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\begin{thebibliography}{10}

\bibitem{AllenEtAl2023}
Patrick~B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao~V.
  Le~Hung, James Newton, Peter Scholze, Richard Taylor, and Jack~A. Thorne,
  \emph{Potential automorphy over {CM} fields}, Annals of Mathematics
  \textbf{197} (2023), no.~3, 897{\textendash}1113.

\bibitem{AtiyahMacdonald1969}
Michael~F. Atiyah and Ian~G. Macdonald, \emph{Introduction to commutative
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\bibitem{BarnetLambGeeGeraghtyTaylor2014}
Thomas Barnet~Lamb, Toby Gee, David Geraghty, and Richard Taylor,
  \emph{Potential automorphy and change of weight}, Annals of Mathematics
  \textbf{179} (2014), no.~2, 501{\textendash}609.

\bibitem{BellaicheChenevier2009}
Jo{\"e}l Bella{\"i}che and Ga{\"e}tan Chenevier, \emph{Families of {Galois}
  representations and {Selmer} groups}, Ast{\'e}risque, vol. 324,
  Soci{\'e}t{\'e} Math{\'e}matique de France, Paris, 2009.

\bibitem{BrauerNesbitt1941}
Richard Brauer and Cecil Nesbitt, \emph{On the modular characters of groups},
  Annals of Mathematics \textbf{42} (1941), no.~2, 556{\textendash}590.

\bibitem{CalegariGeraghty2018}
Frank Calegari and David Geraghty, \emph{Modularity lifting beyond the {Taylor
  Wiles} method}, Inventiones Mathematicae \textbf{211} (2018), no.~1,
  297{\textendash}433.

\bibitem{CaraianiScholze2017}
Ana Caraiani and Peter Scholze, \emph{On the generic part of the cohomology of
  compact unitary {Shimura} varieties}, Annals of Mathematics \textbf{186}
  (2017), no.~3, 649{\textendash}766.

\bibitem{ClozelHarrisTaylor2008}
Laurent Clozel, Michael Harris, and Richard Taylor, \emph{Automorphy for some
  {$l$} adic lifts of automorphic mod {$l$} {Galois} representations},
  Publications Math{\'e}matiques de l'IH{\'E}S \textbf{108} (2008),
  1{\textendash}181.

\bibitem{Diamond1997}
Fred Diamond, \emph{The {Taylor Wiles} construction and multiplicity one},
  Inventiones Mathematicae \textbf{128} (1997), no.~2, 379{\textendash}391.

\bibitem{Diestel2017}
Reinhard Diestel, \emph{Graph theory}, fifth ed., Graduate Texts in
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\bibitem{Eisenbud1995}
David Eisenbud, \emph{Commutative algebra with a view toward algebraic
  geometry}, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995.

\bibitem{Gee2022}
Toby Gee, \emph{Modularity lifting theorems}, Essential Number Theory
  \textbf{1} (2022), no.~1, 73{\textendash}126.

\bibitem{GuralnickHerzigTaylorThorne2012}
Robert Guralnick, Florian Herzig, Richard Taylor, and Jack~A. Thorne,
  \emph{Adequate subgroups}, Journal of the Institute of Mathematics of Jussieu
  \textbf{11} (2012), no.~4, 907{\textendash}920, Appendix to Jack A. Thorne,
  On the automorphy of l-adic Galois representations with small residual image.

\bibitem{Kisin2009}
Mark Kisin, \emph{Moduli of finite flat group schemes, and modularity}, Annals
  of Mathematics \textbf{170} (2009), no.~3, 1085{\textendash}1180.

\bibitem{Matsumura1989}
Hideyuki Matsumura, \emph{Commutative ring theory}, second ed., Cambridge
  Studies in Advanced Mathematics, vol.~8, Cambridge University Press,
  Cambridge, 1989.

\bibitem{Mazur1989}
Barry Mazur, \emph{Deforming {Galois} representations}, Galois Groups over
  {$\mathbb Q$} (Yasutaka Ihara, Kenneth Ribet, and Jean~Pierre Serre, eds.),
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  York, 1989, p.~385{\textendash}437.

\bibitem{MiagkovThorne2023}
Konstantin Miagkov and Jack~A. Thorne, \emph{Automorphy lifting with adequate
  image}, Forum of Mathematics, Sigma \textbf{11} (2023), e8, 31 pages.

\bibitem{Neukirch1999}
J{\"u}rgen Neukirch, \emph{Algebraic number theory}, Grundlehren der
  Mathematischen Wissenschaften, vol. 322, Springer, Berlin, 1999.

\bibitem{NewtonThorne2016}
James Newton and Jack~A. Thorne, \emph{Torsion {Galois} representations over
  {CM} fields and {Hecke} algebras in the derived category}, Forum of
  Mathematics, Sigma \textbf{4} (2016), e21, 88 pages.

\bibitem{RibesZalesskii2010}
Luis Ribes and Pavel Zalesskii, \emph{Profinite groups}, second ed., Ergebnisse
  der Mathematik und ihrer Grenzgebiete, vol.~40, Springer, Berlin, 2010.

\bibitem{Schlessinger1968}
Michael Schlessinger, \emph{Functors of {Artin} rings}, Transactions of the
  American Mathematical Society \textbf{130} (1968), no.~2,
  208{\textendash}222.

\bibitem{Scholze2015}
Peter Scholze, \emph{On torsion in the cohomology of locally symmetric
  varieties}, Annals of Mathematics \textbf{182} (2015), no.~3,
  945{\textendash}1066.

\bibitem{Serre1977}
Jean~Pierre Serre, \emph{Linear representations of finite groups}, Graduate
  Texts in Mathematics, vol.~42, Springer, New York, 1977.

\bibitem{TaylorWiles1995}
Richard Taylor and Andrew Wiles, \emph{Ring theoretic properties of certain
  {Hecke} algebras}, Annals of Mathematics \textbf{141} (1995), no.~3,
  553{\textendash}572.

\bibitem{Thorne2012}
Jack~A. Thorne, \emph{On the automorphy of {$l$} adic {Galois} representations
  with small residual image}, Journal of the Institute of Mathematics of
  Jussieu \textbf{11} (2012), no.~4, 855{\textendash}920.

\bibitem{Weibel1994}
Charles~A. Weibel, \emph{An introduction to homological algebra}, Cambridge
  Studies in Advanced Mathematics, vol.~38, Cambridge University Press,
  Cambridge, 1994.

\bibitem{Wiles1995}
Andrew Wiles, \emph{Modular elliptic curves and {Fermat}'s last theorem},
  Annals of Mathematics \textbf{141} (1995), no.~3, 443{\textendash}551.

\end{thebibliography}