| Description |
I will give a few talks to explain R. Ramakrishna's method of lifting two dimensional, odd, irreducible mod $p$ representations of the absolute Galois group of the rationals to geometric characteristic 0 representations. In symbols, given $\bar{\rho}:G_\Q \rightarrow GL_2(\Z/p\Z)$ that is odd (determinant of image of complex conjugation is -1) and irreducible, we want to construct representations $\rho:G_\Q \rightarrow GL_2(Z_p)$ which reduce mod $p$ to $\bar{\rho}$, and such that $\bar{\rho}$ is ``geometric'' (ramified at finitely many primes and de Rham at $p$). The method generalizes to higher dimensional representations as well. It relies on facts in Galois cohomology which I will state without proof (Poitou-Tate duality, Euler-Poincare formula etc). This also has applications to proving modularity lifting theorems by a method which is different from the one of Wiles and Taylor.
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