Download Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour by David Loeffler, Sarah Livia Zerbes PDF

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 70th 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.

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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).

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