The purpose of this note is to review the material exposed on the Seminar on the $abc$ conjecture and related topics with indications on the published literature. The original talks are in more detail in Spanish, but the main references are in English.
Acknowledgements to the speakers: The talks were given by Matías Alvarado, Francisco Gallardo, Ignacio Henríquez, Héctor Pastén and Rocío Sepúlveda Manzo.
Introduction
History
The $abc$ conjecture was originally formulated independently by David Masser and Joseph Oesterlé in 1985; the statement is as follows:
(Strong) $abc$ conjecture: Fix $\kappa > 1$. For all $a, b, c$ coprime integers with $a + b = c$ we have the bound $$ \max\{ |a|, |b|, |c| \} \ll_\kappa \operatorname{Rad}(abc)^\kappa. $$
Here $\operatorname{Rad}(n)$ denotes the product of the primes $p \mid n$ with multiplicity one; it fits the definition of radical of an ideal in commutative algebra. Many applications actually don’t need the conjecture in this level of generality; if the statement was proven true for a fixed exponent $\kappa$ we would say that the weak $abc$ conjecture holds for exponent $\kappa$.
The conjecture was inspired by an analogous result of Stothers-Mason (1981) for function fields, and it also was inspired by Szpiro’s conjecture for elliptic curves (see below). The $abc$ conjecture is very famous because, if true, it immediately solves many difficult Diophantine problems with ease: some examples are Fermat’s last theorem and Catalan’s conjecture (asymptotically), Pillai’s conjecture, Erdös-Woods conjecture, among others.
Alongside the theorem of Stothers and Mason, there are many more unsolved questions in characteristic zero that are known to hold in the function field case, such as the analogue of Cartan’s second main theorem (for value distribution).
Possible strategies
In the course of this seminar we reviewed some possible formulations of the $abc$ conjecture. The list isn’t comprehensive:
- For elliptic curves, the minimal discriminant behaves like $c$ in the equation $a + b = c$, while the conductor behaves like the radical (in the sense that it has all the same prime factors with controlled exponents). Szpiro’s conjecture is an inequality between these two values, and it’s the most popular reformulation of the $abc$ conjecture.
- In Diophantine geometry, Roth’s approximation theorem occupies a central place; a consequence of it (or Schmidt’s subspace theorem) is the finiteness of solutions to $S$-unit equations. Linear forms in logarithms gives an effective solution to $S$-units in two variables, but an effective Roth is almost as good as $abc$. This also has connections with Vojta’s conjectures in analogy of Cartan’s second main theorem in Nevanlinna theory.
- Speaking of effective results, an effective Faltings’ theorem (or effective Mordell, if the reader prefers so) also implies the $abc$ conjecture. It can be re-stated in terms of an inequality of Faltings heights.
Elliptic curves
Definitions and invariants
Definition: An elliptic curve $E$ over a field $K$ is a regular curve isomorphic to the projective closure of a (long) Weierstrass equation: $$ y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6, \qquad a_1,\dots,a_6 \in K. $$ Notice that there is no $a_5$. The index has the following interpretation: after the change of coordinates $(x, y) \mapsto (u^2x, u^3y)$, after simplifying by $u^{-6}$ changes $a_i \mapsto u^{-i}a_i$. We will refer to this $i$ as the weight of the coefficient.
Proposition: If $\operatorname{char} K \nmid 6$, then any elliptic curve $E$ over $K$ may be given in short Weierstrass form: $$ y^2 = x^3 + d_4x + d_6, \qquad d_4, d_6 \in K. $$ Associated to any equation of this form there is a value called the discriminant, given as $$ \Delta := -16(4d_4^3 + 27d_6^2), $$ which is, up to scalar, the actual discriminant of the cubic polynomial in $x$, and thus measures if it has repeated roots or not; therefore, it is easy to see that a short Weierstrass equation determines an elliptic curve iff $\Delta \ne 0$. The discriminant has weight 12.
Given a short Weierstrass equation, we define the $j$-invariant of $E$ as follows: $$ j(E) := 1728\cdot \frac{d_4^3}{\Delta}, $$ which has now weight 0, and thus it can be verified that is the same for any choice of a Weierstrass equation.
