Elliptic curve over a field. (LMFDB)

An elliptic curve E over a field k is a smooth projective curve of genus 1 together with a distinguished k-rational point O.

The most commonly used model for elliptic curves is a Weierstrass model: a smooth plane cubic with the point O as the unique point at infinity.

Depends on: Definition 1.10 Definition 1.11 Definition 1.12

Additive reduction. (LMFDB)

An elliptic curve E defined over a number field K is said to have additive reduction at a prime \mathfrak{p} of K if the reduction of E modulo \mathfrak{p} has a cuspidal singularity.

Depends on: Definition 3.1 Definition 2.1

Analytic order of \Sha. (LMFDB)

The Tate-Shafarevich group \Sha of an elliptic curve E defined over a number field K is a torsion abelian group, which can be defined in terms of Galois cohomology as

\Sha(E) := \ker\left(H^1(G_K,E) \to \prod_v H^1(G_{K_v},E_{K_v}) \right),

where v runs over all places of K (finite and infinite), K_v is the completion of K at v, E_{K_v} is the base change of E to K_v, and G_K and G_{K_v} denote absolute Galois groups.

The group \Sha is conjectured to be finite, and its order appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for E. The order implied by the conjecture is called the analytic order of Sha and can be defined as the real number

\Sha_{\text{an}} := |D_K|^{1/2}\cdot \frac{L^{(r)}(E,1)} {r!} \cdot \frac{\#E(K)_{\rm tor}^2}{\mathrm{Reg}_{\rm NT}(E/K)} \cdot \frac{1}{ \Omega(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}}}.

Here D_K is the discriminant of K, L(E,s) is the L-function of E/K, r is the analytic rank of E/K, \mathrm{Reg}_{\rm NT}(E/K) is the Neron-Tate (un-normalised) regulator of E/K, E(K)_{\rm tor} is the torsion subgroup of the Mordell-Weil group E(K), \Omega(E/K) is the global period of E/K, and c_{\mathfrak{p}} is the Tamagawa number of E at the prime \mathfrak{p} of K.

It is known that if \Sha is finite then its order is a square, so one expects the real number \Sha_{\text{an}} to always be a square integer.

For elliptic curves defined over \Q of rank 0 or 1, it is a theorem that \Sha_an is a positive rational number, and this rational number can in principle be computed exactly. This exact computation has only been carried out for the curves in the database with rank 0. For curves of rank 2 and above, there is no such theorem, and the values computed are floating point approximate values which are very close to integers. In the LMFDB we store and display the rounded values in this case.

Depends on: Definition 3.1 Definition 3.5 Definition 3.38 Definition 3.45 Definition 3.82 Definition 3.85 Definition 3.93 Definition 3.95 Definition 1.76 Definition 1.84 Definition 1.91 Definition 1.99 Definition 2.1 Definition 2.15 Definition 2.44 Definition 2.47

Bad reduction of an elliptic curve at a prime. (LMFDB)

An elliptic curve E defined over a number field K is said to have bad reduction at a prime \mathfrak{p} of K if the reduction of E modulo \mathfrak{p} is singular. There are three types of bad reduction:

• split multiplicative,

• non-split multiplicative,

• additive.

A curve has bad reduction at \mathfrak{p} if and only if \mathfrak{p} divides its discriminant.

Depends on: Definition 3.1 Definition 3.2 Definition 3.10 Definition 3.42 Definition 3.92 Definition 2.1 Definition 2.47

Base change. (LMFDB)

If E is an elliptic curve defined over a field K, and L is an extension field of K, then the same equation defining E as an elliptic curve over K also defines a curve over L called the base change of E from K to L. Any curve defined over L which is isomorphic to E over L is called a base-change curve from K to L. A sufficient but not necessary condition for a curve to be a base change is that the coefficients of its Weierstrass equation lie in K.

When K=\Q and L is a number field, elliptic curves over L which are base-changes of curves over \Q may simply be called base-change curves. A necessary, but not sufficient, condition for this is that the j-invariant of E should be in \Q.

Depends on: Definition 3.30

Birch Swinnerton-Dyer conjecture. (LMFDB)

The Birch and Swinnerton-Dyer conjecture (BSD) is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing (or analytic rank) and the leading coefficient of the L-function associated to an elliptic curve E defined over a number field K at the central point s=1 to certain arithmetic data, the BSD invariants of E.

