Duke Mathematical Journal

On the p-adic L-function of a modular form at a supersingular prime

Robert Pollack

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we study the two $p$-adic $L$-functions attached to a modular form $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, we are able to decompose both the sum and the difference of the two unbounded distributions attached to $f$ into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of $f$ (and the fact that $a\sb p$ vanishes). From this description we explain how the $p$-adic $L$-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas for the $p$-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.

Article information

Duke Math. J. Volume 118, Number 3 (2003), 523-558.

First available: 23 April 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11R23: Iwasawa theory


Pollack, Robert. On the p -adic L -function of a modular form at a supersingular prime. Duke Mathematical Journal 118 (2003), no. 3, 523--558. doi:10.1215/S0012-7094-03-11835-9. http://projecteuclid.org/euclid.dmj/1082744678.

Export citation


  • A. Abbes and E. Ullmo, À propos de la conjecture de Manin pour les courbes elliptiques modulaires, Compositio Math. 103 (1996), 269--286.
  • Y. Amice and J. Vélu, ``Distributions $p$-adiques associées aux séries de Hecke'' in Journées arithmétiques de Bordeaux (Bordeaux, 1974), Astérisque 24 --.25, Soc. Math. France, Montrouge, 1975, 119--131.
  • D. Bernardi and B. Perrin-Riou, Variante $p$-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 227--232.
  • 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.
  • P. Colmez, Théorie d'Iwasawa des représentations de de Rham d'un corps local, Ann. of Math. (2) 148 (1998), 485--571.
  • J. E. Cremona, Algorithms for Modular Elliptic Curves, 2d ed., Cambridge Univ. Press, Cambridge, 1997.
  • R. Greenberg, ``Iwasawa theory for elliptic curves'' in Arithmetic Theory of Elliptic Curves (Cetraro, Italy, 1997), Lecture Notes in Math. 1716, Springer, Berlin, 1999, 51--144.
  • R. Greenberg and G. Stevens, ``On the conjecture of Mazur, Tate, and Teitelbaum'' in $p$-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, 1991), Contemp. Math. 165, Amer. Math. Soc., Providence, 1994, 183--211.
  • K. Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, preprint, 2000.
  • S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1--36.
  • M. Kurihara, On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction, I, Invent. Math. 149 (2002), 195--224. \CMP1 914 621
  • M. Lazard, Les zéros des fonctions analytiques d'une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47--75.
  • Ju. I. Manin, Cyclotomic fields and modular curves (in Russian), Uspekhi Mat. Nauk 26, no. 6 (1971), 7--71.
  • --. --. --. --., Parabolic points and zeta functions of modular curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19--66.
  • --. --. --. --., Periods of cusp forms, and $p$-adic Hecke series (in Russian), Mat. Sb. (N.S.) 92 (134) (1973), 378--401., 503.
  • B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183--266.
  • --. --. --. --., Rational isogenies of prime degree, Invent. Math. 44 (1978), 129--162.
  • B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1--61.
  • B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1--48.
  • A. G. Nasybullin, $p$-adic $L$-series of supersingular elliptic curves (in Russian), Funkcional. Anal. i Priložen. 8, no. 1 (1974), 82--83.
  • B. Perrin-Riou, Théorie d'Iwasawa $p$-adique locale et globale, Invent. Math. 99 (1990), 247--292.
  • --. --. --. --., Fonctions $L$ $p$-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), 945--995.
  • --------, Arithmétique des courbes elliptiques à réduction supersingulière en $p$, preprint, 2001, http://math.uiuc.edu/Algebraic-Number-Theory/0306
  • D. E. Rohrlich, On $L$-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409--423.
  • K. Rubin, ``Euler systems and modular elliptic curves'' in Galois Representations in Arithmetic Algebraic Geometry (Durham, England, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 351--367.
  • J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, New York, 1992.
  • M. M. Višik, Nonarchimedean measures associated with Dirichlet series (in Russian), Mat. Sb. (N.S.) 99 (141), no. 2 (1976), 248--260., 296.
  • M. M. Višik and Ju. I. Manin, $p$-adic Hecke series of imaginary quadratic fields (in Russian), Mat. Sb. (N.S.) 95 (137) (1974), 357--383., 471.
  • A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443--551.