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 3.37 Definition 3.44 Definition 3.81 Definition 3.84 Definition 3.92 Definition 3.94 Definition 1.76 Definition 1.84 Definition 1.91 Definition 1.99 Definition 2.1 Definition 2.14 Definition 2.41 Definition 2.44

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 3.41 Definition 3.91 Definition 2.1 Definition 2.44

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 3.29

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 3.44 Definition 3.81 Definition 3.84 Definition 3.92 Definition 3.94 Definition 1.108 Definition 1.109 Definition 1.111 Definition 1.118 Definition 2.1 Definition 2.14 Definition 2.44

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 3.37 Definition 2.1 Definition 2.66

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 3.29 Definition 2.6 Definition 2.39 Definition 1.131

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 3.39 Definition 2.1 Definition 2.52

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 3.96 Definition 1.133

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 3.43 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 1.125

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 3.33 Definition 3.96 Definition 2.1 Definition 2.6 Definition 2.23

Good ordinary reduction. (LMFDB)

An elliptic curve E defined over a number field K is said to have ordinary reduction at a prime \mathfrak{p} of K if the reduction E_{\mathfrak{p}} of E modulo \mathfrak{p} is smooth, and E_{\mathfrak{p}} is ordinary.

An elliptic curve E_{\mathfrak{p}} defined over a finite field of characteristic p is ordinary if E_{\mathfrak{p}}(\overline{\F_p}) has nontrivial p-torsion.

Depends on: Definition 3.1 Definition 2.1

Good reduction. (LMFDB)

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

If E has good reduction at every prime of K then E is said to have everywhere good reduction.

Depends on: Definition 1.12 Definition 3.1 Definition 2.1 Definition 2.44

Good supersingular reduction. (LMFDB)

An elliptic curve E defined over a number field K is said to have supersingular reduction at a prime \mathfrak{p} of K if the reduction E_{\mathfrak{p}} of E modulo \mathfrak{p} is smooth, and E_{\mathfrak{p}} is supersingular.

An elliptic curve E_{\mathfrak{p}} defined over a finite field of characteristic p is supersingular if E_{\mathfrak{p}}(\overline{\F_p}) has no p-torsion.

Depends on: Definition 3.1 Definition 2.1

Integral model. (LMFDB)

An integral model for an elliptic curve E defined over a number field K is a Weierstrass equation for E all of whose coefficients are in the ring of integers of K.

Depends on: Definition 3.1 Definition 3.96 Definition 2.52

Elliptic curve invariants. (LMFDB)

The invariants of an elliptic curve E over a number field K are its

• conductor, \mathfrak{N}, which is an integral ideal of K whose norm is the conductor norm N(\mathfrak{N})

• minimal discriminant, \mathfrak{D}, also an integral ideal of K, whose norm is the minimal discriminant norm N(\mathfrak{D})

• j-invariant, j

• endomorphism ring, \text{End}(E)

• Sato-Tate group, \text{ST}(E)

Each Weierstrass model for E also has a discriminant, \Delta, and discriminant norm, N(\Delta), which are not strictly invariants of E since different models have, in general, different discriminants.

Depends on: Definition 3.1 Definition 3.9 Definition 3.10 Definition 3.12 Definition 3.29 Definition 3.36 Definition 3.96 Definition 2.1 Definition 1.136 Definition 1.148

Isogeny between elliptic curves. (LMFDB)

Let E_1 and E_2 be two elliptic curves defined over a field K. An isogeny (over K) between E_1 and E_2 is a non-constant morphism f\colon E_1 \to E_2 defined over K, i.e., a morphism of curves given by rational functions with coefficients in K, such that f(O_{E_1})= O_{E_2}. Elliptic curves E_1 and E_2 are called isogenous if there exists an isogeny f\colon E_1 \to E_2.

An isogeny respects the group laws on E_1 and E_2, and hence determines a group homomorphism E_1(L)\to E_2(L) for any extension L of K. The kernel is a finite group, defined over K; in general the points in the kernel are not individually defined over K but over a finite Galois extension of K and are permuted by the Galois action.

The degree of an isogeny is its degree as a morphism of algebraic curves. For a separable isogeny this is equal to the cardinality of the kernel. Over a field of characteristic 0 such as a number field, all isogenies are separable. In finite characteristic p, isogenies of degree coprime to p are all separable.