• The weak form of the BSD conjecture states just that the analytic rank r_{an} (that is, the order of vanishing of vanishing of L(E,s) at s=1), is equal to the rank r of E/K.

• The strong form of the conjecture states that r=r_{an} and also that the leading coefficient of the L-function is given by the formula

\frac{1}{r!} L^{(r)}(E,1) = \frac{1}{|d_K|^{1/2}}\cdot \frac{\# \Sha(E/K)\cdot \Omega(E/K) \cdot \mathrm{Reg}_{\rm NT}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}}}{\#E(K)_{\rm tor}^2}.

The quantities appearing in this formula are as follows:

• d_K is the discriminant of K;

• r is the rank of E(K);

• \Sha(E/K) is the Tate-Shafarevich group

  of E/K;

• \mathrm{Reg}(E/K) is the regulator of E/K;

• \Omega(E/K) is the global period of E/K;

• c_{\mathfrak{p}} is the Tamagawa number of E at each prime \mathfrak{p} of K;

• E(K)_{\rm tor} is the torsion subgroup of E(K).

Implicit in the strong form of the conjecture is that the Tate-Shafarevich group \Sha(E/K) is finite.

There is a similar conjecture for abelian varieties over number fields.

Depends on: Definition 1.5 Definition 3.1 Definition 3.3 Definition 3.45 Definition 3.82 Definition 3.85 Definition 3.93 Definition 3.95 Definition 1.109 Definition 1.110 Definition 1.112 Definition 1.119 Definition 2.1 Definition 2.15 Definition 2.47

Canonical height on an elliptic curve. (LMFDB)

Let E be an elliptic curve defined over a number field K. The canonical height on E is a function

\hat{h}: E(K) \to \R_{ {}\ge0}

defined on the Mordell-Weil group E(K) which induces a positive definite quadratic form on E(K)\otimes\R.

One definition of \hat{h}(P) is

\hat{h}(P)=\lim_{n\to\infty} n^{-2}h\bigl(x(nP)\bigr), where h(x) is the Weil height of x\in K. This definition gives the non-normalised height. A normalised height which is invariant under base-change is given by

\frac{1}{[K:\Q]} \hat{h}(P).

Related to the canonical height is the height pairing

\langle-,-\rangle : E(K)\times E(K) \to \R

defined by \langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)), which is a positive definite quadratic form on E(K)\otimes\R, used in defining the regulator of E/K.

Depends on: Definition 3.1 Definition 3.38 Definition 2.1 Definition 2.71

Complex multiplication. (LMFDB)

An elliptic curve whose endomorphism ring is larger than \Z is said to have complex multiplication (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the CM discriminant.

An elliptic curve whose geometric endomorphism ring is larger than \Z is said to have potential complex multiplication (potential CM). In the literature, these too are often called CM elliptic curves.

The property of having potential CM depends only on the j-invariant of the curve. In characteristic 0, CM j-invariants are algebraic integers, and there are only finitely many in any given number field. There are precisely 13 CM j-invariants in \Q (all integers), associated to the 13 imaginary quadratic orders of class number 1:

\begin{array}{c|ccccccccccccc} j & -12288000 & 54000 & 0 & 287496 & 1728 & 16581375 & -3375 & 8000 & -32768 & -884736 & -884736000 & -147197952000 & -262537412640768000\\ \text{CM discriminant} &-27 & -12 & -3 & -16 & -4 & -28 & -7 & -8 & -11 & -19 & -43 & -67 & -163 \end{array}

CM elliptic curves are examples of CM abelian varieties.

