On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime. (English) Zbl 1074.11061

Pollack, Robert

Duke Math. J. 118, No. 3, 523-558 (2003).

In 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 Weierstraßpreparation. 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

Keywords:

L-functions; elliptic curves; trivial zeros; Iwasawa invariants

Software:

ecdata

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 · doi:10.1090/S0894-0347-01-00370-8
[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 · doi:10.2307/121003
[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 · doi:10.1007/BFb0093453
[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 · doi:10.1007/s00222-002-0265-4
[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 · doi:10.1007/s002220100206
[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 · doi:10.1007/BF02684326
[13]	Ju. I. Manin, Cyclotomic fields and modular curves (in Russian), Uspekhi Mat. Nauk 26 , no. 6 (1971), 7–71. · Zbl 0266.14012 · doi:10.1070/rm1971v026n06ABEH001272
[14]	–. –. –. –., Parabolic points and zeta functions of modular curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. · Zbl 0248.14010 · doi:10.1070/IM1972v006n01ABEH001867
[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 · doi:10.1007/BF01389815
[17]	–. –. –. –., Rational isogenies of prime degree , Invent. Math. 44 (1978), 129–162. · Zbl 0386.14009 · doi:10.1007/BF01390348
[18]	B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves , Invent. Math. 25 (1974), 1–61. · Zbl 0281.14016 · doi:10.1007/BF01389997
[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 · doi:10.1007/BF01388731
[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 · doi:10.1007/BF02028316
[21]	B. Perrin-Riou, Théorie d’Iwasawa \(p\)-adique locale et globale , Invent. Math. 99 (1990), 247–292. · Zbl 0715.11030 · doi:10.1007/BF01234420
[22]	–. –. –. –., Fonctions \(L\) \(p\)-adiques d’une courbe elliptique et points rationnels , Ann. Inst. Fourier (Grenoble) 43 (1993), 945–995. · Zbl 0840.11024 · doi:10.5802/aif.1362
[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 · doi:10.1007/BF01388636
[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 · doi:10.2307/2118559