An isogeny is cyclic if its kernel is a cyclic group. Every isogeny is the composition of a cyclic isogeny with the multiplication-by-m map for some m\ge1.

Isogeny is an equivalence relation, and the equivalence classes are called isogeny classes. Over a number field, it is a consequence of a theorem of Shafarevich that isogeny classes are finite. Between any two curves in an isogeny class there is a unique degree of cyclic isogeny between them, except when the curves have additional endomorphisms defined over the base field of the curves; in that case there are cyclic isogenies of infinitely many different degrees between any two isogenous curves.

Isogenies from an elliptic curve E to itself are called endomorphisms. The set of all endomrpshisms of E forms a ring under pointwise addition and composition, the endomorphism ring of E.

An isogeny of elliptic curves is a special case of an isogeny of abelian varieties.

Depends on: Definition 1.8 Definition 1.41 Definition 3.1 Definition 3.11 Definition 3.12 Definition 1.87 Definition 2.1 Definition 2.29

Isogeny class of an elliptic curve. (LMFDB)

The isogeny class (over a field K) of an elliptic curve E defined over K is the set of all isomorphism classes of elliptic curves defined over K that are isogenous to E over K. Over a number field K this is always a finite set; over \Q, it has at most 8 elements by a theorem of Kenku .

Depends on: Definition 3.1 Definition 3.23

Isogeny class degree. (LMFDB)

The isogeny class degree of an isogeny class of elliptic curves is the least common multiple of the degrees of all rational cyclic isogenies between elliptic curves in the isogeny class.

Depends on: Definition 3.1 Definition 3.23 Definition 3.24

Isogeny graph of an isogeny class of elliptic curves. (LMFDB)

The isogeny graph of an isogeny class of elliptic curves is the graph whose vertices are the isomorphism classes (over the base field) of elliptic curves in the isogeny class and whose edges are the isogenies of prime degree between the curves representing the vertices.

The vertices of the isogeny graphs in the LMFDB are labeled by the final entry of the LMFDB label of the corresponding (isomorphism classes of) elliptic curves. Their edges, of which there may be several between any two given vertices, are labeled by the prime that is the degree of the corresponding isogeny.

Depends on: Definition 3.1 Definition 3.24

Isogeny matrix of an isogeny class of elliptic curves. (LMFDB)

The isogeny matrix of an isogeny class of elliptic curves is a symmetric matrix with integral entries that records the minimum among the degrees of the cyclic isogenies between the elliptic curves in the isogeny class.

In the LMFDB, the rows and columns of the matrices are ordered by the final entry of the label of the elliptic curves in the isogeny class in question, so that the (i,j)-th entry is the smallest possible degree of a cyclic isogeny between the i-th and j-th curve in the isogeny class.

Depends on: Definition 3.1 Definition 3.24

Isomorphism of elliptic curves. (LMFDB)

An isomorphism between two elliptic curves E, E' defined over a field K is an isogeny f:E\to E' such that there exist an isogeny g:E'\to E with the compositions g\circ f and f\circ g being the identity maps. Equivalently, an isomorphism E\to E' is an isogeny of degree 1.

Isomorphism is an equivalence relation, the equivalnce classes being called isomorphism classes.

When E and E' are defined by Weierstrass models, such an isomorphism is uniquely represented as a Weierstrass isomorphism between these models.

Depends on: Definition 3.1 Definition 3.23

j-invariant of an elliptic curve. (LMFDB)

The j-invariant of an elliptic curve E defined over a field K is an invariant of the isomorphism class of E over \overline{K}. If the Weierstrass equation of E is

y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,

then its j-invariant is given by

j = \frac{c_4^3}{\Delta} 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\\ c_4 &= b_2^2 - 24b_4 \end{aligned}

and

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

is the discriminant of E.

Depends on: Definition 3.1 Definition 3.10 Definition 3.28 Definition 3.96

Kodaira symbol. (LMFDB)

The Kodaira symbol of an elliptic curve E defined over a number field encodes the reduction type of E at a prime \mathfrak{p} of K. It describes the combinatorics of the special fiber of the Ne;ron model of the elliptic curve. The Ne;ron model is obtained from the local minimal model for E at \mathfrak{p} using Tate's algorithm. For an exact definition and properties, consult a text on elliptic curves.

