+227}, ISBN = {0-691-05075-9; 0-691-05076-7}, MRCLASS = {11R23 (11G40 11R34 11R42)}, MRNUMBER = {1749177}, MRREVIEWER = {Jan Nekov{á}{\v{r}}}, ZBLNUMBER = {0977.11001}, }
[modpreps] K. Rubin and A. Silverberg, "Families of elliptic curves with constant mod $p$ representations," in Elliptic Curves, Modular Forms, & Fermat’s Last Theorem, Int. Press, Cambridge, MA, 1995, vol. I, pp. 148-161.

Show bibtex

@incollection {modpreps, MRKEY = {1363500}, AUTHOR = {Rubin, Karl and Silverberg, A.}, TITLE = {Families of elliptic curves with constant mod {$p$} representations}, BOOKTITLE = {Elliptic Curves, Modular Forms, \& {F}ermat's Last Theorem}, VENUE={{H}ong {K}ong, 1993}, SERIES = {Ser. Number Theory}, VOLUME={I}, PAGES = {148--161}, PUBLISHER = {Int. Press, Cambridge, MA}, YEAR = {1995}, MRCLASS = {11G05}, MRNUMBER = {1363500}, MRREVIEWER = {Andrea Mori}, ZBLNUMBER = {0856.11027}, }
[serrecg] . J-P. Serre, Cohomologie Galoisienne, Fifth ed., New York: Springer-Verlag, 1994, vol. 5.

Show bibtex

@book {serrecg, MRKEY = {1324577}, AUTHOR = {Serre, {\relax J-P}}, TITLE = {Cohomologie Galoisienne}, SERIES = {Lecture Notes in Math.}, VOLUME = {5}, EDITION = {Fifth}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1994}, PAGES = {x+181}, ISBN = {3-540-58002-6}, MRCLASS = {12G05 (11R34)}, MRNUMBER = {1324577}, ZBLNUMBER = {0812.12002}, }
[serre1972] . J-P. Serre, "Propriétés galoisiennes des points d’ordre fini des courbes elliptiques," Invent. Math., vol. 15, iss. 4, pp. 259-331, 1972.

Show bibtex

@article {serre1972, MRKEY = {0387283}, AUTHOR = {Serre, {\relax J-P}}, TITLE = {Propriétés galoisiennes des points d'ordre fini des courbes elliptiques}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {15}, YEAR = {1972}, NUMBER = {4}, PAGES = {259--331}, ISSN = {0020-9910}, MRCLASS = {14G25 (14K15)}, MRNUMBER = {0387283}, MRREVIEWER = {J. W. S. Cassels}, ZBLNUMBER = {0235.14012}, }
[sw-d] $Go to document$ P. Swinnerton-Dyer, "The effect of twisting on the 2-Selmer group," Math. Proc. Cambridge Philos. Soc., vol. 145, iss. 3, pp. 513-526, 2008.

Show bibtex

@article {sw-d, MRKEY = {2464773}, AUTHOR = {Swinnerton-Dyer, Peter}, TITLE = {The effect of twisting on the 2-{S}elmer group}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {145}, YEAR = {2008}, NUMBER = {3}, PAGES = {513--526}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {11G05}, MRNUMBER = {2464773}, MRREVIEWER = {Pete L. Clark}, DOI = {10.1017/S0305004108001588}, ZBLNUMBER = {1242.11041}, }
[tate] J. Tate, "Duality theorems in Galois cohomology over number fields," in Proc. Internat. Congr. Mathematicians, Djursholm: Inst. Mittag-Leffler, 1963, pp. 288-295.

Show bibtex

@incollection {tate, MRKEY = {0175892}, AUTHOR = {Tate, John}, TITLE = {Duality theorems in {G}alois cohomology over number fields}, BOOKTITLE = {Proc. {I}nternat. {C}ongr. {M}athematicians}, VENUE={{S}tockholm, 1962}, PAGES = {288--295}, PUBLISHER = {Inst. Mittag-Leffler}, ADDRESS = {Djursholm}, YEAR = {1963}, MRCLASS = {10.66}, MRNUMBER = {0175892}, ZBLNUMBER = {0126.07002}, }
[tatealg] J. Tate, "Algorithm for determining the type of a singular fiber in an elliptic pencil," in Modular Functions of One Variable, IV, New York: Springer-Verlag, 1975, vol. 476, pp. 33-52.

Show bibtex

@incollection {tatealg, MRKEY = {0393039}, AUTHOR = {Tate, John}, TITLE = {Algorithm for determining the type of a singular fiber in an elliptic pencil}, BOOKTITLE = {Modular Functions of One Variable, {IV}}, VENUE={{P}roc. {I}nternat. {S}ummer {S}chool, {U}niv. {A}ntwerp, {A}ntwerp, 1972}, PAGES = {33--52}, SERIES={Lecture Notes in Math.}, VOLUME={476}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1975}, MRCLASS = {14G25 (14K15)}, MRNUMBER = {0393039}, MRREVIEWER = {J. S. Milne}, ZBLNUMBER = {1214.14020}, }

Disparity in Selmer ranks of quadratic twists of elliptic curves

Abstract

Authors