Theorem (Poincaré): Given any elliptic curve $E$ over $K$ and a fixed rational point $o \in E(K)$, the set of rational points admit an algebraic group structure $(E(K), +)$ which commutes with base change and in which $o$ is the neutral element; that is, the operation is locally given by equations in $K$ and given a field extension $L / K$, the inclusion map $E(K) \hookrightarrow E(L)$ is a group homomorphism.
This gives an elliptic curve structure of an algebraic group.
Definition: An abelian variety $A$ over a field $K$ is an algebraic group, which is a smooth, projective variety over the base field.
Actually, one may prove that any algebraic group which is connected, generically reduced and proper is an abelian variety. Abelian varieties are generalizations of elliptic curves which are often ignored a bit, because they’re less explicit; however, they’re necessary and a key tool for Falting’s proof of the Mordell conjecture.
Theorem: Let $X$ be a curve over a field $K$, the following are equivalent:
- $X$ is an elliptic curve.
- $X$ has a structure of abelian variety over $K$.
- $X$ is geometrically integral, smooth, of genus 1 and has at least one rational point $o \in X(K)$.
Going back to the $j$-invariant, the following is easily shown:
Proposition: The $j$-invariant $j \colon \mathscr{A}_1(K) \to K$ is surjective, where $\mathscr{A}_1(K)$ denotes the set of (isomorphism classes of) elliptic curves over $K$. Moreover, two elliptic curves $E_1$ and $E_2$ have the same $j$-invariant if there is a field extension $L/K$ such that $(E_1)_L \cong (E_2)_L$ (that is, they are isomorphic over $L$). In particular, if $K$ is algebraically closed, then $j$ is bijective.
Reduction modulo $p$
Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Taking a Weierstrass equation for $E$ with integral coefficients, we may build an $\mathcal{O}_K$-model $\mathcal{M}$ of $E$ (i.e., an scheme over $\mathcal{O}_K$ whose generic fibre is $E_K$). By using resolution of singularities, we may modify $\mathcal{M}$ to obtain a scheme $\mathcal{E}$ which is regular, projective and of finite type over $\mathcal{O}_K$ (but not smooth, that is, it may have singular fibres!). Given a prime $\mathfrak{p}$ of $\mathcal{O}_K$, we may consider the fibre $\mathcal{E}_\mathfrak{p}$. We’ll say that $E$ has good reduction at $\mathfrak{p}$ if the fibre $\mathcal{E}_\mathfrak{p}$ is smooth; otherwise we say it has bad reduction.
Notice that, since the generic fibre of $\mathcal{E}$ is smooth and $\mathcal{E}$ is regular, the points of bad reduction are closed, and thus finite. Also at each point of bad reduction, the smooth locus of the fibre is open, thus it only has finitely many non-smooth points. By removing all of them, we obtain an $\mathcal{O}_K$-model $\mathcal{E}_0$ of $E_K$ which is smooth and of finite type, known as the Néron model of $E_K$.
Now if $\mathcal{E}_0$ is the Néron model of $E_K$ and $\mathfrak{p}$ is a point of bad reduction, the fibre will be a disconnected algebraic group, whose connected component (not being proper) will be either the multiplicative group $\mathbb{G}_m$ or the additive group $\mathbb{G}_a$; if the former holds, we say $E_K$ has multiplicative reduction modulo $\mathfrak{p}$, or that it has additive reduction if the latter holds. Let $S \subseteq \operatorname{Spec}\mathcal{O}_K$ be an open subscheme, we say $E$ has semi-stable reduction at $S$ if it has good or multiplicative reduction at all the closed points of $S$.
In general, resolution of singularities of (absolute) surfaces always works when the base scheme is excellent. Also, given a general Dedekind scheme $S$ (which we assume connected) and an abelian variety $A_K$ over the generic point of $S$, there is a Néron model $\mathcal{A}_S$ for $A_K$ respect to $S$ which is smooth and of finite type (it must also satisfy the Néron mapping property). Instead of the trichotomy: good, multiplicative and additive reduction, in general, one decomposes the algebraic group given as the fibre of the Néron model into an abelian, multiplicative and unipotent part. We say an abelian variety is of semi-stable reduction if all the fibres of its Néron model have unipotent dimension zero.
