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