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

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.

  • [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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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},
     }

Authors

Benjamin Bakker

Humboldt-Universität zu Berlin

Current address:

University of Georgia, Athens, GA Jacob Tsimerman

University of Toronto, Toronto, Ontario, Canada