There is another way, more simple, of obtaining a replacement for the fibre $(\mathcal{E}_0)_{\mathfrak{p}}$: take a Weierstrass equation for $E_K$ with coefficients in $\mathcal{O}_{K, \mathfrak{p}}$. Associated to the choice of the equation, we get a discriminant $\Delta$ which is still in $\mathcal{O}_{K, \mathfrak{p}}$, and thus it has a non-negative $\mathfrak{p}$-adic valuation $\nu_{\mathfrak{p}}(\Delta) \ge 0$. We say that the equation is minimal at $\mathfrak{p}$ if the valuation of the discriminant is minimal, in which case we will denote by $\nu_{\mathfrak{p}}( \mathfrak{D}_{E/K} )$ its value. Looking back at the equation, it determines a $\mathcal{O}_{K, \mathfrak{p}}$-model for $E_K$ named the minimal Weierstrass model $\mathcal{W}$, which is normal and projective.
If the prime is of good reduction, we may prove that the minimal Weierstrass model agrees with $(\mathcal{E}_0)_{\mathfrak{p}}$; otherwise, they can’t agree since $\mathcal{W}$ is connected and singular; thus, to get a chance we instead consider the smooth locus $\mathcal{W}_{\rm sm}$ and compare it a connected component of $(\mathcal{E}_0)_{\mathfrak{p}}$, and voilà! Now they are isomorphic. This gets interesting in the following way: by the abstract theory of Néron models, one may show that the fibre is an algebraic group, thus all of its connected components are mutually isomorphic, and the component of the identity is a connected subgroup; since it is commutative, smooth, but not proper, this just leaves the additive group $\mathbb{G}_a$ or a torus $T$ (which is, geometrically, the multiplicative group $\mathbb{G}_m$). By a concrete analysis of $\mathcal{W}$ one may show that the only possible singularities lie at the intersection with the line $y = 0$, leaving either a cusp (the additive reduction case) or a node (the multiplicative reduction case).
Anyways, going back to the number $\nu_{\mathfrak{p}}( \mathfrak{D}_{E/K} )$ obtained, we may repeat the process for all primes of $\operatorname{Spec}(\mathcal{O}_K)$. Since just finitely many of them are of back reduction, almost all of $\nu_{\mathfrak{p}}$’s are zero, so we define the minimal discriminant of $E_K$ as the integral ideal $$ \mathfrak{D}_{E/K} := \prod_{\mathfrak{p}} \mathfrak{p}^{ \nu_{\mathfrak{p}}( \mathfrak{D}_{E/K} ) }. $$ This sort-of captures all of the bad reduction in simultaneous. This invariant can be, virtually, whatever ideal it can. For $\Z$ we may associate a single number, but in general, $\mathcal{O}_K$ won’t be a P.I.D., and thus $\mathfrak{D}_{E/K}$ is just an ideal. However, if $\mathfrak{D}_{E/K}$ is principal, then we can actually find a single Weierstrass equation which is minimal at all primes of $\mathcal{O}_K$; this is known as minimal equation. (N.B.: As a matter of fact, the last statement is an “if and only if” by a theorem of Silverman.)
This is all theoretically interesting, but in practice one needs a method for calculating all of these invariants (reduction modulo $p$ and minimal discriminant at $p$). Such method comes as Tate’s algorithm, which is of vital importance in a lot of results.
Szpiro’s conjecture and $abc$
Another invariant which will be of help is the exponent of the conductor $f(E_K, \mathfrak{p})$ which is $0$ (resp. $1$, $2$) if $\mathfrak{p}$ is of good (resp. multiplicative, additive) reduction; with this we construct the conductor which is also an integral ideal that “measures bad reduction” $$ \mathfrak{N}_{E/K} := \prod_{\mathfrak{p}} \mathfrak{p}^{ f(E_K, \mathfrak{p}) }. $$ This is actually the tame part of the exponent of the conductor which agrees with the actual exponent for $\mathfrak{p} \nmid 6$; the full definition involves a wild part, which is somewhat controlled. As a consequence of Ogg’s formula we have $\mathfrak{N}_{E/K} \mid \mathfrak{D}_{E/K}$. In 1983, Lucien Szpiro made the following conjecture:
Szpiro’s conjecture for number fields: For all $\epsilon > 0$ and for all elliptic curves $E_K$ over a fixed number field $K$, the following inequality holds: $$ \mathbf{N}\mathfrak{D} \ll_\epsilon \mathbf{N}(\mathfrak{N}_{E/K})^{6 + \epsilon}. $$
However, by definition, $\mathfrak{N}_{E/K}$ can be, at worst, the radical of $\mathfrak{D}_{E/K}$ squared; thus any comparison between them two has a real $abc$-like flavor. As a matter of fact:
Proposition: The following holds:
- Szpiro’s conjecture for $\mathbb{Q}$ implies the (weak) $abc$ conjecture for exponents $3/2 + \epsilon$.