Depends on: Definition 3.1 Definition 3.33 Definition 3.83 Definition 2.1

Local data of an elliptic curve. (LMFDB)

The local data of an elliptic curve E defined over a number field K at a prime \mathfrak{p} of K consists of

• the Tamagawa number c_{\mathfrak{p}}

• the Kodaira symbol

• the reduction type

• the local root number

• the conductor valuation \text{ord}_{\mathfrak{p}}(\mathfrak{N})

• the discriminant_valuation \text{ord}_{\mathfrak{p}}(\mathfrak{D})

• the j-invariant denominator valuation \text{ord}_{\mathfrak{p}}(j)_{-}

Depends on: Definition 3.1 Definition 3.30 Definition 3.83 Definition 3.92 Definition 2.1

Local minimal discriminant of an elliptic curve. (LMFDB)

Let E be an elliptic curve defined over a number field K, and \mathfrak{p} a prime of K. The local minimal discriminant of E is the ideal \mathfrak{p}^e where e is is the valuation of the discriminant of a local minimal model for E at \mathfrak{p}.

Depends on: Definition 3.1 Definition 3.33 Definition 2.1

Local minimal model. (LMFDB)

A local minimal model for an elliptic curve E defined over a number field K at a prime \mathfrak{p} of K is a Weierstrass equation for E all of whose coefficients are integral at \mathfrak{p}, and whose discriminant has minimal valuation at \mathfrak{p} among all such equations.

Depends on: Definition 3.1 Definition 3.10 Definition 3.96 Definition 2.1 Definition 2.26

Maximal l-adic Galois representation. (LMFDB)

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

\rho_{E,\ell}\colon \Gal(\overline{K}/K)\to \Aut(E[\ell^\infty]) \simeq \GL_2(\Z_{\ell})

be the \ell-adic Galois representation associated to E.

If E does not have potential complex multiplication, then \rho_{E,\ell} is maximal if its image contains \SL_2(\Z_{\ell}).

In general, let \mathcal{O} be the geometric endomorphism ring of E. Then E[\ell^\infty] is an \mathcal{O}-module, and we view \Aut_{\mathcal{O}}(E[\ell^\infty]) as a subgroup of \Aut(E[\ell^\infty]) \simeq \GL_2(\Z_{\ell}) that contains the image of \rho_{E,\ell} whenever K contains \mathcal O. We say that \rho_{E,\ell} is maximal if its image contains \SL_2(\Z_{\ell}) \cap \Aut_{\mathcal{O}}(E[\ell^\infty]), in which case we call \ell a maximal prime for E.

Depends on: Definition 3.1 Definition 3.8 Definition 3.13 Definition 3.16 Definition 2.1

Maximal Galois representation. (LMFDB)

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

\bar\rho_{E,\ell}\colon \Gal(\overline{K}/K)\to \Aut(E[\ell]) \simeq \GL_2(\F_{\ell})

be the mod-\ell Galois representation associated to E.

If E does not have potential complex multiplication, then \bar\rho_{E,\ell} is maximal if its image contains \SL_2(\F_{\ell}).

In general, let \mathcal{O} be the geometric endomorphism ring. Then E[\ell] is an \mathcal{O}-module and we view \Aut_{\mathcal{O}}(E[\ell]) \leq \Aut(E[\ell]) \simeq \GL_2(\F_{\ell}). We say that \bar\rho_{E,\ell} is maximal if its image contains \SL_2(\F_{\ell}) \cap \Aut_{\mathcal{O}}(E[\ell]).

For K=\Q, the image of a maximal \bar\rho_{E,\ell} is \GL_2(\F_{\ell}), a Borel subgroup, the normalizer of a split Cartan subgroup, or the normalizer of a non-split Cartan subgroup, depending on whether \mathcal{O}=\Z or \mathcal{O}\ne \Z and \ell is ramified, split, or inert in \mathcal{O}, respectively.

Depends on: Definition 3.1 Definition 3.8 Definition 3.13 Definition 3.16 Definition 1.63 Definition 1.70 Definition 1.71 Definition 2.1

Minimal discriminant. (LMFDB)

The minimal discriminant (or minimal discriminant ideal) of an elliptic curve E over a number field K is the ideal \mathfrak{D}_{min} of the ring of integers of K given by

\mathfrak{D}_{\mathrm{min}} = \prod_{\mathfrak{p}}\mathfrak{p}^{e_{\mathfrak{p}}},

where the product is over all primes \mathfrak{p} of K, and \mathfrak{p}^{e_{\mathfrak{p}}} is the local minimal discriminant of E at \mathfrak{p}.

