On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime.

*(English)*Zbl 1074.11061In this important paper the author shows what the “arithmetically significant” part of the \(p\)-adic \(L\)-function attached to a modular form \(f\) of weight \(k\) should be in the supersingular case. Actually one has to deal with two functions indexed by + and \(-\). For instance when \(f\) comes from a (modular) elliptic curve \(E\) with \(a_p=0\) (which is, for \(p>3\), tantamount to supersingularity at \(p\)), and \(\alpha\) is a root of \(x^2+p\), then \(L_p(E,\alpha,T) = G^+(T) + G^-(T)\alpha\) where \(G^\pm(T)\) has \(p\)-adic rational coefficients. From the interpolation approach to \(p\)-adic \(L\)-functions it follows that \(G^\pm(T)\) has an infinite series of trivial zeros (in the supersingular case which we assume from now on), and this actually implies that \(L_p(E,\alpha,T)\) has infinitely many zeros as well, even though we don’t see them immediately. (These observations are due to B. Perrin-Riou.) A similar result holds for \(L\)-functions involving twists. The task is to eliminate these predictable zeros.

The paper carefully reviews the construction of \(p\)-adic \(L\)-functions, following Vishik and others, in §2. Then the author proceeds to modify \(G^\pm(T)\) so as to eliminate the trivial zeros. To make this beautiful idea work, he defines two power series (of necessity, with unbounded denominators in their coefficients) \(\log^+_p\) and \(\log^-_p\), the so-called half-logarithms, whose zeros are exactly the trivial zeros of \(G^\pm(T)\). For \(k=2\), these half-logs live in \({\mathbb Q}_p[[T]]\). In general, they involve an extra integral parameter \(j=0,\ldots,k-2\). The definition of the half-logs is quite elegant: one takes the product of all \(p^n\)-th cyclotomic polynomials with \(n\) of a fixed parity, each polynomial divided by \(p\). Recall that these cyclotomic polynomials are Eisenstein in \(T\) with constant coefficient exactly \(p\); so the division by \(p\) changes this into 1, and one sees at least that the constant term of the infinite product makes sense. It is then shown that \(L^\pm_p(f,T)=G^\pm(T)/\log_p^\pm(T)\) has bounded coefficients (we over-simplify notation), and hence only finitely many zeros, by Weierstraß preparation. This is the main result (§5). It is shown that if \(f\) comes from the elliptic curve \(E\), then \(L^\pm_p(f,T)=L^\pm_p(E,T)\) has no denominators at all. In §6 this is applied in order to determine the growth of the analytical order of the \(p\)-part of the Tate-Shafarevich groups in a cyclotomic tower. (Under the B-SD conjecture, the analytical order is just the order.) This is Proposition 6.10. The long sums of \(p\)-powers correspond to the trivial zeros of the \(L\)-functions and are responsible for the guaranteed doubly exponential growth, and the \(\lambda\) and \(\mu\) terms are the usual Iwasawa invariants of \(L^\pm_p(E,T)\). In the particular case that the quotient \(L(E,1)/\Omega_E\) is a \(p\)-unit, these invariants all vanish and one recaptures (part of) a result of Kurihara. Here, remarkably enough, the growth no longer depends on the curve \(E\)! Actually Kurihara shows more in this situation: he proves the relevant case of B-SD, equating analytic and algebraic order.

The paper under review concludes with some very interesting numerical results obtained with the system MAGMA. They describe nontrivial roots and \(\mu\)-invariants of the \(3\)-adic \(L\)-functions attached to twists of the curve \(X_0(32)\).

The paper carefully reviews the construction of \(p\)-adic \(L\)-functions, following Vishik and others, in §2. Then the author proceeds to modify \(G^\pm(T)\) so as to eliminate the trivial zeros. To make this beautiful idea work, he defines two power series (of necessity, with unbounded denominators in their coefficients) \(\log^+_p\) and \(\log^-_p\), the so-called half-logarithms, whose zeros are exactly the trivial zeros of \(G^\pm(T)\). For \(k=2\), these half-logs live in \({\mathbb Q}_p[[T]]\). In general, they involve an extra integral parameter \(j=0,\ldots,k-2\). The definition of the half-logs is quite elegant: one takes the product of all \(p^n\)-th cyclotomic polynomials with \(n\) of a fixed parity, each polynomial divided by \(p\). Recall that these cyclotomic polynomials are Eisenstein in \(T\) with constant coefficient exactly \(p\); so the division by \(p\) changes this into 1, and one sees at least that the constant term of the infinite product makes sense. It is then shown that \(L^\pm_p(f,T)=G^\pm(T)/\log_p^\pm(T)\) has bounded coefficients (we over-simplify notation), and hence only finitely many zeros, by Weierstraß preparation. This is the main result (§5). It is shown that if \(f\) comes from the elliptic curve \(E\), then \(L^\pm_p(f,T)=L^\pm_p(E,T)\) has no denominators at all. In §6 this is applied in order to determine the growth of the analytical order of the \(p\)-part of the Tate-Shafarevich groups in a cyclotomic tower. (Under the B-SD conjecture, the analytical order is just the order.) This is Proposition 6.10. The long sums of \(p\)-powers correspond to the trivial zeros of the \(L\)-functions and are responsible for the guaranteed doubly exponential growth, and the \(\lambda\) and \(\mu\) terms are the usual Iwasawa invariants of \(L^\pm_p(E,T)\). In the particular case that the quotient \(L(E,1)/\Omega_E\) is a \(p\)-unit, these invariants all vanish and one recaptures (part of) a result of Kurihara. Here, remarkably enough, the growth no longer depends on the curve \(E\)! Actually Kurihara shows more in this situation: he proves the relevant case of B-SD, equating analytic and algebraic order.

