Set up FLT example

example/01_introduction.typ

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+#import "common.typ": *
+
+= Introduction
+
+Fermat's Last Theorem is the statement that if $a, b, c, n$ are positive whole
+numbers with $n >= 3$, then $a^n + b^n != c^n$. It is thus a statement about a
+family of Diophantine equations ($a^3 + b^3 = c^3, a^4 + b^4 = c^4, ...$).
+Diophantus was a Greek mathematician who lived around 1800 years ago, and he
+would have been able to understand the statement of the theorem (he knew about
+positive integers, addition and multiplication).
+
+Fermat's Last Theorem was explicitly raised by Fermat in 1637, and was proved by
+Wiles (with the proof completed in joint work with Taylor) in 1994. There are
+now several proofs but all of them go broadly in the same direction, using
+elliptic curves and modular forms.
+
+Explaining a proof of Fermat's Last Theorem to Lean is in some sense like
+explaining the proof to Diophantus; for example, the proof starts by observing
+that before we go any further it's convenient to first invent/discover zero and
+negative numbers, and one can point explicitly at places in Lean's source code
+#link(
+  "https://github.com/leanprover/lean4/blob/260eaebf4e804c9ac1319532970544a4e157c336/src/Init/Prelude.lean#L1049",
+)[here] and #link(
+  "https://github.com/leanprover/lean4/blob/260eaebf4e804c9ac1319532970544a4e157c336/src/Init/Data/Int/Basic.lean#L45",
+)[here] where these things happen. However we will adopt a more efficient
+approach: we will assume all of the theorems both in Lean and in its mathematics
+library #link("https://github.com/leanprover-community/mathlib4")[mathlib], and
+proceed from there. To give some idea of what this entails: mathlib at the time
+of writing contains most of an undergraduate mathematics degree and parts of
+several relevant Masters level courses (for example, the definitions and basic
+properties of #link(
+  "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.html",
+)[elliptic curves] and #link(
+  "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Basic.html",
+)[modular forms] are in mathlib). Thus our task can be likened to teaching a
+graduate level course on Fermat's Last Theorem to a computer.
+
+== Which proof is being formalised?
+
+At the time of writing, these notes do not contain anywhere near a proof of FLT,
+or even a sketch proof. Over the next few years, we will be building parts of
+the argument, following a strategy constructed by Taylor, taking into account
+Buzzard's comments on what would be easy or hard to do in Lean. The proof uses
+refinements of the original Taylor--Wiles method by Diamond/Fujiwara,
+Khare--Wintenberger, Skinner--Wiles, Kisin, Taylor and others --- one could call
+it a 21st century proof of the theorem. We shall furthermore be assuming many
+nontrivial theorems without proof, at least initially. For example we shall be
+assuming the existence of Galois representations attached to weight 2 Hilbert
+modular forms, Langlands' cyclic base change theorem for $"GL"_2$, Mazur's
+theorem bounding the torsion subgroup of an elliptic curve over the rationals,
+and several other nontrivial results which were known by the end of the 1980s.
+The first main goal of the project is to state, and then prove, a modularity
+lifting theorem; the proof of such a theorem was the key breakthrough introduced
+in Wiles' 1994 paper which historically completed the proof of FLT. We will not
+be proving the original Wiles--Taylor--Wiles modularity lifting theorem, but
+instead will prove a more general result which works for Hilbert modular forms.
+
+We shall say more about the exact path we're taking in #todo[Chapter 4].
+
+== The structure of this blueprint
+
+The following chapter of the blueprint, @first-reduction, explains how to reduce
+FLT to two highly nontrivial statements about the $p$-torsion in a certain
+elliptic curve (the Frey curve). It mostly comprises of some basic reductions
+and the introduction of the Frey curve.
+
+The chapter after that, #[chapter 3], goes into more details about elliptic
+curves, however we need so much material here that somehow the top-down approach
+felt quite overwhelming at this point. For example even the basic claim that the
+$p$-torsion in the Frey curve curve is a 2-dimensional mod $p$ Galois
+representation feels like quite a battle to formalise, and proving that it is
+semistable and unramified outside $2p$ is even harder. Several people are
+working on the basic theory of elliptic curves, and hopefully we will be able to
+make more progress on this "top-down" approach later.
+
+The next chapter, #todo[chapter 4], is an _extremely_ sketchy overview of how
+the rest of the proof goes. There is so much to do here that I felt that there
+was little point continuing with this; definition after definition is missing.
+
+All of the remaining chapters are experiments, and most of them are what I am
+currently calling "mini-projects". A mini-project is a bottom-up project,
+typically at early graduate student level, with a concrete goal. The ultimate
+goal of many of these projects is to actually get some result into mathlib. We
+have had one success so far --- the Frobenius mini-project is currently being
+PRed to mathlib by Thomas Browning. Currently most of my efforts are going into
+running mini-projects, with the two most active ones currently being the adeles
+mini-project and the quaternion algebra mini-project. These projects do not
+logically depend on each other for the most part, and one can pick and choose
+how one reads them.
+
+We finish with an appendix, which is again very sketchy, and comprises mostly of
+a big list of nontrivial theorems many of which we will be assuming without
+proof in the FLT project, or at least not prioritising until we have proved the
+modularity lifting theorem assuming them.