Iwasawa theory for elliptic curves at supersingular primes

A pair of main conjectures

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

Text: We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case a p≠0, where a p is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions L p # and L p with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when a p=0 and p is odd. We then generalize Kobayashi's methods to define two Selmer groups Sel # and Sel and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our p-adic L-functions L p # and L p . We then use results by Kato to prove a divisibility statement. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.

Original languageEnglish (US)
Pages (from-to)1483-1506
Number of pages24
JournalJournal of Number Theory
Volume132
Issue number7
DOIs
StatePublished - Jul 1 2012
Externally publishedYes

Fingerprint

Iwasawa Theory
P-adic L-function
Elliptic Curves
Selmer Group
Divisibility
Frobenius
Odd
Trace
Generalise
Formulation

Keywords

  • Elliptic curves
  • Iwasawa theory
  • Supersingular primes

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Iwasawa theory for elliptic curves at supersingular primes : A pair of main conjectures. / Sprung, Florian.

In: Journal of Number Theory, Vol. 132, No. 7, 01.07.2012, p. 1483-1506.

Research output: Contribution to journalArticle

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AB - Text: We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case a p≠0, where a p is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions L p # and L p ≠ with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when a p=0 and p is odd. We then generalize Kobayashi's methods to define two Selmer groups Sel # and Sel ≠ and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our p-adic L-functions L p # and L p ≠. We then use results by Kato to prove a divisibility statement. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.

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