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On the density of modular points in universal deformation spaces. (English) Zbl 0984.11025
Using isomorphism theorems between universal deformation rings and Hecke algebras obeying strong conditions at the prime number \(\ell\), which are due to Wiles, Taylor and Diamond, the author extends the validity of results of F. Gouvêa and B. Mazur [in Buell, D. A. (ed.) et al., Computational perspectives in number theory. AMS/IP Stud. Adv. Math. 7, 127–142 (1998; Zbl 1134.11324)].
He proves that for an irreducible continuous representation of the absolute Galois group \(G_{\mathbb Q}\) in \(\text{GL}(2,\overline{\mathbb F}_\ell)\) satisfying mild technical assumptions, the universal deformation space corresponds to a certain Hecke-algebra of \(\ell\)-adic modular forms. Moreover, these spaces are complete intersections, flat over \({\mathbb Z}_\ell\) of relative dimension three, and the set of modular points is Zariski-dense.
As always, the author’s style is very precise and the paper gives a clear introduction to the whole matter.

MSC:
11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
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