essay.bib

@article{BDJ,
  title={On {S}erre's conjecture for mod $\ell$ {G}alois representations over totally real fields},
  author={Buzzard, Kevin and Diamond, Fred and Jarvis, Frazer},
  journal={Duke Mathematical Journal},
  volume={155},
  number={1},
  pages={105--161},
  year={2010},
  publisher={Duke University Press}
}

@article{BCDT,
  title={On the modularity of elliptic curves over {$\QQ$}: wild 3-adic exercises},
  author={Breuil, Christophe and Conrad, Brian and Diamond, Fred and Taylor, Richard},
  journal={Journal of the American Mathematical Society},
  pages={843--939},
  year={2001},
  publisher={American Mathematical Society}
}

@inproceedings{Birch,
  title={Cyclotomic fields and {K}ummer extensions},
  author={Birch, Brian J.},
  booktitle={Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, DC},
  pages={85--93},
  year={1967}
}

@unpublished{Cais,
  title={Serre's Conjectures},
  author={Cais, Bryden},
  note={\url{http://math.arizona.edu/~cais/Papers/Expos/Serre05.pdf}},
  month={November},
  year={2009},
}

@article{Carayol,
  title={Sur les repr{\'e}sentations $ l $-adiques associ{\'e}es aux formes modulaires de {H}ilbert},
  author={Carayol, Henri},
  journal={Annales Scientifiques de l'Ecole Normale Sup{\'e}rieure},
  volume={19},
  number={3},
  pages={409--468},
  year={1986},
  organization={Soci{\'e}t{\'e} math{\'e}matique de France}
}

@book{Cohen,
  title={Number Theory: Volume I: Tools and Diophantine Equations},
  author={Cohen, Henri},
  isbn={9780387499239},
  lccn={2007925736},
  series={Graduate Texts in Mathematics},
  url={https://books.google.co.uk/books?id=qxCWoFO-oxYC},
  year={2008},
  publisher={Springer New York}
}

@inproceedings{Darmon,
  title={Serre's conjectures},
  author={Darmon, Henri},
  booktitle={Seminar on Fermat's Last Theorem: 1993-1994, the Fields Institute for Research in the Mathematical Sciences, Toronto, Ontario, Canada},
  volume={17},
  pages={135--153},
  year={1995},
  editor={Vijaya Kumar Murty},
  isbn={9780821803134},
  lccn={95036469},
  series={CMS conference proceedings},
  url={https://books.google.co.uk/books?id=m1Pe7XK1e5MC},
  publisher={American Mathematical Society}
}

@inproceedings{DDT,
  title={Fermat's last theorem},
  author={Darmon, Henri and Diamond, Fred and Taylor, Richard},
  pages={1--154},
  booktitle={Elliptic curves, modular forms, and Fermat's last theorem},
  editor={J. Coates and S. Yau},
  year={1995}
}

@article{DeligneSerre,
	author={Deligne, Pierre and Serre, Jean-Pierre},
	journal={Annales scientifiques de l'École Normale Supérieure},
	language={fre},
	number={4},
	pages={507-530},
	publisher={Soci\'et\'e math\'ematique de France},
	title={Formes modulaires de poids $1$},
	url={http://eudml.org/doc/81946},
	volume={7},
	year={1974},
}

@article{Deligne,
	author={Deligne, Pierre},
	journal={S\'eminaire Bourbaki},
	language={fre},
	pages={139-172},
	publisher={Springer-Verlag},
	title={Formes modulaires et repr\'esentations $\ell$-adiques},
	url={http://eudml.org/doc/109756},
	volume={11},
	year={1968-1969},
}

@unpublished{DeligneEng,
	title={Formes modulaires et repr\'esentations $\ell$-adiques (translation)},
	author={Deligne, Pierre},
  note={\url{http://publications.ias.edu/sites/default/files/5_FormesModulaires.pdf}},
  month={June},
  year={2004},
}

@inproceedings{Diamond,
  title={The refined conjecture of {S}erre},
  author={Diamond, Fred},
  pages={172--186},
  booktitle={Elliptic curves, modular forms, and Fermat's last theorem},
  editor={J. Coates and S. Yau},
  year={1995}
}

@incollection{Edixhoven,
  year={1997},
  isbn={978-0-387-98998-3},
  booktitle={Modular Forms and Fermat's Last Theorem},
  editor={Cornell, Gary and Silverman, Joseph H. and Stevens, Glenn},
  doi={10.1007/978-1-4612-1974-3_7},
  title={Serre's Conjecture},
  url={http://dx.doi.org/10.1007/978-1-4612-1974-3_7},
  publisher={Springer New York},
  author={Edixhoven, Bas},
  pages={209-242},
  language={English},
}

