$p$-torsion monodromy representations of elliptic curves over geometric function fields

Benjamin Bakker; Jacob Tsimerman

doi:https://doi.org/10.4007/annals.2016.184.3.2

Abstract

Given a complex quasiprojective curve $B$ and a nonisotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow \mathrm{GL}_2(\mathbb{F}_p)$. We prove that if $\rho_{\mathcal E}[p]\cong \rho_{\mathcal E’}[p]$, then $\mathcal{E}$ and $\mathcal E’$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $B$. This can be viewed as a function field analog of the Frey–Mazur conjecture, which states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p> 17$. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.

Show/hide bibliography for this article

[Abramovich] $Go to document$ D. Abramovich, "A linear lower bound on the gonality of modular curves," Internat. Math. Res. Notices, iss. 20, pp. 1005-1011, 1996.

Show bibtex

@ARTICLE{Abramovich, mrkey = {1422373}, number = {20}, issn = {1073-7928}, author = {Abramovich, Dan}, mrclass = {11G18 (11F32 14G35)}, doi = {10.1155/S1073792896000621}, journal = {Internat. Math. Res. Notices}, zblnumber = {0878.14019}, mrnumber = {1422373}, fjournal = {International Mathematics Research Notices}, mrreviewer = {M. Ram Murty}, title = {A linear lower bound on the gonality of modular curves}, year = {1996}, pages = {1005--1011}, }
[beardon] A. F. Beardon, The Geometry of Discrete Groups, New York: Springer-Verlag, 1995.

Show bibtex

@BOOK{beardon, mrkey = {1393195}, number = {91}, author = {Beardon, Alan F.}, mrclass = {22E40 (11F06 20H15 30F35 57N10)}, series = {Grad. Texts in Math.}, address = {New York}, isbn = {0-387-90788-2}, publisher = {Springer-Verlag}, mrnumber = {1393195}, note = {corrected reprint of the 1983 original}, title = {The Geometry of Discrete Groups}, year = {1995}, pages = {xii+337}, }
[billerey] N. Billerey, Private communication.

Show bibtex

@MISC{billerey, title = {Private communication}, author = {Billerey, N.}, }
[brooks] $Go to document$ R. Brooks, "Platonic surfaces," Comment. Math. Helv., vol. 74, iss. 1, pp. 156-170, 1999.

Show bibtex

@ARTICLE{brooks, mrkey = {1677565}, number = {1}, issn = {0010-2571}, author = {Brooks, Robert}, mrclass = {58G25 (11F72 30F10 30F45)}, doi = {10.1007/s000140050082}, journal = {Comment. Math. Helv.}, zblnumber = {0920.30037}, volume = {74}, mrnumber = {1677565}, fjournal = {Commentarii Mathematici Helvetici}, mrreviewer = {Christopher M. Judge}, coden = {COMHAX}, title = {Platonic surfaces}, year = {1999}, pages = {156--170}, }
[BuSa] $Go to document$ P. Buser and P. Sarnak, "On the period matrix of a Riemann surface of large genus," Invent. Math., vol. 117, iss. 1, pp. 27-56, 1994.

Show bibtex

@ARTICLE{BuSa, mrkey = {1269424}, number = {1}, issn = {0020-9910}, author = {Buser, P. and Sarnak, P.}, mrclass = {22E40 (14H15 14H42 32G20)}, doi = {10.1007/BF01232233}, journal = {Invent. Math.}, zblnumber = {0814.14033}, volume = {117}, mrnumber = {1269424}, note = {with an appendix by J. H. Conway and N. J. A. Sloane}, fjournal = {Inventiones Mathematicae}, mrreviewer = {Jos{é} M. Mu{ñ}oz Porras}, coden = {INVMBH}, title = {On the period matrix of a {R}iemann surface of large genus}, year = {1994}, pages = {27--56}, }
[BT] B. Bakker and J. Tsimerman, On the Frey-Mazur conjecture over low genus curves, 2013.

Show bibtex

@MISC{BT, author = {Bakker, B. and Tsimerman, J.}, arxiv = {1309.6568}, title = {On the {F}rey-{M}azur conjecture over low genus curves}, year = {2013}, }
[carlton] $Go to document$ D. Carlton, "Moduli for pairs of elliptic curves with isomorphic $N$-torsion," Manuscripta Math., vol. 105, iss. 2, pp. 201-234, 2001.

Show bibtex

@ARTICLE{carlton, mrkey = {1846618}, number = {2}, issn = {0025-2611}, author = {Carlton, David}, mrclass = {11G18 (11F25 11F41)}, doi = {10.1007/s002290170003}, journal = {Manuscripta Math.}, zblnumber = {1019.11009}, volume = {105}, mrnumber = {1846618}, fjournal = {Manuscripta Mathematica}, mrreviewer = {Andrea Mori}, coden = {MSMHB2}, title = {Moduli for pairs of elliptic curves with isomorphic {$N$}-torsion}, year = {2001}, pages = {201--234}, }
[fisher] $Go to document$ T. A. Fisher, "On families of 7 and 11-congruent elliptic curves," LMS J. Comput. Math, vol. 17, iss. 1, pp. 536-564, 2014.

Show bibtex

@article{fisher, author = {Fisher, T. A.}, title = {On families of 7 and 11-congruent elliptic curves}, journal={LMS J. Comput. Math}, FJOURNAL = {LMS Journal of Computation and Mathematics}, VOLUME = {17}, YEAR = {2014}, NUMBER = {1}, PAGES = {536--564}, ISSN = {1461-1570}, MRCLASS = {11F80 (11G05)}, MRNUMBER = {3356045}, MRREVIEWER = {Jordi Gu{à}rdia}, DOI = {10.1112/S1461157014000059}, }
[frey] $Go to document$ G. Frey, "On ternary equations of Fermat type and relations with elliptic curves," in Modular Forms and Fermat’s Last Theorem, New York: Springer-Verlag, 1997, pp. 527-548.

