Dependencies

If \(E\) is an elliptic curve over a field \(k\), and \(n\) is a positive integer which is nonzero in \(k\), then the determinant of the 2-dimensional representation of \(\operatorname{Gal}(k^{\operatorname{sep}}/k)\) on \(E(k^{\operatorname{sep}})[n]\) is the mod \(n\) cyclotomic character.

Let \(n\) be a positive integer, let \(F\) be a separably closed field with \(n\) nonzero in \(F\), and let \(E\) be an elliptic curve over \(F\). Then the \(n\)-torsion \(E(F)[n]\) in the \(F\)-points of \(E\) is a finite group isomorphic to \((\mathbb {Z}/n\mathbb {Z})^2\).

If \(p\) is a prime and if \(E\) is an elliptic curve over a field \(K\) of characteristic not equal to \(p\), and if \(C\subseteq E(K^{\operatorname{sep}})[p]\) is a Galois-stable subgroup of order \(p\), then there’s an elliptic curve \(E':=\)“\(E/C\)” over \(K\) and an isogeny of elliptic curves \(E\to E'\) over \(K\) inducing a Galois-equivariant surjection \(E(K^{\operatorname{sep}})\to E'(K^{\operatorname{sep}})\) with kernel precisely \(C\).

Let \(E\) be an elliptic curve over the field of fractions \(K\) of a valuation ring \(R\) with maximal ideal \(\mathfrak {m}\). We say \(E\) has good reduction over \(R\) if \(E\) has a model with coefficients in \(R\) and the reduction mod \(\mathfrak {m}\) is still non-singular. If \(E\) is an elliptic curve over a number field \(N\) and \(P\) is a maximal ideal of its integer ring \(\mathcal{O}_N\), then one says that \(E\) has good reduction at \(P\) if \(E\) has good reduction over the \(\mathcal{O}_{N,P}\), the localisation of \(\mathcal{O}_N\) at \(P\).

Let \(E\) be an elliptic curve over the field of fractions \(K\) of a valuation ring \(R\) with maximal ideal \(\mathfrak {m}\). We say \(E\) has multiplicative reduction over \(R\) if \(E\) has a model with coefficients in \(R\) and which reduces mod \(R/\mathfrak {m}\) to a plane cubic with one singularity, which is an ordinary double point. We say that the reduction is split if the two tangent lines at the ordinary double point are both defined over \(R/\mathfrak {m}\), and non-split otherwise.

If \(R\) is a commutative ring, then a finite flat group scheme over \(R\) is the spectrum of a commutative Hopf algebra \(H/R\) which is finite and flat as an \(R\)-module.

If \((a,b,c,\ell )\) is a Frey package and the \(j\)-invariant of the corresponding Frey curve is \(j\), and if \(2{\lt}p\mid abc\), then the \(p\)-adic valuation \(v_p(j)\) of \(j\) is a multiple of \(\ell \).

With notation as above, the characters \(\alpha \) and \(\beta \) are unramified at \(p\) for all primes \(p\not=\ell \).

One of \(\alpha \) and \(\beta \) is unramified at \(\ell \).

One of \(\alpha \) and \(\beta \) is trivial.

If \((a,b,c,\ell )\) is a Frey package, then the semisimplification of the restriction of the \(\ell \)-torsion \(\rho \) in the associated Frey curve to \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) is unramified.

If \(p\not=\ell \) is a prime not dividing \(abc\) then \(\rho \) is unramified at \(p\).

If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\), and if \(p\) is a prime not dividing \(abc\) (and in particular if \(p{\gt}2\)), then \(E\) has good reduction at \(p\).

Let \(\rho \) be the Galois representation on the \(\ell \)-torsion of the Frey curve coming from a Frey package \((a,b,c,\ell )\). Then \(\rho \) is hardly ramified.

The \(\ell \)-torsion in the Frey curve associated to a Frey package \((a,b,c,\ell )\) is irreducible.

If \((a,b,c,\ell )\) is a Frey package then the \(j\)-invariant of the corresponding Frey curve is \(2^8(C^2-AB)^3/A^2B^2C^2\), where \(A=a^\ell \), \(B=b^\ell \) and \(C=c^\ell \).

