By Robert Alexander Adams

ISBN-10: 0120441500

ISBN-13: 9780120441501

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**Additional info for Sobolev Spaces (Pure and Applied Mathematics (Academic Press)) **

**Example text**

On anticyclotomic μ-invariants of modular forms. Compositio Math. 147(5) (2011) 19. : On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(2) (1990) 20. : Euler Systems (Hermann Weyl Lectures). In: Annals of Mathematics Studies, vol. 147. Princeton University Press (2000) 21. : Multiplicative reduction and the cyclotomic main conjecture for GL2 . Preprint (2014) 22. : The Iwasawa main conjecture for GL2 . Invent. Math. 195(1) (2014) 23. : Special values of anticyclotomic L-functions.

Since 1 , 2 and 3 are 2n-admissible primes, they are also r-admissible primes relative to ( f , K, p). For every divisor m of 1 2 3 write Selpr (K, fm ) ⊂ H 1 (K, Tf ,r ); Selpr (K, f ) := Sel(K, f{r} ) to denote the Selmer group attached to the reduction modulo pr of the mod-p2n form fm . e. the Selmer group defined by the same local conditions used to define Selpr (K, fm ) at every prime of K which does not divide L, and by imposing no local condition at every prime of K dividing L. As explained Congruences Between Modular Forms and the Birch .

Moreover, A/K = DK · A · AK , where ∗ is the real Néron period of the elliptic curve ∗/Q [11, P. 312]. Using (28), the preceding equation gives L (A/Q, 1) L(AK /Q, 1) · · = [A(K) : ZPK ]2 , NT A · h (P) AK (29) · where = denotes equality up to multiplication by a p-adic unit, and P is a generator of A(Q)/torsion. ) Step III. 3 (see in particular the discussion following Eq. (13)), the residual representation ρ A,p = ρ f is ramified at every prime q|N, hence ρ AK ,p is also ramified at every prime q|N.

### Sobolev Spaces (Pure and Applied Mathematics (Academic Press)) by Robert Alexander Adams

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