Show bibtex

@INCOLLECTION{frey, mrkey = {1638494}, author = {Frey, Gerhard}, mrclass = {11G05 (11D41)}, address = {New York}, publisher = {Springer-Verlag}, zblnumber = {0976.11027}, mrnumber = {1638494}, booktitle = {Modular Forms and {F}ermat's Last Theorem}, venue = {{B}oston, {MA}, 1995}, title = {On ternary equations of {F}ermat type and relations with elliptic curves}, pages = {527--548}, year = {1997}, doi = {10.1007/978-1-4612-1974-3_20}, }
[hermann] $Go to document$ C. F. Hermann, "Modulflächen quadratischer Diskriminante," Manuscripta Math., vol. 72, iss. 1, pp. 95-110, 1991.

Show bibtex

@ARTICLE{hermann, mrkey = {1107455}, number = {1}, issn = {0025-2611}, author = {Hermann, Carl Friedrich}, mrclass = {11F41 (14G35 14J29)}, doi = {10.1007/BF02568268}, journal = {Manuscripta Math.}, volume = {72}, mrnumber = {1107455}, fjournal = {Manuscripta Mathematica}, mrreviewer = {K.-B. Gundlach}, coden = {MSMHB2}, title = {Modulflächen quadratischer {D}iskriminante}, year = {1991}, pages = {95--110}, zblnumber = {0749.14016}, }
[hwangto1] $Go to document$ J. Hwang and W. To, "Volumes of complex analytic subvarieties of Hermitian symmetric spaces," Amer. J. Math., vol. 124, iss. 6, pp. 1221-1246, 2002.

Show bibtex

@ARTICLE{hwangto1, mrkey = {1939785}, number = {6}, issn = {0002-9327}, author = {Hwang, Jun-Muk and To, Wing-Keung}, mrclass = {32Q15 (53C55)}, journal = {Amer. J. Math.}, zblnumber = {1024.32013}, volume = {124}, mrnumber = {1939785}, fjournal = {American Journal of Mathematics}, mrreviewer = {Khalid Koufany}, coden = {AJMAAN}, title = {Volumes of complex analytic subvarieties of {H}ermitian symmetric spaces}, year = {2002}, pages = {1221--1246}, doi = {10.1353/ajm.2002.0038}, }
[hwangto2] $Go to document$ J. Hwang and W. To, "Injectivity radius and gonality of a compact Riemann surface," Amer. J. Math., vol. 134, iss. 1, pp. 259-283, 2012.

Show bibtex

@ARTICLE{hwangto2, mrkey = {2876146}, number = {1}, issn = {0002-9327}, author = {Hwang, Jun-Muk and To, Wing-Keung}, mrclass = {30F45 (14H51)}, doi = {10.1353/ajm.2012.0007}, journal = {Amer. J. Math.}, zblnumber = {1241.53032}, volume = {134}, mrnumber = {2876146}, fjournal = {American Journal of Mathematics}, mrreviewer = {Dawei Chen}, coden = {AJMAAN}, title = {Injectivity radius and gonality of a compact {R}iemann surface}, year = {2012}, pages = {259--283}, }
[kani] $Go to document$ E. Kani and W. Schanz, "Modular diagonal quotient surfaces," Math. Z., vol. 227, iss. 2, pp. 337-366, 1998.

Show bibtex

@ARTICLE{kani, mrkey = {1609061}, number = {2}, issn = {0025-5874}, author = {Kani, E. and Schanz, W.}, mrclass = {14G35 (11F41 11G18)}, doi = {10.1007/PL00004379}, journal = {Math. Z.}, zblnumber = {0996.14012}, volume = {227}, mrnumber = {1609061}, fjournal = {Mathematische Zeitschrift}, mrreviewer = {Yoshinori Hamahata}, coden = {MAZEAX}, title = {Modular diagonal quotient surfaces}, year = {1998}, pages = {337--366}, }
[mazur] $Go to document$ B. Mazur, "Rational isogenies of prime degree (with an appendix by D. Goldfeld)," Invent. Math., vol. 44, iss. 2, pp. 129-162, 1978.

Show bibtex

@ARTICLE{mazur, mrkey = {0482230}, number = {2}, issn = {0020-9910}, author = {Mazur, B.}, mrclass = {14K07 (10D35 14G25)}, doi = {10.1007/BF01390348}, journal = {Invent. Math.}, zblnumber = {0386.14009}, volume = {44}, mrnumber = {482230}, fjournal = {Inventiones Mathematicae}, mrreviewer = {V. V. Shokurov}, coden = {INVMBH}, title = {Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)}, year = {1978}, pages = {129--162}, }
[zograf] $Go to document$ P. G. Zograf, "Small eigenvalues of automorphic Laplacians in spaces of cusp forms," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$, vol. 134, pp. 157-168, 1984.

Show bibtex

@ARTICLE{zograf, mrkey = {741858}, issn = {0373-2703}, author = {Zograf, P. G.}, mrclass = {58G25 (11F72)}, journal = {Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$}, zblnumber = {0536.10018}, volume = {134}, mrnumber = {741858}, note = {Automorphic functions and number theory, II}, fjournal = {Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)}, mrreviewer = {G. M. Lyan}, title = {Small eigenvalues of automorphic {L}aplacians in spaces of cusp forms}, year = {1984}, pages = {157--168}, url = {http://mi.mathnet.ru/eng/znsl/v134/p157}, }

$p$-torsion monodromy representations of elliptic curves over geometric function fields

Abstract

Authors