Talk given by Kenji Terao
Abstract: Modular curves are objects of central importance in arithmetic geometry, parametrizing elliptic curves with particular Galois representations. They form a key part of the proof of results such as Fermat’s Last Theorem and Mazur’s torsion theorem. On the other hand, isolated points are “exceptional” low-degree points on curves, which lie outside the infinite families of low-degree points effected by the geometry of the curve. In this talk, I will aim to give a gentle introduction to these two concepts, the former via a stroll through the world of moduli spaces, the latter via an examination of Faltings’s proof of the Mordell conjecture. In particular, little to no prior knowledge will be assumed. Time permitting, I will conclude with some recent advances on the intersection of these two notions.
A moduli space is a variety such that
The points on variety parametrize equivalence classes of some set of geometric objects.
The arithmetic of the variety characterizes the arithmetic of my objects.
Example. Moduli space of right-angled triangles.
\[\{x^2 + y^2 = z^2\}\]
Two right-angled triangles are equivalent if they are similar.
\((x,y,z)\sim (\lambda x, \lambda y, \lambda z)\)
representative \((x,y,1)\).
We are lead to consider the unit circle \(x^2 + y^2 = 1\). Let’s consider some sub-fields \(K\subset \mathbb{R}\), and we can look at the \(K\) points
\[C(K):=\{(x,y)\in K^2 : x^2 + y^2 = 1\}\]
These are exactly the set of right-angled triangles up to similarity with side lengths in \(K\).
Let’s take some elliptic curve \(E/K\), \(\mathrm{char}K = 0\). There is a group law and we can consider the set of \(n\)-torsion points
\[E[n] = \{P\in E(\overline{k}) : nP = 0\}\cong (\mathbb{Z}/n\mathbb{Z})^2\]
Let \(G_k=\mathrm{Gal}(\overline{k}/k)\) be the absolute Galois group of \(k\), have the map
\[\sigma : E[n]\to E[n], \quad P=(x,y)\mapsto \sigma P=(\sigma x, \sigma y)\]
Which induces a representation
\[\overline{\rho}_{E,\{P,Q\}}: G_k \to \mathrm{Aut}(E[n])\cong \mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})\]
Where we have fixed \(P,Q\) a basis (you can actually choose an symplectic basis) of \(E[n]\) to obtain a matrix.
The image of \(\overline{\rho}_{E,\{P,Q\}}\) is a subgroup of \(\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})\) tells us a lot about the arithmetic of \(E\).
Consider the subgroup \(B_1(n)=\left\{\begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix}\right\}\)
The action is like \(P\mapsto P\), \(Q\mapsto aP+bQ\), we note that any element of the representation fixes \(P\) (sub-representation! owo), i.e. \(P\) is a rational point on \(E\) that is also a \(n\)-torsion point.
Suppose now the representation image is contained in \(B_0(n)=\left\{\begin{pmatrix} * & * \\ 0 & * \end{pmatrix}\right\}\)
there is still a sub-representation \(\langle P\rangle\), this cyclic subgroup defined over \(k\), which gives a rational \(n\)-isogeny \(E\to E/\langle P\rangle\).
Let \(H\le \mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})\) be a subgroup, we can consider the modular curve \(X_H\) parametrizing the set of
\[X_H(K) = \{(E,P,Q) : \overline{\rho}_{E,(P,Q)}(G_k)\le H\}\]
Example. \(H=B_1(n)\), we have \(X_H(K)=X_1(n)(K)\)
\[X_1(n)(K) = \{(E,P) : P\in E[n](K)\}\]
it is really parametrizing pair of elliptic curves with a rational point of order \(n\).
Let \(H=B_0(n)\), the curve is denoted \(X_0(n)\)
\[X_0(n)(K) = \{(E,E') : \exists \text{ rational isogeny } E\to E' \text{ of degree } n\}\]
Theorem (Mazur). Take an elliptic curve \(E/\mathbb{Q}\), \(P\in E(\mathbb{Q})\) which is a torsion. Then \(|P|\) is one of \(\{1,2,3,4,5,6,7,8,9,10,12\}\).
This theorem can be rephrased as saying that \(X_1(n)(\mathbb{Q})\) is empty for any \(n\) outside the list above.
Theorem. (Mordell conjecture, Faltings) Let \(C/k\) be a curve defined over a number field \(k\), suppose the genus \(g(C)\ge 2\), then \(C(k)\) is finite.
