## 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

Zbl 0978.11020
Full Text:

### References:

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