Let \(\rho \) be the \(\ell \)-torsion in the Frey curve associated to a Frey package \((a,b,c,\ell )\). Then the restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\) comes from a finite flat group scheme.

If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\), and if \(p\) is an odd prime which divides \(abc\), then \(E\) has multiplicative reduction at \(p\).

If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\) then \(E\) has multiplicative reduction at 2.

\(\rho \) cannot have a trivial 1-dimensional quotient.

\(\rho \) cannot have a trivial 1-dimensional submodule.

If \(\rho \) is reducible, then either \(\rho \) has a trivial 1-dimensional submodule or a trivial 1-dimensional quotient (here “trivial” means that the Galois group \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\) acts trivially).

If \(\rho \) has a trivial 1-dimensional submodule then the Frey curve has a non-trivial point of order \(\ell \).

If \((a,b,c,\ell )\) is a Frey package, if \(2{\lt}p\mid abc\) is a prime with \(p\not=\ell \), then the \(\ell \)-torsion in the Frey curve is unramified at \(p\).

If \((a,b,c,\ell )\) is a Frey package, then the \(\ell \)-torsion in the Frey curve is unramified at all primes \(p\not=2,\ell \).

If \(E\) is an elliptic curve over a number field \(N\) and \(E\) has good reduction at a maximal ideal \(P\) of \(\mathcal{O}_N\) containing the prime number \(p\), then the Galois representation on the \(p\)-torsion of \(E\) comes from a finite flat group scheme over the localisation \(\mathcal{O}_{N,P}\).

If \(E\) is an elliptic curve over a number field \(N\) and \(E\) has good reduction at a maximal ideal \(P\) of \(\mathcal{O}_N\), and if furthermore \(n\not\in P\), then the Galois representation on the \(n\)-torsion of \(E\) is unramified.

Say \(n\) is a positive integer, \(r\) is a natural, and \(A\) is an abelian group. Assume that for all \(d\mid n\), the \(d\)-torsion \(A[d]\) of \(A\) has size \(d^r\). Then \(A[n]\cong (\mathbb {Z}/n\mathbb {Z})^r\).

Let \(\ell \geq 5\) be a prime and let \(V\) be a 2-dimensional vector space over \(\mathbb {Z}/\ell \mathbb {Z}\). A representation \(\rho : \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}(V)\) is said to be hardly ramified if it satisfies the following four axioms:

1. \(\det (\rho )\) is the mod \(\ell \) cyclotomic character;

2. \(\rho \) is unramified outside \(2\ell \);

3. The semisimplification of the restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) is unramified;

4. The restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\) comes from a finite flat group scheme.

Let \(E\) be an elliptic curve over \(\mathbb {Q}\). Then the torsion subgroup of \(E\) has size at most 16.

If \(E\) is an elliptic curve over a field \(K\) complete with respect to a nontrivial nonarchimedean (real-valued) norm and with perfect residue field, and if \(E\) has multiplicative reduction, then there’s an unramified character \(\chi \) of \(\operatorname{Gal}(K^{\operatorname{sep}}/K)\) whose square is 1, such that for all positive integers \(n\) with \(n\not=0\) in \(K\), the \(n\)-torsion \(E(K^{\operatorname{sep}})[n]\) is an extension of \(\chi \) by \(\epsilon \chi \), where \(\epsilon \) is the cyclotomic character. Furthermore, the element of \(K^\times /(K^\times )^\ell \) corresponding to this extension is given by the \(q\)-invariant of the curve.

If \(E\) is an elliptic curve over a field complete with respect to a nontrivial nonarchimedean (real-valued) norm \(K\) and if \(E\) has split multiplicative reduction, then there is a Galois-equivariant injection \((K^{\operatorname{sep}})^\times /q^{\mathbb {Z}}\to E(K^{\operatorname{sep}})\), where \(q\in K^\times \) satisfies \(|q|=|j(E)|^{-1}\).

Let \(n\) be a positive integer, let \(F\) be a separably closed field with \(n\) nonzero in \(F\), and let \(E\) be an elliptic curve over \(F\). Then the \(n\)-torsion \(E(K)[n]\) in the \(F\)-points of \(E\) is a finite group of size \(n^2\).