- The (strong) $abc$ conjecture implies Szpiro’s conjecture for $\mathbb{Q}$.
If one looks for full equivalence one has to improve slightly Szpiro’s conjecture:
Theorem: The following statements are equivalent:
- Masser-Oesterlé’s $abc$ conjecture.
- The generalized Szpiro’s conjecture: for all $\epsilon > 0$, let $E_\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ given with a minimal Weierstrass equation, then the following inequality holds: $$ \max\{ |\Delta|, |d_4|^3 \} \ll_\epsilon N_{E, K}^{6+\epsilon}. $$
It is worth noticing that there’s a big deal in looking for generalizations of the $abc$ conjecture for number fields, and Szpiro’s conjecture is quite natural in that regards.
Linear forms in logarithms
Linear forms in logarithms is a method developed inside the branch of transcendence theory, dedicated to the study of algebraic/transcendence numbers. A known result in this area was the Lindemann-Weierstrass theorem which says that given distinct algebraic numbers $\alpha_1, \dots, \alpha_n$ the exponentials $e^{\alpha_1}, \dots, e^{\alpha_n}$ are algebraically independent (over $\mathbb{Q}$). A proposed improvement was Hilbert’s seventh problem which says that $\alpha^b$ is transcendent whenever $\alpha \notin \{0, 1\}$ is algebraic and $b$ is irrational; this was proven independently by Gelfond and Schneider in 1934.
A restatement is that if $\log\alpha_1$ and $\log\alpha_2$ are linearly independent over $\mathbb{Q}$ then they are algebraically independent. This formulation may be generalized to $n$ terms, which was proven by Alan Baker in 1966, what Baker actually found was that the linear form $$ \Lambda = \beta_0 + \beta_1\log\alpha_1 + \cdots + \beta_n\log\alpha_n $$ has a lower bound in terms of the $\beta_i \in \Z$.
After Baker’s works, many people improved Baker’s bounds. In the 2000’s, Matveev obtained the best one up to this date; and in 1986 Yu obtained a similar bound for the $p$-adic absolute value of $\Lambda$ above. This has many applications for heights bounds in algebraic groups (the most direct example can be by taking exponentials of $\Lambda$, thus obtaining approximations in the multiplicative group). In 2001 Stewart and Yu applied the theory of linear forms in logarithms to obtain the inconditional $abc$-like bound: $$ \log c \ll_\varepsilon \operatorname{Rad}(abc)^{\frac{1}{3}+\varepsilon}. $$ If we suppose that $a < c^{1 - \eta}$ for a fixed $\eta > 0$ (at choice), H. Pastén obtained the recent improvement: $$ \log c \ll_{\eta, \varepsilon} \operatorname{Rad}(abc)^\varepsilon. $$ And, as stated in the beginning, the $abc$ conjecture has tons of applications, so approaches to $abc$ also have many corollaries, including: Tijdeman’s theorem (Catalan’s equation has at most finitely many solutions), Siegel’s theorem on finiteness of integer points in curves of geometric genus $\ge 1$, finiteness of solutions of Thue’s equations, etc. Another of the big theorems in transcendence theory are effective bounds for $S$-unit equations.
Faltings’ proof of the Mordell conjecture
Heights and diophantine geometry
Classically, given a rational number $x = u/v$ with $u$ and $v$ prime to each other (or if you prefer, an affine point $x \in \mathbb{A}^1(\mathbb{Q})$) we may define its (logarithmic) height as $$ h(x) = \max\{ |u|, |v| \}, $$ here $|\,|$ denotes the usual archimedian absolute value. In general, by using all of the places in a number field, we may extend the definition and obtain Weil’s (logarithmic) height for algebraic points in $\mathbb{P}^1(\mathbb{Q})$.
