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The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
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Explicitly computing D(V)^{psi=1} for (phi,Gamma)-modules
In the paper Construction of some families of 2-dimensional crystalline representations by Berger-Li-Zhu, a very explicit description is given for the $(\phi,\Gamma)$-module attached to some 2-dimensi …
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Companion forms
What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? Ar …