## Duke Mathematical Journal

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

#### Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ attached to a newform $f$ of weight 2 on $\Gamma_0(N)$, and let $K$ be a real quadratic field in which all the primes dividing $N$ are split. This paper relates the canonical $\mathbb{R}/\mathbb{Z}$-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

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

Dates
First available in Project Euclid: 24 March 2004

https://projecteuclid.org/euclid.dmj/1080137206

Digital Object Identifier
doi:10.1215/S0012-7094-04-12216-X

Mathematical Reviews number (MathSciNet)
MR2046811

Zentralblatt MATH identifier
1072.11036

#### Citation

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

#### References

• 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.