Heights are at the heart of diophantine geometry, a discipline which mixes number theory and algebraic geometry by turning functorial properties of divisors (or sheafs) into numerical properties of heights. One key theorem is the following:
Theorem (Northcott’s property): Given a real number $B$ and a positive integer $d$, there are finitely many projective points of height $\le B$ and degree $\le d$.
This innocent-looking result ended up being key for the Mordell-Weil theorem that $A(K)$ is a finitely generated abelian group when $K$ is a number field and $A$ is an abelian variety.
The reader may think that Faltings’ final purpose is to prove certain finitude property on abelian varieties, for which we look how to apply Northcott’s property. This will come for free by taking a Weil height in a coarse moduli space, but won’t give the desired outcome, because it has the effect of counting abelian varieties that are $\mathbb{C}$-isomorphic. This is known as the moduli height and, in the case of elliptic curves $E$, this is equivalent to taking $h(j(E))$, where $j(E)$ denotes the $j$-invariant. I want to stretch again the fact that this doesn’t work because a single elliptic curve $E$ over $K$ may have infinitely many twists (i.e., a non-isomorphic $E^\prime$ which becomes isomorphic after extending the base field).
So, how to obtain finitude then? Shafarevich’s conjecture (already known for elliptic curves) predicts that after fixing an abelian variety $A_K$ over $K$, it has only finitely many twists of controlled good reduction. Namely, we need to create a “height function” which takes into account bad reduction, and which isn’t too far from the moduli height. Here enters Faltings height $h_{\rm Fal}(E)$ which is defined as the Arakelov-theoretic self-intersection of a Néron differential. The comparison with the moduli height is, as expected, not of bounded difference, but of logarithmic difference, as shown by Faltings-Silverman’s theorem: $$ O(1) \le h(j_E) + \frac{1}{[K : \mathbb{Q}]}\log \mathbf{N}\Upsilon_{E/K} - 12h_{\rm Fal}(E_K), $$ here, $\Upsilon_{E/K}$ is a term known as the unstable minimal discriminant; if $E_K$ is semi-stable, then $\Upsilon_{E/K} = 1$ and thus the corresponding term vanishes. Therefore, $\Upsilon_{E/K}$ here does control the amount of bad reduction allowed; by bounding both the moduli height and the unstable minimal discriminant, we obtain finitely many elliptic curves of bounded Faltings height.
Faltings’ proof now proceeds in a sort-of inverse path to that we have just described, the preliminaries are the following: given an abelian variety $A_K$ over a number field $K$, there is a finite extension where $A_L$ is semi-stable (known as the semi-stable reduction theorem); and after any base change $L/K$, the Faltings height may drop $h_{\rm Fal}(A_L) \le h_{\rm Fal}(A_K)$, but equality is attained whenever $A_K$ had semi-stable reduction. Thus, we define Faltings’ geometric height $h_{\rm geom}(A)$ as the height of $A_L$ where $L/\mathbb{Q}$ is a number field in which $A_L$ has semi-stable reduction. In consequence, $h_{\rm geom}$ is comparable (somehow) to the moduli height (as in Falting-Silverman’s theorem), but $h_{\rm Fal}$ always carries an “error measuring bad reduction”. Then $h_{\rm Fal}$ has the Northcott property, and thus, a single abelian variety can only have finitely many twists with bad reduction in a fixed set of finite primes; this is Shafarevich’s conjecture. That the Mordell conjecture follows from Shafarevich’s conjecture was well known prior to Faltings’ works.
Frey’s conjecture on heights
Say we are given a fractionary ideal $\mathfrak{a}$ in $\mathcal{O}_K$, then we may expand it as a product of primes $\mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_n^{e_n}$, where the exponents are (possibly negative) integers $e_j \in \Z$. Then we define its degree as $$ \deg\mathfrak{a} = \sum_{j=1}^n e_j \, \textbf{N}\mathfrak{p}_j. $$ In 1989, Frey made the following conjecture:
Frey’s height conjecture ($H$): Let $K$ be a number field, then for all elliptic curves $E_K$ over $K$ the following inequality holds: $$ h_{\rm Fal}(E_K) \ll \deg(\mathfrak{N}_E). $$
Now, the definition of the degree makes it comparable to the logarithm of $\textbf{N} \mathfrak{N}_E$, and the Faltings height is the logarithm of an integral that captures, in some way, the complexity of the elliptic curve. Thus the following isn’t too surprising:
Proposition: Frey’s height conjecture implies Szpiro’s conjecture.