The paper under review concludes with some very interesting numerical results obtained with the system MAGMA. They describe nontrivial roots and \(\mu\)-invariants of the \(3\)-adic \(L\)-functions attached to twists of the curve \(X_0(32)\).

Reviewer: Cornelius Greither (Neubiberg)

##### MSC:

11R23 | Iwasawa theory |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

##### Software:

ecdata**OpenURL**

##### References:

[1] | A. Abbes and E. Ullmo, À propos de la conjecture de Manin pour les courbes elliptiques modulaires , Compositio Math. 103 (1996), 269–286. · Zbl 0865.11049 |

[2] | 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. · Zbl 0332.14010 |

[3] | 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. · Zbl 0804.14009 |

[4] | 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. JSTOR: · Zbl 0982.11033 |

[5] | P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local , Ann. of Math. (2) 148 (1998), 485–571. JSTOR: · Zbl 0928.11045 |

[6] | J. E. Cremona, Algorithms for Modular Elliptic Curves , 2d ed., Cambridge Univ. Press, Cambridge, 1997. · Zbl 0872.14041 |

[7] | 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. · Zbl 0946.11027 |

[8] | 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. · Zbl 0846.11030 |

[9] | K. Kato, \(p\)-adic Hodge theory and values of zeta functions of modular forms , preprint, 2000. |

[10] | S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes , Invent. Math. 152 (2003), 1–36. · Zbl 1047.11105 |

[11] | 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 · Zbl 1033.11028 |

[12] | 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. · Zbl 0119.03701 |

[13] | Ju. I. Manin, Cyclotomic fields and modular curves (in Russian), Uspekhi Mat. Nauk 26 , no. 6 (1971), 7–71. · Zbl 0266.14012 |

[14] | –. –. –. –., Parabolic points and zeta functions of modular curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. · Zbl 0248.14010 |

[15] | –. –. –. –., Periods of cusp forms, and \(p\)-adic Hecke series (in Russian), Mat. Sb. (N.S.) 92 ( 134 ) (1973), 378–401., 503. · Zbl 0293.14007 |

[16] | B. Mazur, Rational points of abelian varieties with values in towers of number fields , Invent. Math. 18 (1972), 183–266. · Zbl 0245.14015 |

[17] | –. –. –. –., Rational isogenies of prime degree , Invent. Math. 44 (1978), 129–162. · Zbl 0386.14009 |

[18] | B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves , Invent. Math. 25 (1974), 1–61. · Zbl 0281.14016 |

[19] | 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. · Zbl 0699.14028 |

[20] | A. G. Nasybullin, \(p\)-adic \(L\)-series of supersingular elliptic curves (in Russian), Funkcional. Anal. i Priložen. 8 , no. 1 (1974), 82–83. · Zbl 0304.14011 |

[21] | B. Perrin-Riou, Théorie d’Iwasawa \(p\)-adique locale et globale , Invent. Math. 99 (1990), 247–292. · Zbl 0715.11030 |

[22] | –. –. –. –., Fonctions \(L\) \(p\)-adiques d’une courbe elliptique et points rationnels , Ann. Inst. Fourier (Grenoble) 43 (1993), 945–995. · Zbl 0840.11024 |

[23] | ——–, Arithmétique des courbes elliptiques à réduction supersingulière en \(p\) , preprint, 2001, |

[24] | D. E. Rohrlich, On \(L\)-functions of elliptic curves and cyclotomic towers , Invent. Math. 75 (1984), 409–423. · Zbl 0565.14006 |

[25] | 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. |

[26] | J. H. Silverman, The Arithmetic of Elliptic Curves , Grad. Texts in Math. 106 , Springer, New York, 1992. |

[27] | M. M. Višik, Nonarchimedean measures associated with Dirichlet series (in Russian), Mat. Sb. (N.S.) 99 ( 141 ), no. 2 (1976), 248–260., 296. |

[28] | 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. |

[29] | A. Wiles, Modular elliptic curves and Fermat’s last theorem , Ann. of Math. (2) 141 (1995), 443–551. JSTOR: · Zbl 0823.11029 |

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