If E has a Weierstrass model which is a global minimal model then \mathfrak{D}_{\mathrm{min}} = (\Delta), the principal ideal generated by the discriminant \Delta of this model. In general, \mathfrak{D}_{\mathrm{min}} differs from the ideal generated by the discriminant of any Weierstrass model by the 12th power of an ideal.

Depends on: Definition 3.1 Definition 3.10 Definition 3.17 Definition 3.32 Definition 3.96 Definition 2.1 Definition 2.44 Definition 2.52

Mordell-Weil group. (LMFDB)

For an elliptic curve E defined over a field K, the Mordell-Weil group of E/K is the group E(K) of K-rational points of E. It is a finitely-generated Abelian group.

This is a special case of the Mordell-Weil group of an abelian variety.

The Mordell-Weil Theorem, first proved by Mordell for elliptic curves defined over \Q and later generalized by Weil to abelian varieties A over general number fields K, states that, if K is a number field, then A(K) is a finitely generated abelian group. Its rank is called the Mordell-Weil rank of A over K.

The Mordell-Weil theorem implies in particular that the torsion subgroup E(K)_{\mathrm{tor}} of E(K) is finite, and thus that the torsion order of E, one of the BSD invariants, is finite.

Depends on: Definition 1.5 Definition 1.22 Definition 3.1 Definition 2.1

Mordell-Weil theorem. (LMFDB)

For an elliptic curve E defined over a number field F, the Mordell-Weil theorem states that the set E(F) of F-rational points on E is a finitely generated Abelian group.

This group is called the Mordell-Weil group of E/K.

Depends on: Definition 3.1 Definition 3.37 Definition 2.1

Multiplicative reduction. (LMFDB)

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

The case of multiplicative reduction is further subdivided into split multiplicative reduction and nonsplit multiplicative reduction.

Depends on: Definition 3.1 Definition 3.41 Definition 3.91 Definition 2.1

Mordell-Weil generators. (LMFDB)

The Mordell-Weil group E(K) of an elliptic curve E over a number field K is a finitely generated abelian group, explicitly described by giving a \Z-basis for the group, equivalently, a (minimal) set of Mordell-Weil generators, each of which is a rational point on the curve.

The generators consist of r non-torsion generators, where r is the rank of E(K), and up to two torsion generators, which generate the torsion subgroup E(K)_{\textrm{tor}}.

Depends on: Definition 3.1 Definition 3.37 Definition 3.81 Definition 3.94 Definition 1.76 Definition 2.1

Non-split multiplicative reduction. (LMFDB)

An elliptic curve E defined over a number field K is said to have non-split multiplicative reduction at a prime \mathfrak{p} of K if the reduction of E modulo \mathfrak{p} has a nodal singularity with tangent slopes not defined over the residue field at \mathfrak{p}.

Depends on: Definition 3.1 Definition 2.1

Obstruction class of an elliptic curve. (LMFDB)

Let E be an elliptic curve defined over a number field K. The obstruction class of E is an ideal class of K which is trivial if and only if E has a global minimal model.

Depends on: Definition 3.1 Definition 3.17 Definition 2.1 Definition 2.23

Tate module of an elliptic curve. (LMFDB)

Let p \in \mathbb{Z}_{\geq 0} be a prime and E an elliptic curve defined over a field K. The p-adic Tate module of E is the inverse limit

T_p(E) = \lim_{\xleftarrow[n \in \mathbb{N}]{}} E[p^n].

Here for m\in\mathbb{Z}_{\geq 0}, E[m] denotes the m-torsion subgroup of E, which is the kernel of the multiplication-by-m endomorphism of E.

If K has characteristic not equal to p, then T_p(E) is a free \Z_p-module of rank 2. It carries an action of the absolute Galois group of K, and thus has an associated Galois representation.

This is a special case of the Tate module of an abelian variety.