Modularity
We previously discussed the use of the coarse moduli space $\mathscr{A}_g$ of abelian varieties of dimension $g$, but it’s necessary a more precise description of this object, which may be given by the $j$-invariant.
Over $\mathbb{C}$, all abelian varieties are, as complex manifolds, of the form $\mathbb{C}^g/\Lambda$, where $\Lambda \le \mathbb{C}^g$ is a lattice (i.e., a discrete $\Z$-submodule of rank $2g$); so, for $g = 1$, the $j$-invariant determines a map from lattices in $\mathbb{C}$ to complex numbers. Every lattice is homothetic to one of the form $\mathbb{Z} + \tau \mathbb{Z}$, where $\tau$ is in the upper half-plane $\mathfrak{H}$, so it determines a bijection between $\mathbb{C}$ and points of $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathfrak{H} =: Y(1)$. By using $\mathfrak{H}^* := \mathfrak{H} \cup \mathbb{P}^1(\mathbb{Q})$ instead, the quotient $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathfrak{H}^* =: X(1)$ is a compact complex manifold, which can be given as a projective smooth curve over $\mathbb{Q}$. If one uses a subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbb{Z})$ instead, one may define other kinds of modular curves, which often are compactifications of coarse moduli spaces. Among these, a special fuchsian group is the special modular subgroup of level $N$
$$ \Gamma_0(N) = \left\{ \begin{bmatrix}a & b \\ c & d\end{bmatrix} \in \operatorname{SL}_2\mathbb{Z} : c \equiv 0 \pmod{N} \right\}, $$ whose modular curve is denoted $X_0(N)$.
An elliptic curve $E_K$ over a number field $K$ is said to be modular if there exists an (algebraic) morphism $\phi\colon X_0(N) \to E$ over $\mathbb{C}$ which is non-constant; such $\phi$ are called modular parametrizations. One can prove that if such morphism exist, then it is defined in a number field (possibly larger than $K$). By the Shimura-Taniyama conjecture (proven by Taylor, Wiles and others), every elliptic curve is modular. In this context, we have the following conjecture:
Frey’s degree conjecture ($D_d$): Fix $S \subseteq M_\mathbb{Q}$ a finite set of places with $\infty \in S$. For all $\epsilon > 0$, and for every elliptic curve $E$ over $\mathbb{Q}$ semi-stable outside $S$, it has a modular parametrization $\phi$ such that $$ \deg\phi \ll_\epsilon \prod_{\ell\mid N_E} \ell^{d+\epsilon}. $$
As a corollary, Frey’s height conjecture follows.
References
For the basics on elliptic curves, it was used Silverman’s book The arithmetic
of elliptic curves [Si]; the equivalent geometric definitions for an elliptic
curve are given in Görtz and Wedhorn’s Algebraic Geometry (Vols. I and II).
Néron models are exposed in Artin’s article in Arithmetic geometry (ed. by
Cornell and Silverman) [AG], in the self-titled book by Bosch, Lütkehbohmert
and Raynaud; and also on Liu’s Algebraic geometry and arithmetic curves, the
latter also compares the three associated models related to an elliptic curve
(proper minimal, Néron and Weierstrass models).
The relationship between Szpiro’s conjecture and $abc$ is treated in both [Si]
and Bombieri and Gubler’s Heights in diophantine geometry [BoGu].
There are plenty of books on linear forms in logarithms. To name a few we have: Baker’s Transcendental number theory; Evertse and Győry’s Unit equations in diophantine number theory, and Waldschmidt’s Diophantine approximation on linear algebraic groups.
The last section was almost entirely covered by [AG], although the basics on heights are better exposed in [BoGu]. For modular functions we refer to both Serre’s A course of arithmetic and Diamond and Shurman’s A first course in modular forms; modular curves are, again, exposed in [AG]. Also see the volume Modular Forms and Fermat’s Last Theorem (ed. by Cornell, Silverman and Stevens). Frey’s conjectures are presented in his original article Links between solutions of $A-B = C$ and elliptic curves in the volume Number theory (ed. by Schlickewei and Wirsing).