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Creators/Authors contains: "Lang, Jaclyn"

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  1. ; (, Proceedings of the American Mathematical Society, Series B)
    We show that for primes N , p ≥<#comment/> 5 N, p \geq 5 with N ≡<#comment/> −<#comment/> 1 mod p N \equiv -1 \bmod p , the class number of Q ( N 1 / p ) \mathbb {Q}(N^{1/p}) is divisible by p p . Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when N ≡<#comment/> −<#comment/> 1 mod p N \equiv -1 \bmod p , there is always a cusp form of weight 2 2 and level Γ<#comment/> 0 ( N 2 ) \Gamma _0(N^2) whose ℓ<#comment/> \ell th Fourier coefficient is congruent to ℓ<#comment/> + 1 \ell + 1 modulo a prime above p p , for all primes ℓ<#comment/> \ell . We use the Galois representation of such a cusp form to explicitly construct an unramified degree- p p extension of Q ( N 1 / p ) \mathbb {Q}(N^{1/p}) . 
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  2. ; ; (, Mathematische Annalen)
    Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation ρ of a profinite group on a local pro- p domain A contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for GL2-Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms. 
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  3. ; (, Transactions of the American Mathematical Society, Series B)
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