Depends on: Definition 1.9 Definition 3.1 Definition 3.12 Definition 3.16 Definition 3.30 Definition 2.6 Definition 2.42 Definition 1.132

Conductor of an elliptic curve. (LMFDB)

The conductor of an elliptic curve E defined over a number field K is an ideal of the ring of integers of K that is divisible by the prime ideals of bad reduction and no others. It is defined as

\mathfrak{n} = \prod_{\mathfrak{p}}\mathfrak{p}^{e_{\mathfrak{p}}}

where the exponent e_{\mathfrak{p}} is as follows:

• e_{\mathfrak{p}}=0 if E has good reduction at \mathfrak{p};

• e_{\mathfrak{p}}=1 if E has multiplicative reduction at \mathfrak{p};

• e_{\mathfrak{p}}=2 if E has additive reduction at \mathfrak{p} and \mathfrak{p} does not lie above either 2 or 3; and

• 2\leq e_{\mathfrak{p}}\leq 2+6v_{\mathfrak{p}}(2)+3v_{\mathfrak{p}}(3), where v_{\mathfrak{p}} is the valuation at \mathfrak{p}, if E has additive reduction and \mathfrak{p} lies above 2 or 3.

For \mathfrak{p}=2 and 3, there is an algorithm of Tate that simultaneously creates a minimal Weierstrass equation and computes the exponent of the conductor. See:

<UL> <LI> J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52. <EM>Lecture Notes in Math.</EM>, Vol. <B>476</B>, Springer, Berlin, 1975.

<LI> J.H. Silverman, <EM>Advanced topics in the arithmetic of elliptic curves</EM>, GTM <B>151</B>, Springer-Verlag, New York, 1994.

</UL>

The conductor norm is the norm [\mathcal{O}_K:\mathfrak{n}] of the ideal \mathfrak{n}.

Depends on: Definition 3.1 Definition 3.2 Definition 3.4 Definition 3.19 Definition 3.40 Definition 2.1 Definition 2.56

Discriminant of a Weierstrass equation. (LMFDB)

The discriminant \Delta of a Weierstrass equation over a field K is an element of K defined in terms of the Weierstrass coefficients. If the Weierstrass equation is y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, then \Delta is given by a polynomial expression in a_1,\dots,a_6, namely,

\Delta=-b_2^2b_8 - 8 b_4^3 -27 b_6 ^2 + 9 b_2 b_4 b_6 where

\begin{aligned} b_2 &= a_1^2 + 4 a_2\\ b_4 &= 2a_4 + a_1 a_3\\ b_6 &= a_3^2 + 4 a_6 \\ b_8 &= a_1^2 a_6 + 4 a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2. \end{aligned}

Then \Delta\neq 0 if and only if the equation defines a smooth curve, in which case its projective closure gives an elliptic curve.

Depends on: Definition 1.12 Definition 3.1 Definition 3.97 Definition 1.134

Endomorphism of an elliptic curve. (LMFDB)

An endomorphism of an elliptic curve defined over a field K is a homomorphism \varphi\colon E\to E defined over K. The set of all endomorphisms of E forms a ring called the endomorphism ring of E, denoted \End(E), a special case of the endomorphism ring of an abelian variety.

Depends on: Definition 1.5 Definition 1.15 Definition 3.1

Endomorphism ring of an elliptic curve. (LMFDB)

The endomorphism ring \End(E) of an elliptic curve E over a field K is the ring of all endomorphisms of E defined over K. For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of E.

For elliptic curves defined over fields of characteristic zero, this ring is isomorphic to \Z, unless the curve has complex multiplication (CM) defined over the ground field, in which case the endomorphism ring is an order in an imaginary quadratic field; for curves defined over \Q, this order is one of the 13 orders of class number one.

\End(E) always contains a subring isomorphic to \Z, since for m\in\Z there is the multiplication-by-m map [m] \colon E\to E.

This is a special case of the endomorphism ring of an abelian variety.

Depends on: Definition 1.5 Definition 1.15 Definition 3.1 Definition 3.11

Galois representations attached to an elliptic curve. (LMFDB)

If E is an elliptic curve defined over a field K and m is a positive integer, then the mod-m Galois representation attached to E is the continuous homomorphism

\overline{\rho}_{E,m}: \Gal(\overline{K}/K) \to \Aut(E[m])

describing the action of the absolute Galois group of K on the m-torsion subgroup E[m].

When the characteristic of K does not divide m > 1, we may identify the finite abelian group E[m] with (\Z/m\Z)^2 and hence view the representation as a map

\overline{\rho}_{E,m}: \Gal(\overline{K}/K) \to \GL(2,\Z/m\Z)

defined up to conjugation. In particular, when m=\ell is a prime different from the characteristic of K, we have the mod-\ell Galois representation

\overline{\rho}_{E,\ell}: \Gal(\overline{K}/K) \to \GL(2,\Z/\ell\Z).

