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Iwasawa theory for elliptic curves at supersingular primes. (English) Zbl 1047.11105
Let $$p$$ 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.

##### MSC:
 11R23 Iwasawa theory 11G05 Elliptic curves over global fields
##### Keywords:
supersingular Main Conjecture
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