The main conjecture for CM elliptic curves at supersingular primes
By Robert H. Pollack, Karl Rubin. 2005
58 citations recorded by the source corpus.
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 similar in structure to the ordinary case. Namely, Kobayashi's conjecture relates modified Selmer groups, which he defined, with modified padic L-functions defined by the first author. In this paper we prove Kobayashi's conjecture for elliptic curves with complex multiplication.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2fe24dc501b635ab5048bd9d3d03be302b66e581
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Iwasawa theory for modular forms at supersingular primes (2009, methodology): “We will now follow the strategy of [14] to prove the “main conjecture” for the case of f defined over Q and ǫ = 1 in the CM case.”
- Wach modules and Iwasawa theory for modular forms (2009, background): “c Galois representation of Eis not surjective), establishing half of the main conjecture. (When the elliptic curve has complex multiplication, the full conjecture has been proved by Pollack and Rubin [PoR04].) Sprung [Spr09] has extended the results of Kobayashi to elliptic curves with supersingular”
- Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures (2009, methodology): “prove our main theorem. In Section 7, we give a formulation of a pair of main conjectures for any odd supersingular prime, each in the spirit of Mazur’s. Each is equivalent Kato’s [Ka], Perrin-Riou’s [PR], or Kurihara’s [Ku] main conjecture. Let pbe the prime ideal of Q∞above p, and Sel♯(E/Q ∞) := K”
- Iwasawa Main Conjecture for Supersingular Elliptic Curves (2014, background): “Before this work the only result for ±-main conjecture is due to Pollack-Rubin ([15]) for CM elliptic curves.”