By David Loeffler, Sarah Livia Zerbes

Celebrating one of many prime figures in modern quantity thought – John H. Coates – at the celebration of his seventieth birthday, this number of contributions covers a number subject matters in quantity concept, focusing on the mathematics of elliptic curves, modular types, and Galois representations. a number of of the contributions during this quantity have been provided on the convention *Elliptic Curves, Modular varieties and Iwasawa Theory*, held in honour of the 70^{th } birthday of John Coates in Cambridge, March 25-27, 2015. the most unifying topic is Iwasawa conception, a box that John Coates himself has performed a lot to create.

This assortment is necessary analyzing for researchers in Iwasawa concept, and is attention-grabbing and worthwhile for these in lots of similar fields.

**Read Online or Download Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015 PDF**

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**Additional info for Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015**

**Example text**

1, we need the following lifting theorem, proved in Part 3 of [6]. The notations and assumptions are as in the previous sections; in particular p deg(πA ). 7 Let 1 be a 2n-admissible prime relative to ( f , K, p). Assume that f 1 cannot be lifted to a true modular form. Then there exists infinitely many pairs ( 2 , 3 ) of 2n-admissible primes such that: 1. f 1 2 3 ∈ S2 (N, 1 2 3 ; Z/p2n Z) can be lifted to a true modular form g := g 1 2 3 ∈ S2 (N, 1 2 3 ; Zp ), 2. v 3 κ( 1 2 ) = 0 if and only if κ( 1 2 ) = 0.

5 Algebraicity of Special L-Values In this section we present some algebraicity results on the special values of the L-function introduced above, which were obtained in [6]. Results of this kind have been obtained by Shimura [30], but over the algebraic closure of Q, and in [6] we worked out the precise field of definition, as well as, the reciprocity properties. There is also work by Harris [18, 19] and we refer to [6] for a discussion of how the results there compare with the ones presented here.

2]). 6]). For a q ∈ G L n (K )A and an s ∈ SA we have f q s qˆ 0 qˆ cf (τ, q)eA (τ s). 2] and for the definition of eA to [30, p. 127]. We also note that sometimes we may write c(τ, q; f) for cf (τ, q). 38 Th. Bouganis For a subfield L of C we will be writing Mk ( q , ψ, L) for the Hermitian modular forms in Mk ( q , ψ) whose Fourier expansion at infinity, that is γ is the identity in Eq. 1, has coefficients in L. For a fixed set B as above we will be writing Mk (D, ψ, L) for the subspace of Mk (D, ψ) consisting of elements whose image under the above isomorphism lies in ⊕b∈B Mk ( b , ψb , L).