Duke Mathematical Journal

A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing

Massimo Bertolini and Henri Darmon

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Let E be an elliptic curve over Q attached to a newform f of weight 2 on Γ 0 (N) , and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical / -valued "circle pairing" on E(K) defined by Mazur and Tate [MT1] to a period integral I (f,k) defined in terms of f and k . The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which I (f,k) replaces the derivative of the complex L -series L(f,K,s) and the circle pairing replaces the Néron-Tate height. It emerges naturally as an archimedean fragment of the theory of anticyclotomic p-adic L-functions developed in [BD], and has been tested numerically in a variety of situations. The last section formulates a conjectural variant of a formula of Gross, Kohnen, and Zagier [GKZ] for the Mazur-Tate circle pairing.

Article information

Duke Math. J., Volume 122, Number 1 (2004), 181-204.

First available in Project Euclid: 24 March 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F04 11G05: Elliptic curves over global fields [See also 14H52] 11G50: Heights [See also 14G40, 37P30]


Bertolini, Massimo; Darmon, Henri. A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing. Duke Math. J. 122 (2004), no. 1, 181--204. doi:10.1215/S0012-7094-04-12216-X. https://projecteuclid.org/euclid.dmj/1080137206

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  • M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves, Invent. Math. 126 (1996), 413--456.
  • M. Bertolini, H. Darmon, and P. Green, "Periods and points attached to quadratic algebras" to appear in Proceedings of the MSRI Workshop on Special Values of Rankin $L$-Series, H. Darmon and S. Zhang, eds., Cambridge Univ. Press, Cambridge.
  • E. D. Bone, The circle pairing on elliptic curves, bachelor's thesis, Amherst College, Amherst, Mass., 1995.
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over $\mathbf Q$: Wild $3$-adic exercises, J. Amer. Math. Soc. 14 (2001), 843--939.
  • G. S. Call, Local heights on families of abelian varieties, Ph.D. dissertation, Harvard University, Cambridge, Mass., 1986.
  • J. E. Cremona, Algorithms for Modular Elliptic Curves, 2nd ed., Cambridge Univ. Press, Cambridge, 1997.
  • H. Darmon, "Heegner points, Heegner cycles, and congruences" in Elliptic Curves and Related Topics, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, 1994, 45--59.
  • B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series, II, Math. Ann. 278 (1987), 497--562.
  • S. R. Hamblen, Calculating the circle pairing over families of elliptic curves, bachelor's thesis, Amherst College, Amherst, Mass., 1998.
  • Ju. I. Manin, Parabolic points and zeta functions of modular curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19--66.; English translation in Math. USSR-Izv. 6 (1972), 19--64.
  • B. Mazur and J. Tate, "Canonical height pairings via biextensions" in Arithmetic and Geometry, Vol. I, Progr. Math. 35, Birkhäuser, Boston, 1983, 195--237.
  • --. --. --. --., Refined conjectures of the "Birch and Swinnerton-Dyer type," Duke Math. J. 54 (1987), 711--750.
  • A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. (2) 82 (1965) 249--331.
  • A. A. Popa, Central values of Rankin $L$-series over real quadratic fields, Ph.D. dissertation, Harvard University, Cambridge, Mass., 2003.
  • R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553--572.
  • A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443--551.
  • D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), 372--384.