Taking the inverse limit over prime powers m=\ell^n yields the \ell-adic Galois representation attached to E,

\rho_{E,\ell}: \Gal(\overline{K}/K) \to \Aut(T_{\ell}(E)) \cong \GL(2,\Z_{\ell}),

which describes the action of the absolute Galois group of K on T_{\ell}(E), the \ell-adic Tate module of E.

When K has characteristic zero one can take the inverse limit over all positive integers m (ordered by divisibility) to obtain the adelic Galois representation

\rho_{E}: \Gal(\overline{K}/K) \to \GL(2,\hat{\Z}).

If E is an elliptic curve without complex multiplication that is defined over a number field, then the image of \rho_E is an open subgroup of \GL(2,\hat{\Z}) that has an associated level, index, and genus.

Depends on: Definition 3.1 Definition 3.8 Definition 3.44 Definition 1.66 Definition 1.67 Definition 1.72 Definition 1.84 Definition 2.1

Image of the adelic Galois representation. (LMFDB)

The image of the adelic Galois representation associate to an elliptic curve E over a number field K that does not have potential complex multiplication is an open subgroup H of \GL(2,\widehat{\Z}). The subgroup H has the following invariants:

• The level of H is the least positive integer N such that H is the full inverse image of its projection to \GL(2,\Z/N\Z).

• The index of H is the positive integer [\GL(2,\Z/N\Z):H].

• The genus of H is the genus of the corresponding modular curve X_H.

Depends on: Definition 3.8 Definition 3.13 Definition 1.72 Definition 1.126

Image of mod-l Galois representation. (LMFDB)

Let \ell be a prime and let E be an elliptic curve defined over a number field K.

Subgroups G of \GL(2,\F_{\ell}) that can arise as the image of the mod-\ell Galois representation

\bar\rho_{E,\ell}\colon {\Gal}(\overline{K}/K)\to \GL(2,\F_{\ell})

attached to E that do not contain \SL(2,\F_{\ell}) are identified using the labels introduced by Sutherland in . For groups with surjective determinant map (necessarily the case when K=\Q), these labels have the form

\texttt{LS.a.b.c},

where \texttt{L} is the prime \ell, \texttt{S} is one of G, B, Cs, Cn, Ns, Nn, A4, S4, A5, and \texttt{a}, \texttt{b}, \texttt{c} are optional positive integers. When the determinant map is not surjective the label has "\texttt{[d]}", where d is the index of the determinant image in \F_{\ell}^\times.

When \bar\rho_{E,\ell} does not contain \SL(2,\F_{\ell}) the possibilities for \texttt{S} are: Borel B, split Cartan Cs, normalizer of the split Cartan Ns, nonsplit Cartan Cn, normalizer of the nonsplit Cartan Nn, exceptional A4, S4, A5. The cases A4 and A5 cannot occur when K=\Q.

Depends on: Definition 3.1 Definition 3.13 Definition 1.63 Definition 1.65 Definition 1.68 Definition 1.70 Definition 1.71 Definition 1.74 Definition 2.1

Geometric endomorphism ring. (LMFDB)

The geometric endomorphism ring of an elliptic curve E over a field K is \End(E_{\overline{K}}), the endomorphism ring of the base change of E to an algebraic closure \overline{K} of K.

This is a special case of the geometric endomorphism ring of an abelian variety.

Depends on: Definition 1.5 Definition 1.16 Definition 3.12

Global minimal model. (LMFDB)

A global minimal model for an elliptic curve E defined over a number field K is a Weierstrass equation for E which is integral and is a local minimal model at all primes of K.

When K has class number 1 all elliptic curves over K have global minimal models. In general, there is an obstruction to the existence of a global minimal model for each elliptic curve E defined over K, which is an ideal class of K. In case this class is nontrivial for E, there is a semi-global minimal model for E, which is minimal at all primes except one, the ideal class of that one prime being the obstruction class.

Depends on: Definition 3.1 Definition 3.21 Definition 3.34 Definition 3.97 Definition 2.1 Definition 2.6 Definition 2.24