Depends on: Definition 1.43 Definition 3.11 Definition 1.84

Global period of an elliptic curve. (LMFDB)

The global period \Omega(E/K) of an elliptic curve defined over a number field K is a product of local factors \Omega_v(E_v/K_v), one for each infinite place v of K. Here, K_v denotes the completion of K at v (so K_v=\R for a real place and K_v=\C for a complex place), and E_v denotes the base change of E to K_v.

Fixing a Weierstrass model for E with coefficients a_i\in K, a model for E_v is given by the Weierstrass equation with coefficients a_{i,v}, the images of a_i under v in K_v. Associated to this model we have a discriminant \Delta(E_v) and an invariant differential \omega_v=dx/(2y+a_{1,v}x+a_{3,v}).

For a real place given by an embedding v:K\to\R, we define

\Omega_v(E_v) = \left|\int_{E_v(\R)} \omega_E\right|.

In terms of a basis of the period lattice of E_v of the form [x,yi] (when \Delta(E_v)>0) or [2x,x+yi] (when \Delta(E_v)<0), where x and y are positive real numbers, we have \Omega_v(E_v)=2x.

For a complex place given by an embedding v:K\to\C, we define

\Omega_v(E_v) = \left|\int_{E_v(\C)} \omega_E\wedge\overline{\omega_E}\right|.

In terms of a basis [w_1,w_2] of the period lattice of E_v, where \Im(w_2/w_1)>0, we have \Omega_v(E_v)=2\Im(\overline{w_1}w_2), which is double the covolume of the period lattice.

When E has a global minimal model, we have

\Omega(E/K) = \prod_{v}\Omega_v(E_v).

In general, given an arbitrary model for E with discriminant \Delta(E), we have

\Omega(E/K) = \left|\frac{N(\Delta(E))}{N(\mathfrak{d}(E))}\right|^{1/12}\prod_{v}\Omega_v(E_v),

where \mathfrak{d} is the minimal discriminant ideal of E and N(\mathfrak{d}) denotes its norm. This quantity is independent of the model of E.

Depends on: Definition 3.1 Definition 3.5 Definition 3.10 Definition 3.17 Definition 3.36 Definition 3.70 Definition 3.96 Definition 2.1 Definition 2.16 Definition 2.41

Potential good reduction. (LMFDB)

An elliptic curve E defined over a number field K is said to have potential good reduction if E has everywhere good reduction over a finite extension of K.

This is equivalent to the j-invariant of E being integral.

Depends on: Definition 3.1 Definition 3.19 Definition 3.29 Definition 2.1 Definition 2.26

Elliptic curve over \mathbb Q. (LMFDB)

An elliptic curve E over \mathbb{Q} has a Weierstrass equation of the form

E : y^2 = x^3 + ax + b with a, b \in \mathbb{Z} such that its discriminant

\Delta := -16(4a^3 + 27b^2 ) \not = 0.

Note that such an equation is not unique and E has a unique minimal Weierstrass equation.

Depends on: Definition 3.54 Definition 3.65

abc quality. (LMFDB)

Given a triple a,b,c of nonzero coprime integers, the quality of the triple is defined as

Q = \frac{\log \max(|a|, |b|, |c|)}{\log \operatorname{rad}(abc)},

where \operatorname{rad}(abc) is the product of the primes dividing abc. The abc conjecture stipulates that for any \epsilon > 0 there are only finitely many relatively prime triples a,b,c with quality larger than 1+\epsilon.

The abc quality of an elliptic curve E is the quality of an a,b,c triple determined by its j-invariant, namely the one defined by writing \frac{j}{1728} = \frac{a}{c} in lowest terms and setting b = c - a. Note that the abc quality is undefined for j=0 and j=1728.

The reason for defining the quality of E in this way comes from the equivalence of the abc conjecture with the modified Szpiro conjecture. For elliptic curves with small conductor, j-invariants often have unusually large quality.

Depends on: Definition 3.1 Definition 3.29 Definition 3.52

Analytic rank of an elliptic curve over \Q. (LMFDB)

The analytic rank of an elliptic curve E is the analytic rank of its L-function L(E,s). The weak form of the BSD conjecture implies that the analytic rank is equal to the rank of the Mordell-Weil group of E.

For elliptic curves E over \Q, it is known that L(E,s) satisfies the Hasse-Weil conjecture, and hence that the parity of the analytic rank is always compatible with the sign of the functional equation.

