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Register a new userGalois representations, $(varphi, Gamma)$-modules and prismatic F-crystals
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Authors
Zhiyou Wu
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We prove that both local Galois representations and $(varphi,Gamma)$-modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and $(varphi,Gamma)$-modules.
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We give a new construction of $(varphi, hat G)$-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a different proof about the equivalence between the category of prismatic $F$-crystals in finite locally free $mathcal{O}_{Delta}$-modules over $mathrm{Spf}(mathcal{O}_K)$ and the category of lattices in crystalline representations of $G_K$, where $K$ is a complete discretely valued field of mixed characteristic with perfect residue field. We also propose a possible generalization of this result for semi-stable representations using the absolute logarithmic prismatic site.
Let $F$ be a finite extension of $mathbb{Q}_p$. We determine the Lubin-Tate $(varphi,Gamma)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${rm Gal}(bar{F}/F)$.
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].
We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
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