Iwasawa theory for elliptic curves at supersingular primes: a pair of Main Conjectures. (English) Zbl 1284.11147

Sprung, Florian E. Ito

J. Number Theory 132, No. 7, 1483-1506 (2012).

Let \(E/\mathbb Q\) be an elliptic curve and let \(p\) be a prime. The aim of the paper under review is to formulate a pair of main conjectures in the case, where \(p\) is a good supersingular prime, i.e. \(p\) divides \(a_p = p+1-|E(\mathbb F_p)|\).
When \(a_p = 0\), R. Pollack [Duke Math. J. 118, No. 3, 523–558 (2003; Zbl 1074.11061)] has constructed two \(p\)-adic \(L\)-functions \(L_p^{\pm}(E,X) \in \mathbb Z_p[[X]]\) such that (for \(p\) odd) \[ L_p(E,\alpha, X) = L_p^+(E,X) \mathrm{log}_p^+(1+X) + L_p^-(E,X) \mathrm{log}_p^-(1+X) \alpha \] for each root \(\alpha\) of the Hecke polynomial \(Y^2 - a_p Y + p\). Here \(\mathrm{log}_p^{\pm}\) are Pollack’s half-logarithms. Then S.-i. Kobayashi [Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)] has constructed two modified Selmer groups \(\mathrm{Sel}^{\pm}\) and has formulated a pair of main conjectures: each of Pollack’s \(L_p^{\pm}(E,X)\) should generate the characteristic ideal of the Pontryagin dual of \(\mathrm{Sel}^{\pm}\).
In this paper the author allows the case \(a_p \not=0\) as well (but note that the Hasse-Weil bound \(|a_p| \leq 2 \sqrt p\) then forces \(p=2\) or \(p=3\), and the former case is excluded in the last section of the paper, where the main conjecture is formulated). The key construction is that of two Coleman maps \(\mathrm{Col}^{\sharp}\) and \(\mathrm{Col}^{\flat}\) which generalize Kobayashi’s Coleman maps. These send K. Kato’s zeta elements [Astérisque 295, 117–290 (2004; Zbl 1142.11336)] to new \(p\)-adic \(L\)-functions \(L_p^{\sharp}(E,X)\) and \(L_p^{\flat}(E,X)\) which generalize Pollack’s \(p\)-adic \(L\)-functions. At least one of these two is nonzero, and conjecturally both are.
Now let \(p\) be an odd prime. The main conjecture then asserts that \(L_p^{\sharp}(E,X)\) and \(L_p^{\flat}(E,X)\) (when nonzero) generate the characteristic ideals of the Pontryagin dual of two modified Selmer groups \(\mathrm{Sel}^{\sharp}(E / \mathbb Q_{\infty})\) and \(\mathrm{Sel}^{\flat}(E / \mathbb Q_{\infty})\), respectively. When \(E\) does not have complex multiplication, and the \(p\)-adic representation \(Gal(\overline{\mathbb Q} / \mathbb Q) \rightarrow GL_{\mathbb Z_p}(T)\) on the automorphism group of the \(p\)-adic Tate module \(T\) is surjective, then the divisibility statement \[ (L_p^{\ast}(E,X)) \subseteq \mathrm{char}((\mathrm{Sel}^{\ast}(E/\mathbb Q_{\infty})^{\vee}) \] is proven, where \(\ast \in \{\sharp, \flat \}\) so that \(L_p^{\ast}(E,X) \not= 0\).
For an application on the asymptotic growth of the \(p\)-primary component of the Shafarevich-Tate group in the cyclotomic tower see F. Sprung [J. Reine Angew. Math. 681, 199–218 (2013; Zbl 1288.11060)].

Reviewer: Andreas Nickel (Bielefeld)

MSC:

11R23	Iwasawa theory
11G05	Elliptic curves over global fields

Keywords:

elliptic curve; supersingular prime; Iwasawa main conjecture

Citations:

Zbl 1074.11061; Zbl 1047.11105; Zbl 1142.11336; Zbl 1288.11060

References:

