Iwasawa theory for elliptic curves at supersingular primes.

*(English)*Zbl 1047.11105Let \(p\) be an odd prime, \({\mathbb Q}_{\infty} = \bigcup_{n}\;F_{n}\) the cyclotomic \({\mathbb Z}_{p}\)-extension of \({\mathbb Q},\) \(\wedge\) the usual Iwasawa algebra. In the Iwasawa theory of elliptic curves at good ordinary primes, the Main Conjecture states that the Selmer group over \({\mathbb Q}_{\infty}\) is \(\wedge\)-cotorsion and the characteristic ideal is generated by the Mazur and Swinnerton-Dyer \(p\)-adic \(L\)-function. But at supersingular primes, the Selmer group is no longer cotorsion and the \(p\)-adic \(L\)-function does not live in \(\wedge \otimes \overline{\mathbb Q}_{p}.\)

In this paper, the author proposes a new formulation at good supersingular primes by modifying both the Selmer group and the \(p\)-adic \(L\)-function. More precisely, let \(E\) be an elliptic curve over \({\mathbb Q}\) with good supersingular reduction at \(p,\) and such that \(a_{p} =0\) (this is automatically satisfied for a supersingular \(p > 3).\) The modified Selmer groups \(\text{Sel}^{\pm}(E/F_{n})\) are defined by replacing the groups of local points by the modified groups \(E^{\pm}(F_{n, p})\) introduced by B. Perrin-Riou [Invent. Math. 99, No. 2, 247–292 (1990; Zbl 0715.11030)] and H. Knospe [Manuscr. Math. 87, 225–258 (1995; Zbl 0847.14026)]. The relevant analytical objects are Pollack’s \(p\)-adic \(L\)-functions \({\mathcal L}^{\pm}_{p} (E, X),\) which interpolate special values of the Hasse-Weil \(L\)-function and live in \(\wedge.\) The author shows that the modified Selmer groups \(\text{Sel}^{\pm}(E/{\mathbb Q}_{\infty})\) are \(\wedge\)-cotorsion, and he conjectures that char \((\text{Sel}^{\pm}(E/{\mathbb Q}_{\infty})^{\vee}) = ({\mathcal L}_{p}^{\pm} (E, X)).\) He shows that this supersingular Main Conjecture (say SSMC) is equivalent to the conjecture formulated in cohomological terms (for general motives) by K. Kato [Arithmetic algebraic geometry. Lect. Notes Math. 1553, 50–163 (1993; Zbl 0815.11051)] and Perrin-Riou (loc. cit.)].

The SSMC was recently proved in the CM case by R. Pollack and K. Rubin [Ann. Math. (2) 159, No. 1, 447–464 (2004; Zbl 1082.11035)]. Here, in the non-CM case, the author shows half of the SSMC, namely that \({\mathcal L}^{\pm}_{p}(E, X)\) lives in the corresponding characteristic ideal. He also derives an asymptotic formula for the \(p\)-adic order of the Tate-Shafarevich groups \(\text{ Ш} (E/F_{n})\) (assuming their finiteness) involving the \(\lambda\)-and \(\mu\)-invariants of \(\text{ Sel}^{\pm}(E/F_{\infty})^{\vee}.\) This improves upon similar formulas obtained previously by Kurihara and Perrin-Riou with unspecified rational numbers \(\lambda\) and \(\mu\); here, the invariants are specified as \(\lambda_{\pm}\) and \(\mu_{\pm}\) as in Pollack’s analytic counterpart.

The key point is the construction of \(\wedge\)-valued \(\pm\) Coleman maps which send Kato’s zeta element to Pollack’s \(p\)-adic \(L\)-function. Once these maps are constructed, the proofs proceed in the same way as in the good ordinary case.

In this paper, the author proposes a new formulation at good supersingular primes by modifying both the Selmer group and the \(p\)-adic \(L\)-function. More precisely, let \(E\) be an elliptic curve over \({\mathbb Q}\) with good supersingular reduction at \(p,\) and such that \(a_{p} =0\) (this is automatically satisfied for a supersingular \(p > 3).\) The modified Selmer groups \(\text{Sel}^{\pm}(E/F_{n})\) are defined by replacing the groups of local points by the modified groups \(E^{\pm}(F_{n, p})\) introduced by B. Perrin-Riou [Invent. Math. 99, No. 2, 247–292 (1990; Zbl 0715.11030)] and H. Knospe [Manuscr. Math. 87, 225–258 (1995; Zbl 0847.14026)]. The relevant analytical objects are Pollack’s \(p\)-adic \(L\)-functions \({\mathcal L}^{\pm}_{p} (E, X),\) which interpolate special values of the Hasse-Weil \(L\)-function and live in \(\wedge.\) The author shows that the modified Selmer groups \(\text{Sel}^{\pm}(E/{\mathbb Q}_{\infty})\) are \(\wedge\)-cotorsion, and he conjectures that char \((\text{Sel}^{\pm}(E/{\mathbb Q}_{\infty})^{\vee}) = ({\mathcal L}_{p}^{\pm} (E, X)).\) He shows that this supersingular Main Conjecture (say SSMC) is equivalent to the conjecture formulated in cohomological terms (for general motives) by K. Kato [Arithmetic algebraic geometry. Lect. Notes Math. 1553, 50–163 (1993; Zbl 0815.11051)] and Perrin-Riou (loc. cit.)].

The SSMC was recently proved in the CM case by R. Pollack and K. Rubin [Ann. Math. (2) 159, No. 1, 447–464 (2004; Zbl 1082.11035)]. Here, in the non-CM case, the author shows half of the SSMC, namely that \({\mathcal L}^{\pm}_{p}(E, X)\) lives in the corresponding characteristic ideal. He also derives an asymptotic formula for the \(p\)-adic order of the Tate-Shafarevich groups \(\text{ Ш} (E/F_{n})\) (assuming their finiteness) involving the \(\lambda\)-and \(\mu\)-invariants of \(\text{ Sel}^{\pm}(E/F_{\infty})^{\vee}.\) This improves upon similar formulas obtained previously by Kurihara and Perrin-Riou with unspecified rational numbers \(\lambda\) and \(\mu\); here, the invariants are specified as \(\lambda_{\pm}\) and \(\mu_{\pm}\) as in Pollack’s analytic counterpart.

The key point is the construction of \(\wedge\)-valued \(\pm\) Coleman maps which send Kato’s zeta element to Pollack’s \(p\)-adic \(L\)-function. Once these maps are constructed, the proofs proceed in the same way as in the good ordinary case.

Reviewer: Thong Nguyen Quang Do (Besançon)