# Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

@article{Fakhruddin2021RelativeDT, title={Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations}, author={Najmuddin Fakhruddin and Chandrashekhar Khare and Stefan Patrikis}, journal={Duke Mathematical Journal}, year={2021} }

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\bar{\rho}$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

#### One Citation

Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory

- Mathematics
- 2020

Let E/Q be an elliptic curve of conductor N and let K be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for E using K-CM points and… Expand