For this theorem, the Jacobian of \(C\) is used. It can be described as a moduli space:
Divisor on \(C\) is a formal sum of points on \(C\).
\[\sum_{P\in C} n_P P\]
where \(n_P\in \mathbb{Z}\) and \(n_P=0\) for all but finitely many \(P\).
Whose degree is \(\deg D = \sum n_P\).
It is called effective if \(n_P\ge 0\) for all \(P\).
The equivalence relation we need is linear equivalence.
Let \(f:C\to \mathbb{P}^1\) be a rational function, we define the divisor of \(f\) as
\[\mathrm{div}(f) = \sum_{P\in C(\overline{k})} \mathrm{ord}_P(f)P\]
where \(\mathrm{ord}_P(f)\) is the order of vanishing of \(f\) at \(P\), it is the valuation of the function at \(P\) in the local ring \(\mathcal{O}_{C,P}\).
We say that \(D,D'\) are linearly equivalent if \(D-D'=\mathrm{div}(f)\) for some rational function \(f\).
Now the Jacobian of \(C\) is defined as
\[J(C)(K) = \mathrm{Div}^0(C)(K)/\sim\]
where \(\mathrm{Div}^0(C)(K)\) is the group of divisors of degree \(0\) defined over \(K\) and \(\sim\) is linear equivalence. It is an abelian variety of dimension \(g(C)\).
Suppose \(C(k)\neq \varnothing\), pick \(P_0\in C(k)\) and define a map \(\phi:C\to J(C)\) by
\[P\mapsto [P-P_0]\]
\(\varphi\) is a morphism and its image is a subvariety of Jacobian. It could be not injective, suppose \(\varphi\) is not injective, then there are two points \(P,P'\) such that
\[\varphi(P)=\varphi(P')\implies [P-P_0]=[P'-P_0]\implies P-P'\sim \mathrm{div}(f)\]
which means \(f\) is a degree one function between \(C\) and \(\mathbb{P}^1\), this forces \(C\) to be a rational curve. Therefore when we set \(g(C)\ge 2\), the map \(\varphi\) is injective, we can view its image being identified with \(C\).
Theorem.(Faltings) Let \(A\) be an abelian variety over \(k\), and \(X\subset A\) a subvariety, then there exists finitely many abelian sub-varieties \(A_i\le A\) and finitely many rational points, \(y_i\in X(k)\) such that
\[y_i+A_i \subset X\]
and \(X(k)=\bigcup (y_i+A_i)\).
It basically tells you that all rational points are contained in a finite union of translates of abelian sub-varieties.
Suppose \(C(k)\) is infinite, then there exists \(y_i+A_i\subset \varphi(C)(k)\) and it means
\[\dim A_i\ge 1\]
this means \(y_i+A_i \cong \varphi(C)\). So \(A_i\) is an elliptic curve whose genus is \(1\), which is a contradiction that \(g(C)\ge 2\).
Take \(d\ge 1\), and a field extension \(K/k\) of degree \(d\), then we can look at the set of \(k\) rational points \(C(k)\) which is finite, but if you take union
\[\Sigma_d = \bigcup_{[K:k]=d} C(K)\]
is it finite?
Let’s define the symmetric power \(\mathrm{Sym}^d C = C^{(d)}:= C^d/S_d\).
\(C^{(d)}(K)\) parametrizes degree \(d\) effective divisors on \(C\) defined over \(K\).
for any such point \(P\) you can add all its conjugates and get a rational thing
\[\sigma_1(P)+\dots + \sigma_d(P) \in C^{(d)}(k)\]
this gives some kind of finite kernel map \(\Sigma_d\to C^{(d)}(k)\).
\[\varphi_d : C^{(d)}\to J(C)\quad P_1+\dots+P_d \mapsto [P_1+\dots+P_d-dP_0]\]
If \(\varphi_d\) is not injective, then there exists \(P_1+\dots+P_d = Q_1+\dots+Q_d + \mathrm{div}(f)\), which means \(f\) is a degree \(d\) function \(C\to\mathbb{P}^1\) and you can get \(\mathbb{P}^1\)-parametrized family of \(d\)-points
If \(\varphi_d\) is injective, we can get \(A\subset J(C)\) and a translate \(y+A\subset \varphi_d(C^{(d)})\), \(A\) has positive rank.