@article{EdixhovenWeight,
  title={The weight in {S}erre's conjectures on modular forms},
  author={Edixhoven, Bas},
  journal={Inventiones mathematicae},
  volume={109},
  number={1},
  pages={563--594},
  year={1992},
  issn={0020-9910},
  doi={10.1007/BF01232041},
  url={http://dx.doi.org/10.1007/BF01232041},
  publisher={Springer-Verlag},
  language={English}
}

@inproceedings{Frohlich,
  title={Local fields},
  author={Fr{\"o}hlich, A},
  booktitle={Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, DC},
  pages={1--41},
  year={1967}
}

@unpublished{Ghitza,
  title={On the two-dimensional modular representations of {$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$} (translation)},
  author={Ghitza, Alex},
  note={\url{http://www.ms.unimelb.edu.au/~aghitza/nuts/serre.pdf}},
}

@article{Gross,
  title={A tameness criterion for {G}alois representations associated to modular forms (mod $p$)},
  author={Gross, Benedict H.},
  journal={Duke Mathematical Journal},
  ajournal={Duke Math. J.},
  volume={61},
  number={2},
  pages={445--517},
  year={1990},
  publisher={Duke University Press}
}

@article{Jochnowitz,
  title={A study of the local components of the {H}ecke algebra mod $l$},
  author={Jochnowitz, Naomi},
  journal={Transactions of the American Mathematical Society},
  volume={270},
  number={1},
  pages={253--267},
  year={1982}
}

@article{KWI,
  title={Serre's modularity conjecture ({I})},
  author={Khare, Chandrashekhar and Wintenberger, Jean-Pierre},
  journal={Inventiones mathematicae},
  volume={178},
  number={3},
  pages={485--504},
  year={2009},
  publisher={Springer}
}

@article{KWII,
  title={Serre's modularity conjecture ({II})},
  author={Khare, Chandrashekhar and Wintenberger, Jean-Pierre},
  journal={Inventiones mathematicae},
  volume={178},
  number={3},
  pages={505--586},
  year={2009},
  publisher={Springer}
}

@article{Kisin,
  title={Modularity of 2-adic {B}arsotti-{T}ate representations},
  author={Kisin, Mark},
  journal={Inventiones mathematicae},
  volume={178},
  number={3},
  pages={587--634},
  year={2009},
  publisher={Springer}
}

@article{Livne,
  title={On the conductors of mod $l$ {G}alois representations coming from modular forms},
  author={Livn{\'e}, Ron},
  journal={Journal of Number Theory},
  volume={31},
  number={2},
  pages={133--141},
  year={1989},
  publisher={Elsevier}
}

@article{Mascot,
  title={Computing modular {G}alois representations},
  author={Mascot, Nicolas},
  journal={Rendiconti del Circolo Matematico di Palermo},
  volume={62},
  number={3},
  pages={451--476},
  year={2013},
  publisher={Springer}
}

@article{Ribet90,
  title={On modular representations of {$\Gal(\Qb/\QQ)$} arising from modular forms},
  author={Ribet, Kenneth A.},
  journal={Inventiones mathematicae},
  volume={100},
  number={1},
  pages={431--476},
  year={1990},
  publisher={Springer}
}

@inproceedings{Ribet94,
  title={Report on mod $l$ representations of {$\Gal(\Qb/\QQ)$}},
  author={Ribet, Kenneth A.},
  booktitle={Proceedings of Symposia in Pure Mathematics},
  volume={55},
  pages={639--676},
  year={1994}
}

@inproceedings{Ribet95,
  title={Irreducible {G}alois representations arising from component groups of {J}acobians},
  author={Ribet, Kenneth A.},
  booktitle={Elliptic curves, modular forms, and Fermat's last theorem},
  editor={J. Coates and S. Yau},
  pages={131--147},
  year={1995}
}

@incollection{Ribet04,
  year={2004},
  isbn={978-3-0348-9621-4},
  booktitle={Modular Curves and Abelian Varieties},
  volume={224},
  series={Progress in Mathematics},
  editor={Cremona, John E. and Lario, Joan-Carles and Quer, Jordi and Ribet, Kenneth A.},
  doi={10.1007/978-3-0348-7919-4_15},
  title={Abelian {V}arieties over {$\QQ$} and {M}odular {F}orms},
  url={http://dx.doi.org/10.1007/978-3-0348-7919-4_15},
  publisher={Birkhäuser Basel},
  author={Ribet, Kenneth A.},
  pages={241-261},
  language={English}
}

@incollection{RibetStein,
  year={1999},
  author={Ribet, Kenneth A. and Stein, William A.},
  booktitle={Arithmetic Algebraic Geometry},
  editor={Conrad, B.D. and Rubin, K.},
  isbn={9780821886915},
  language={English},
  pages={143-232},
  publisher={American Mathematical Soc.},
  series={IAS Park City mathematics series: Institute for Advanced Study},
  title={Lectures on {S}erre's conjectures},
  url={https://books.google.co.uk/books?id=PgHjLgIVidgC},
}

