Fix typos in blueprint

Author: pitmonticone

blueprint/src/chapter/ch02reductions.tex

@@ -1,7 +1,7 @@
 \chapter{First reductions of the problem.}
 
 \section{Overview}
-The proof of Fermat's Last Theorem is by contradiction. We assume that we have a counterexample $a^n+b^n=c^n$, and manipulate it until it satsfies the axioms of a ``Frey package''. From the Frey package we build a Frey curve -- an elliptic curve defined over the rationals. We then look at a certain representation of a Galois group coming from this elliptic curve, and finally using two very deep and independent theorems (one due to Mazur, the other due to Wiles) we show that this representation is neither reducible or irreducible, a contradiction.
+The proof of Fermat's Last Theorem is by contradiction. We assume that we have a counterexample $a^n+b^n=c^n$, and manipulate it until it satisfies the axioms of a ``Frey package''. From the Frey package we build a Frey curve -- an elliptic curve defined over the rationals. We then look at a certain representation of a Galois group coming from this elliptic curve, and finally using two very deep and independent theorems (one due to Mazur, the other due to Wiles) we show that this representation is neither reducible or irreducible, a contradiction.
 
 \section{Reduction to \texorpdfstring{$n\geq5$}{ngeq5} and prime}
 
@@ -100,7 +100,7 @@ \section{Galois representations and elliptic curves}\label{twopointfour}
   and if $n$ is a natural number, hen the group of $k$-automorphisms of $K$ acts on the additive abelian group $E(K)[n]$ of $n$-torsion points on the curve.
 \end{definition}
 
-If furthermore $n=p$ is prime, then $A[p]$ is naturally a vector space over the field $\Z/p\Z$, and thus it inherits the stucture of a mod $p$ representation of $G$. Applying this to the above situation, we deduce that if $E$ is an elliptic curve over $\Q$ then $\GQ$ acts on $E(\Qbar)[p]$ and this is the \emph{mod $p$ Galois representation} attached to the curve $E$.
+If furthermore $n=p$ is prime, then $A[p]$ is naturally a vector space over the field $\Z/p\Z$, and thus it inherits the structure of a mod $p$ representation of $G$. Applying this to the above situation, we deduce that if $E$ is an elliptic curve over $\Q$ then $\GQ$ acts on $E(\Qbar)[p]$ and this is the \emph{mod $p$ Galois representation} attached to the curve $E$.
 
 In the next section we apply this theory to an elliptic curve coming from a counterexample to Fermat's Last theorem.
 

blueprint/src/chapter/ch03frey.tex

@@ -8,7 +8,7 @@ \section{Overview}
 \section{The arithmetic of elliptic curves}
 
 We give an overview of the results we need, citing the literature for proofs. Everything here is
- standard, and mostof it dates back to the 1970s or before.
+ standard, and most of it dates back to the 1970s or before.
 
 \begin{theorem}\label{EllipticCurve.n_torsion_card}\lean{EllipticCurve.n_torsion_card}\notready
   Let $n$ be a positive integer, let $F$ be a separably closed

blueprint/src/chapter/ch04overview.tex

@@ -31,7 +31,7 @@ \section{Potential modularity.}
     \item A modularity lifting theorem.
 \end{itemize}
 
-Almost verything here dates back to the the 1980s or before.
+Almost everything here dates back to the the 1980s or before.
 The exception is the modularity lifting theorem, which we now state explicitly.
 
 \section{A modularity lifting theorem}
@@ -82,7 +82,7 @@ \section{A modularity lifting theorem}
 In the minimal case, the argument is the usual Taylor--Wiles trick, using refinements due to Kisin and others.
 \end{proof}
 
-Given this modularity lifting theorem, the strategy to show potential modularity of $\rho$ is to use Moret--Bailly to find an appropriate totally real field $F$, an auxilary prime $p$, and an auxiliary elliptic curve over $F$ whose mod $\ell$ Galois representation is $\rho$ and whose
+Given this modularity lifting theorem, the strategy to show potential modularity of $\rho$ is to use Moret--Bailly to find an appropriate totally real field $F$, an auxiliary prime $p$, and an auxiliary elliptic curve over $F$ whose mod $\ell$ Galois representation is $\rho$ and whose
 mod $p$ Galois representation is induced from a character. By converse theorems (for example)
 the mod $p$ Galois representation is associated to an automorphic representation of
 $\GL_2/F$ and hence by Jacquet--Langlands it is modular. Now we use the

blueprint/src/chapter/chtopbestiary.tex

@@ -9,7 +9,7 @@ \section{Results from class field theory}
 Let $K$ be a finite extension of $\Q_p$. We write $\widehat{\Z}$ for the profinite completion of $\Z$; it is isomorphic to $\prod_p\Z_p$ where $\Z_p$ is the $p$-adic integers and the product is over all primes.
 
 \begin{theorem}\label{maximal_unramified_extension_of_p-adic_field}\notready The maximal unramified extension $K^{un}$ in a given algebraic closure of $K$
-    is Galois over $K$ with Galois group ``canonically'' isomorphic to $\widehat{\Z}$ in two ways; one of these two isomorphisms identifies $1\in\widehat{\Z}$ with an arithmetic Frobenius (the endomorphism inducing $x\mapsto x^q$ on the residue field of $K^{un}$, where $q$ is the size of the residue field of $K$). The other identifies 1 with geometric Frobenius (defined to be the inverse of arithematic Frobenius).
+    is Galois over $K$ with Galois group ``canonically'' isomorphic to $\widehat{\Z}$ in two ways; one of these two isomorphisms identifies $1\in\widehat{\Z}$ with an arithmetic Frobenius (the endomorphism inducing $x\mapsto x^q$ on the residue field of $K^{un}$, where $q$ is the size of the residue field of $K$). The other identifies 1 with geometric Frobenius (defined to be the inverse of arithmetic Frobenius).
 \end{theorem}
 
 It is impossible to say which of the two canonical isomorphisms is ``the most canonical''; people working in different areas make different choices in order to locally minimise the number of minus signs in their results.

blueprint/src/chapter/global_langlands.tex

@@ -12,7 +12,7 @@ \section{Statement of the conjecture}
 
   An \emph{affine group scheme over $R$} is a group object in the category of affine schemes over $R$.
 
-  \begin{definition}\label{Hopf_algebra}\lean{TODO}% we have Hopf alegbras in mathlib
+  \begin{definition}\label{Hopf_algebra}\lean{TODO}% we have Hopf algebras in mathlib
 
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