get LaTeX compiling again

blueprint/src/FLT.bib

@@ -64,5 +64,146 @@ @article {Taylor-Wiles
        URL = {https://doi.org/10.2307/2118560},
 }
 
-% silverman, silverman2, ddt, edix, serrepropgal, BLGGT
+@article {taylor-mero-cont,
+    AUTHOR = {Taylor, Richard},
+     TITLE = {On the meromorphic continuation of degree two {$L$}-functions},
+   JOURNAL = {Doc. Math.},
+  FJOURNAL = {Documenta Mathematica},
+      YEAR = {2006},
+     PAGES = {729--779},
+      ISSN = {1431-0635,1431-0643},
+   MRCLASS = {11R39 (11F80 11G40)},
+  MRNUMBER = {2290604},
+MRREVIEWER = {M.\ Ram\ Murty},
+}
+
+@article {moret-bailly,
+    AUTHOR = {Moret-Bailly, Laurent},
+     TITLE = {Groupes de {P}icard et probl\`emes de {S}kolem. {I}, {II}},
+   JOURNAL = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
+  FJOURNAL = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure.
+              Quatri\`eme S\'{e}rie},
+    VOLUME = {22},
+      YEAR = {1989},
+    NUMBER = {2},
+     PAGES = {161--179, 181--194},
+      ISSN = {0012-9593},
+   MRCLASS = {11G35 (11D72 14G25)},
+  MRNUMBER = {1005158},
+MRREVIEWER = {Philippe\ Satg\'{e}},
+       URL = {http://www.numdam.org/item?id=ASENS_1989_4_22_2_161_0},
+}
+@article {BLGGT,
+    AUTHOR = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David and
+              Taylor, Richard},
+     TITLE = {Potential automorphy and change of weight},
+   JOURNAL = {Ann. of Math. (2)},
+  FJOURNAL = {Annals of Mathematics. Second Series},
+    VOLUME = {179},
+      YEAR = {2014},
+    NUMBER = {2},
+     PAGES = {501--609},
+      ISSN = {0003-486X,1939-8980},
+   MRCLASS = {11F33},
+  MRNUMBER = {3152941},
+MRREVIEWER = {Wen-Wei\ Li},
+       DOI = {10.4007/annals.2014.179.2.3},
+       URL = {https://doi.org/10.4007/annals.2014.179.2.3},
+}
+
+@article {edix,
+    AUTHOR = {Edixhoven, Bas},
+     TITLE = {The weight in {S}erre's conjectures on modular forms},
+   JOURNAL = {Invent. Math.},
+  FJOURNAL = {Inventiones Mathematicae},
+    VOLUME = {109},
+      YEAR = {1992},
+    NUMBER = {3},
+     PAGES = {563--594},
+      ISSN = {0020-9910,1432-1297},
+   MRCLASS = {11R39 (11F80 11G18)},
+  MRNUMBER = {1176206},
+MRREVIEWER = {Douglas\ L.\ Ulmer},
+       DOI = {10.1007/BF01232041},
+       URL = {https://doi.org/10.1007/BF01232041},
+}
 
+@article {serrepropgal,
+    AUTHOR = {Serre, Jean-Pierre},
+     TITLE = {Propri\'{e}t\'{e}s galoisiennes des points d'ordre fini des
+              courbes elliptiques},
+   JOURNAL = {Invent. Math.},
+  FJOURNAL = {Inventiones Mathematicae},
+    VOLUME = {15},
+      YEAR = {1972},
+    NUMBER = {4},
+     PAGES = {259--331},
+      ISSN = {0020-9910,1432-1297},
+   MRCLASS = {14G25 (14K15)},
+  MRNUMBER = {387283},
+MRREVIEWER = {J.\ W. S. Cassels},
+       DOI = {10.1007/BF01405086},
+       URL = {https://doi.org/10.1007/BF01405086},
+}
+@book {silverman1,
+    AUTHOR = {Silverman, Joseph H.},
+     TITLE = {The arithmetic of elliptic curves},
+    SERIES = {Graduate Texts in Mathematics},
+    VOLUME = {106},
+   EDITION = {Second},
+ PUBLISHER = {Springer, Dordrecht},
+      YEAR = {2009},
+     PAGES = {xx+513},
+      ISBN = {978-0-387-09493-9},
+   MRCLASS = {11-02 (11G05 11G20 14H52 14K15)},
+  MRNUMBER = {2514094},
+MRREVIEWER = {Vasil\cprime \ \={I}.\ Andr\={\i}\u{\i}chuk},
+       DOI = {10.1007/978-0-387-09494-6},
+       URL = {https://doi.org/10.1007/978-0-387-09494-6},
+}
+
+@book {silverman2,
+    AUTHOR = {Silverman, Joseph H.},
+     TITLE = {Advanced topics in the arithmetic of elliptic curves},
+    SERIES = {Graduate Texts in Mathematics},
+    VOLUME = {151},
+ PUBLISHER = {Springer-Verlag, New York},
+      YEAR = {1994},
+     PAGES = {xiv+525},
+      ISBN = {0-387-94328-5},
+   MRCLASS = {11G05 (11G07 11G15 11G40 14H52)},
+  MRNUMBER = {1312368},
+MRREVIEWER = {Henri\ Darmon},
+       DOI = {10.1007/978-1-4612-0851-8},
+       URL = {https://doi.org/10.1007/978-1-4612-0851-8},
+}
+
+@incollection {ddt,
+    AUTHOR = {Darmon, Henri and Diamond, Fred and Taylor, Richard},
+     TITLE = {Fermat's last theorem},
+ BOOKTITLE = {Current developments in mathematics, 1995 ({C}ambridge, {MA})},
+     PAGES = {1--154},
+ PUBLISHER = {Int. Press, Cambridge, MA},
+      YEAR = {1994},
+      ISBN = {1-57146-029-2},
+   MRCLASS = {11G18 (11D41 11F80 11G05)},
+  MRNUMBER = {1474977},
+MRREVIEWER = {M.\ Ram\ Murty},
+}
+
+@article {toby-modularity,
+    AUTHOR = {Gee, Toby},
+     TITLE = {Modularity lifting theorems},
+   JOURNAL = {Essent. Number Theory},
+  FJOURNAL = {Essential Number Theory},
+    VOLUME = {1},
+      YEAR = {2022},
+    NUMBER = {1},
+     PAGES = {73--126},
+      ISSN = {2834-4626,2834-4634},
+   MRCLASS = {11S37},
+  MRNUMBER = {4573253},
+MRREVIEWER = {Sazzad\ Ali\ Biswas},
+       DOI = {10.2140/ent.2022.1.73},
+       URL = {https://doi.org/10.2140/ent.2022.1.73},
+}