@unpublished{RibetSteinBook,
  title={Lectures on modular forms and {H}ecke operators},
  author={Ribet, Kenneth A. and Stein, William A.},
  note={\url{http://wstein.org/books/ribet-stein/main.pdf}},
  year={2011},
  month={November}
}

@manual{Sage,
  Key      = {Sage},
  Author     = {W.\thinspace{}A. Stein and others},
  Organization = {The Sage Development Team},
  Title    = {{S}age {M}athematics {S}oftware ({V}ersion 6.5)},
  note     = {{\tt http://www.sagemath.org}},
  Year     = {2015},
}

@article{Serre67,
  title={Une interpr{\'e}tation des congruences relatives {\`a} la fonction $\tau$ de {R}amanujan},
  author={Serre, Jean-Pierre},
  journal={S{\'e}minaire Delange-Pisot-Poitou. Th{\'e}orie des nombres},
  volume={9},
  number={1},
  pages={1--17},
  year={1967},
  publisher={Secr{\'e}tariat math{\'e}matique}
}

@article{Serre72,
  title={Propri{\'e}t{\'e}s galoisiennes des points d'ordre fini des courbes elliptiques},
  author={Serre, Jean-Pierre},
  journal={Inventiones mathematicae},
  volume={15},
  number={4},
  pages={259--331},
  year={1972},
  publisher={Springer}
}

@incollection{Serre73,
  title={Congruences et formes modulaires (d'apr\`es {H.P.F. S}winnerton-{D}yer)},
  author={Serre, Jean-Pierre},
  booktitle={S{\'e}minaire Bourbaki vol. 1971/72 Expos{\'e}s 400--417},
  pages={319--338},
  year={1973},
  publisher={Springer}
}

@incollection{Serre73Formes,
  title={Formes modulaires et fonctions z{\^e}ta $p$-adiques},
  author={Serre, Jean-Pierre},
  booktitle={Modular functions of one variable III},
  editors={W. Kuijk and J.-P. Serre},
  pages={191--268},
  year={1973},
  publisher={Springer}
}

@article{Serre75,
  title={Valeurs propres des op{\'e}rateurs de {H}ecke modulo $l$},
  author={Serre, Jean-Pierre},
  journal={Ast{\'e}risque},
  volume={24},
  pages={109--117},
  year={1975}
}

@article{Serre87,
  ajournal={Duke Math. J.},
  author={Serre, Jean-Pierre},
  doi={10.1215/S0012-7094-87-05413-5},
  journal={Duke Mathematical Journal},
  number={1},
  pages={179-230},
  publisher={Duke University Press},
  title={Sur les repr\'esentations modulaires de degr\'e $2$ de {$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$}},
  url={http://dx.doi.org/10.1215/S0012-7094-87-05413-5},
  volume={54},
  year={1987},
}

@book{SerreLF,
  title={Local Fields},
  author={Greenberg, Marvin J. and Serre, Jean-Pierre},
  isbn={9780387904245},
  lccn={lc79012643},
  series={Graduate Texts in Mathematics},
  url={http://books.google.co.uk/books?id=DAxlMdw\_QloC},
  year={1995},
  publisher={Springer New York}
}

@book{SerreLinear,
  title={Linear Representations of Finite Groups},
  author={Scott, Leonhard L. and Serre, Jean-Pierre},
  isbn={9780387901909},
  lccn={76012585},
  series={Graduate Texts in Mathematics},
  url={http://books.google.co.uk/books?id=NCfZgr54TJ4C},
  year={1996},
  publisher={Springer New York}
}

@incollection{SD,
  title={On $\ell$-adic representations and congruences for coefficients of modular forms},
  author={Swinnerton-Dyer, Peter},
  booktitle={Modular functions of one variable III},
  editors={W. Kuijk and J.-P. Serre},
  pages={3--55},
  year={1973},
  publisher={Springer}
}

@unpublished{WieseGal,
  title={{G}alois {R}epresentations},
  author={Gabor Wiese},
  note={\url{http://math.uni.lu/~wiese/notes/GalRep.pdf}},
  month={February},
  year={2012}
}

@unpublished{WieseMod,
  title={Modular {G}alois {R}epresentations and {A}pplications},
  author={Gabor Wiese},
  note={\url{http://math.uni.lu/~wiese/notes/2013-ModGalRepApp.pdf}},
  month={April},
  year={2013}
}

@article{Tate,
  title={The non-existence of certain {G}alois extensions of {$\QQ$} unramified outside 2},
  author={Tate, John},
  journal={Contemporary Mathematics},
  volume={174},
  pages={153--153},
  year={1994},
  publisher={AMERICAN MATHEMATICAL SOCIETY}
}