[1]	Amice, Y.; Vélu, J., Distributions \(p\)-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux. Journées Arithmétiques de Bordeaux, Bordeaux, 1974. Journées Arithmétiques de Bordeaux. Journées Arithmétiques de Bordeaux, Bordeaux, 1974, Astérisque, 24-25, 119-131 (1975) · Zbl 0332.14010
[2]	Bhargava, M.; Shankar, A., Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 · Zbl 1317.11038
[3]	Coates, J.; Greenberg, R., Kummer theory for abelian varieties over local fields, Invent. Math., 124, 1-3, 129-174 (1996) · Zbl 0858.11032
[4]	Coates, J.; Sujatha, R., Galois Cohomology of Elliptic Curves, Tata Inst. Fund. Res. Stud. Math., vol. 88 (2000), Narosa Publishing House: Narosa Publishing House New Delhi · Zbl 0973.11059
[5]	Colmez, P., Théorie dʼIwasawa des représentations de de Rham dʼun corps local, Ann. of Math., 148, 2, 485-571 (1998) · Zbl 0928.11045
[6]	Elkies, N., The existence of infinitely many supersingular primes for every elliptic curve over \(Q\), Invent. Math., 89, 3, 561-567 (1987) · Zbl 0631.14024
[7]	Hazewinkel, M., On norm maps for one dimensional formal groups I: The cyclotomic \(Γ\)-extension, J. Algebra, 32, 89-108 (1974) · Zbl 0288.12011
[8]	Honda, T., On the theory of commutative formal groups, J. Math. Soc. Japan, 22, 213-246 (1970) · Zbl 0202.03101
[9]	Iovita, A.; Pollack, R., Iwasawa theory of elliptic curves at supersingular primes over \(Z_p\)-extensions of number fields, J. Reine Angew. Math., 598, 71-103 (2006) · Zbl 1114.11053
[10]	Kato, K., \(p\)-Adic Hodge theory and values of zeta functions of modular forms, Astérisque, 295, 117-290 (2004) · Zbl 1142.11336
[11]	Kobayashi, S., Iwasawa theory for elliptic curves at supersingular primes, Invent. Math., 152, 1, 1-36 (2003) · Zbl 1047.11105
[12]	Kurihara, M., On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, Invent. Math., 149, 195-224 (2002) · Zbl 1033.11028
[13]	Lazard, M., Les zéros des fonctions analytiques dʼune variable sur un corps valué complet, Publ. Math. Inst. Hautes Etudes Sci., 14, 47-75 (1962) · Zbl 0119.03701
[14]	Lei, A., Iwasawa theory for modular forms at supersingular primes, Compos. Math., 147, 3, 803-838 (May 2011) · Zbl 1234.11148
[15]	Lei, A.; Loeffler, D.; Zerbes, S., Wach modules and Iwasawa theory for modular forms, Asian J. Math., 14, 4, 475-528 (December 2010) · Zbl 1281.11095
[16]	Mazur, B., Courbes elliptiques et symboles modulaires, (Séminaire Bourbaki 414, 1971/1972. Séminaire Bourbaki 414, 1971/1972, Lecture Notes in Math., vol. 317 (1972), Springer) · Zbl 0276.14012
[17]	Mazur, B., Rational points of abelian varieties with values in towers of number fields, Invent. Math., 18, 183-266 (1972) · Zbl 0245.14015
[18]	Mazur, B.; Swinnerton-Dyer, P., Arithmetic of Weil curves, Invent. Math., 25, 1-61 (1974) · Zbl 0281.14016
[19]	Mazur, B.; Tate, J.; Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., 84, 1-48 (1986) · Zbl 0699.14028
[20]	Perrin-Riou, B., Fonctions L p-adiques des représentations \(p\)-adiques, Astérisque, 229 (1995), 198 pp · Zbl 0845.11040
[21]	Pollack, R., The \(p\)-adic \(L\)-function of a modular form at a supersingular prime, Duke Math. J., 118, 3, 523-558 (2003) · Zbl 1074.11061
[22]	Pollack, R., An algebraic version of a theorem of Kurihara, J. Number Theory, 110, 164-177 (2005) · Zbl 1143.11326
[23]	Pollack, R.; Rubin, K., The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math., 159, 1, 447-464 (2004) · Zbl 1082.11035
[24]	Rohrlich, D. E., On \(L\)-functions of elliptic curves and cyclotomic towers, Invent. Math., 75, 409-423 (1984) · Zbl 0565.14006
[25]	Silverman, J. H., The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106 (1992), Springer: Springer New York
[26]	C. Skinner, E. Urban, The Iwasawa main conjectures for \(\operatorname{GL}_2\); C. Skinner, E. Urban, The Iwasawa main conjectures for \(\operatorname{GL}_2\) · Zbl 1301.11074
[27]	Višik, M. M., Nonarchimedean measures associated with Dirichlet series, Mat. Sb., 99 (141), 2, 248-260 (1976), 296 · Zbl 0358.14014
[28]	Washington, L., Introduction to Cyclotomic Fields, Grad. Texts in Math., vol. 83 (1980), Springer: Springer New York