THE ANTICYCLOTOMIC MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES

The anticyclotomic Main Conjecture for elliptic curves at supersingular primes

The Main Conjecture of Iwasawa theory for an elliptic curve E over Q and the anticyclotomic Zp-extension of an imaginary quadratic field K was studied in [BD2], in the case where p is a prime of ordinary reduction for E. Analogous results are formulated, and proved, in the case where p is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by...

The main conjecture for CM elliptic curves at supersingular primes

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is ...

Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic Zp-extensions

The main conjecture of Iwasawa theory for elliptic curves with complex multiplication over abelian extensions at supersingular primes

We develop the plus/minus p-Selmer group theory and plus/minus padic L-function theory for an elliptic curve E with complex multiplication over an abelian extension F of the imaginary quadratic field K given by the complex multiplication of E when p is a prime inert over K/Q (i.e. supersingular). As a result, we prove that the characteristic ideal of the Pontryagin dual of the plus/minus p-Selm...

Tate–Shafarevich groups in anticyclotomic p-extensions at supersingular primes

Let E/Q be an elliptic curve and p a prime of supersingular reduction for E. Denote by K∞ the anticyclotomic Zp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/Q, we prove that X(K∞, E)p∞ has trivial Λ-corank and, in the process, also show that HSel(K∞, Ep∞) and E(K∞)⊗Qp/Zp both have Λ-corank two.