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Base change and a problem of Serre. (English) Zbl 1016.11017
Let \(F\) be a totally real number field and \(f\) a Hilbert modular newform of (parallel) weight 2 and level n, associated to \(F\). Let \(\rho\) be the two-dimensional mod-\(p\)-representation of the absolute Galois group of \(F\) attached to \(f\). It is shown that if \(\rho\) is irreducible, there exists a solvable totally real extension \(F'\) of \(F\) such that the restriction \(\rho'\) of \(\rho\) to the absolute Galois group of \(F'\) still is irreducible and associated to a Hilbert modular newform \(g\) for \(F'\) with level dividing the product of the \(p\)-part of n and the primes at which \(\rho'\) ramifies. A similar result is said to hold in arbitrary parallel weight (which often can be replaced by weight 2 and treated that way). This is a weak version of level lowering for Hilbert modular forms. An alternative approach is pursued, e.g., by F. Jarvis [Math. Ann. 313, 141-160 (1999; Zbl 0978.11020)]. Nevertheless, the present result seems to be much simpler to prove and still sufficient for most arithmetic applications.

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