blueprint/src/chapter/ch03frey.tex

@@ -42,7 +42,7 @@ \section{The arithmetic of elliptic curves}
 \end{proof}
 
 We saw in section~\ref{twopointfour} that if if $E$ is an elliptic curve over a field $K$,
-then $\GK$ acts naturally on the abelian group $E(\Kbar)[n]$. If furthermore $n\not=0$ in $K$ then from the above corollary, we now know that this abelian group is free of rank 2 over $\Z/n\Z$. If we choose a basis (this is traditionally done in the literature, although we do not ever seem to actually use such a choice), then $E(\Kbar)[n]$ gives us a \emph{Galois representation} $\Gal(\Kbar/K)\to\GL_2(\Z/n\Z)$.
+then $\GK$ acts naturally on the abelian group $E(K^{\sep})[n]$. If furthermore $n\not=0$ in $K$ then from the above corollary, we now know that this abelian group is free of rank 2 over $\Z/n\Z$. If we choose a basis (this is traditionally done in the literature, although we do not ever seem to actually use such a choice), then $E(K^{\sep})[n]$ gives us a \emph{Galois representation} $\GK\to\GL_2(\Z/n\Z)$.
 
 A fundamental fact about this Galois representation is that its determinant is the
 cyclotomic character.
@@ -57,16 +57,18 @@ \section{The arithmetic of elliptic curves}
 \section{Good reduction}
 
 We give a brief overview of the theory of good and multiplicative reduction of elliptic curves.
-For more details one can consult the standard sources such as~\cite{silverman}. **TODO** more
+For more details one can consult the standard sources such as~\cite{silverman1}. **TODO** more
 precise ref. We stick with the low-level approach, thinking of elliptic curves as plane cubics; whilst we cannot do this forever, it will suffice for these initial results.
 
 \begin{definition}\label{good_reduction} Let $E$ be an elliptic curve over the field of fractions $K$ of a DVR
 $R$ with maximal ideal $\m$. We say $E$ has \emph{good reduction} if $E$ has a model with 
 coefficients in $R$ and the reduction mod $\m$ is still non-singular. If $E$ is an elliptic curve 
 over a number field $N$ and $P$ is a finite place of $N$, then one says that $E$ has \emph{good reduction at $P$} if 
 the base extension of $E$ to the completion $N_P$ of $N$ at $P$ has good reduction.
+\end{definition}
 
 \begin{remark} From this point on, our Frey curves and Frey packages will use notation $(a,b,c,\ell)$, with $\ell\geq 5$ a prime number. This frees up $p$ for use as another prime.
+\end{remark}
 
 \begin{example} If $E$ is the Frey curve $Y^2=X(X-a^\ell)(X+b^\ell)$ associated to a
   Frey package $(a,b,c,\ell)$, and if $p$ is a prime
@@ -274,7 +276,7 @@ \section{The l-torsion in the Frey curve is hardly ramified.}
 \end{theorem}
 \begin{proof}
   This follows from the results above. The fact that $\ell\geq 5$ follows from the definition of
-  a Frey package. The first condition is theorem~\ref{Elliptic_curve_det_p_torsion},
+  a Frey package. The first condition is theorem~\ref{Elliptic_curve_det_n_torsion},
   and the second is theorem~\ref{frey_curve_unramified}. The third condition is
   theorem~\ref{frey_curve_at_2}, and the fourth is theorem~\ref{Frey_curve_mod_ell_rep_at_ell}.
 \end{proof}

blueprint/src/content.tex

@@ -3,4 +3,5 @@
 \input{chapter/ch01introduction}
 \input{chapter/ch02reductions}
 \input{chapter/ch03frey}
+\input{chapter/ch04overview}
 \input{chapter/biblio}

blueprint/src/macro/common.tex

@@ -12,6 +12,7 @@
 \newcommand{\GQp}{\Gal(\Qpbar/\Qp)}
 \newcommand{\GQl}{\Gal(\Qlbar/\Ql)}
 \newcommand{\m}{\mathfrak{m}}
+\newcommand{\GK}{\Gal(K^{\sep}/K)}
 \DeclareMathOperator{\Gal}{Gal}
 \DeclareMathOperator{\Aut}{Aut}
 \DeclareMathOperator{\GL}{GL}