Galois representations and elliptic curves

To continue, we need some of the theory of elliptic curves over \(\mathbb{Q}\). So let \(f(X)\) denote any monic cubic polynomial with rational coefficients and whose three complex roots are distinct, and let us consider the equation \(E:Y^2=f(X)\), which defines a curve in the \((X,Y)\) plane. This curve (or strictly speaking its projectivisation) is a so-called elliptic curve (or an elliptic curve over \(\mathbb{Q}\) if we want to keep track of the field where the coefficients of \(f(X)\) lie).

If \(E:Y^2=f(X)\) is an elliptic curve over \(\mathbb{Q}\), and if \(K\) is any characteristic zero field (and hence a \(\mathbb{Q}\)-algebra), then we write \(E(K)\) for the set of solutions to \(y^2=f(x)\) with \(x,y\in K\), together with an additional “point at infinity” corresponding morally to \(x=y=\infty\). It is an extraordinary fact, and not at all obvious, that \(E(K)\) naturally has the structure of an additive abelian group, with the point at infinity being the zero element (the identity). Fortunately this fact is already in mathlib. This additive group structure has the property that three distinct points \(P,Q,R\in K^2\) which are in \(E(K)\) will sum to zero if and only if they are collinear.

The group structure behaves well under change of field: if \(E\) is an elliptic curve over \(\mathbb{Q}\) and if \(K\to L\) is a homomorphism of characteristic zero fields then the induced map \(E(K)\to E(L)\) is a group homomorphism. Thus if \(f:K\to L\) is an isomorphism of characteristic zero fields, the induced map \(E(K)\to E(L)\) is an isomorphism of groups, with the inverse isomorphism being the map \(E(L)\to E(K)\) induced by \(f^{-1}\). This construction thus gives us an action of the multiplicative group \(\operatorname{Aut}(K)\) of automorphisms of the field \(K\) on the additive abelian group \(E(K)\), and hence also on the \(n\)-torsion of this group for any positive integer \(n\). In particular, if \(\overline{\mathbb{Q}}\) denotes an algebraic closure of the rationals (for example, the algebraic numbers in \(\mathbb{C}\)) and if \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) denotes the group of field isomorphisms \(\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}\), then for any elliptic curve \(E\) over \(\mathbb{Q}\) we have an action of \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the additive abelian group \(E(\overline{\mathbb{Q}})\), and hence on its \(n\)-torsion subgroup \(E(\overline{\mathbb{Q}})[n]\).

If furthermore \(n=p\) is prime, then \(E(\overline{\mathbb{Q}})[p]\) is naturally a vector space over the field \(\mathbb{Z}/p\mathbb{Z}\), and thus it inherits the structure of a mod \(p\) representation of \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). This is the mod \(p\) Galois representation attached to the elliptic curve \(E\). It is well-known to be 2-dimensional. We call this representation \(\rho_{E,p}\).

In the next section we apply this theory to an elliptic curve coming from a counterexample to Fermat's Last theorem.

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