Iwasawa theory of elliptic curves at supersingular primes over ℤ p -extensions of number fields

2006 ◽  
Vol 2006 (598) ◽  
Author(s):  
Adrian Iovita ◽  
Robert Pollack
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Supersingular Primes

scholarly journals Iwasawa theory for modular forms at supersingular primes

2011 ◽  
Vol 147 (3) ◽  
pp. 803-838 ◽  
Author(s):  
Antonio Lei
Keyword(s):  
Elliptic Curves ◽  
Modular Forms ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Main Conjecture ◽  
Supersingular Primes

AbstractWe generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

scholarly journals Growth of in towers for isogenous curves

2015 ◽  
Vol 151 (11) ◽  
pp. 1981-2005 ◽  
Author(s):  
Tim Dokchitser ◽  
Vladimir Dokchitser
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Growth Types

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

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