In general, analytic ranks stored in the LMFDB are only upper bounds on the true analytic rank (they could be incorrect if L(E,s) had a zero very close to but not on the central point). For elliptic curves over \Q of analytic rank less than 2 this upper bound is necessarily tight, due to parity; for analytic ranks 2 and 3 is also tight due to results of Kolyvagin; Murty and Murty; Bump, Friedberg and Hoffstein; Coates and Wiles; Gross and Zagier which together say that when the analytic rank is 0 or 1 then it equals the Mordell-Weil rank.

Depends on: Definition 1.22 Definition 3.1 Definition 3.37 Definition 1.108 Definition 1.111 Definition 1.116 Definition 1.122

Analytic order of \Sha. (LMFDB)

The Tate-Shafarevic group \Sha of an elliptic curve E defined over \mathbb{Q} is a torsion group defined in terms of Galois cohomology, which is conjectured to be finite. Its order \#\Sha appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for E. The value of the order which is predicted by the conjecture is called the Analytic Order of Sha, \Sha_{an}. <!--Note that the value of \Sha_an predicted by the conjecture is always a square.-->

For elliptic curves of rank 0 or 1 it is a theorem that \Sha_an is a positive rational number, and this rational number can be computed exactly; this exact computation has only been carried out for the curves in the database with rank 0 and conductor N\le 500000. These values are always in fact integer squares in all cases computed to date. For curves of rank 2 and above, there is no such theorem, and the values computed are simply floating point approximate values which happen to be very close to integers. In the LMFDB we store and display the rounded values in this case.

Depends on: Definition 3.1 Definition 1.91

Birch and Swinnerton-Dyer conjecture. (LMFDB)

The Birch and Swinnerton-Dyer conjecture is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing and the first non-zero Taylor series coefficient of the L-function associated to an elliptic curve E defined over \Q at the central point s=1 to certain arithmetic data, the BSD invariants of E.

Specifically, the BSD conjecture states that the order r of vanishing of L(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(\Q), and that

<!-- comment: if you make the following display into a normal one using .. or .. then something about the html code for Sha stops it displaying properly-->

<p align="center"> \displaystyle \frac{1}{r!} L^{(r)}(E,1)= \displaystyle \frac{\# \Sha(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2}. </p> The quantities appearing in this formula are the BSD invariants of E:

• r is the rank of E(\Q) (a non-negative integer);

• \#\Sha(E/\Q) is the order of the Tate-Shafarevich group

  of E (which is conjectured to always be finite, a positive integer);

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

• \Omega_E is the real period of E/\Q (a positive real number);

• c_p is the Tamagawa number of E at each prime p (a positive integer which is 1 for all but at most finitely many primes);

• E(\Q)_{\rm tor} is the torsion order of E(\Q) (a positive integer).

There is a similar conjecture for abelian varieties, in which the real period is replaced by the covolume of the period lattice.

Depends on: Definition 1.5 Definition 3.1 Definition 3.3 Definition 3.38 Definition 3.71 Definition 3.81 Definition 3.84 Definition 3.92 Definition 3.93 Definition 1.108 Definition 1.111

Canonical height. (LMFDB)

Let E be an elliptic curve defined over \mathbb{Q}. The canonical height of a rational point P\in E(\mathbb{Q}) is computed by writing the x-coordinate x(nP)=A_n(P)/D_n(P) as a fraction in lowest terms and setting

\hat{h}(P) = \lim_{n\to\infty} \frac{1}{n^2}\log \max\bigl\{|A_n(P)|,|D_n(P)|\bigr\}.

(<EM>Note</EM>. Some sources define \hat{h} to be \frac12 of this quantity.)

Properties of \hat{h}: <UL> <LI> \hat{h}(P)=\log \max\bigl\{|A_1(P)|,|D_1(P)|\bigr\}+O(1) as P ranges over E(\mathbb{Q}). <LI> \hat{h}(P)\ge0; and \hat{h}(P)=0 if and only if P is a torsion point. <LI> \hat{h}:E(\mathbb{Q})\to\mathbb{R} extends to a positive definite quadratic form on E(\mathbb{Q})\otimes\mathbb{R}. </UL> The height pairing on E is the associated bilinear form \langle P,Q\rangle=\frac{1}{2}\bigl(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)\bigr), which is used to compute the elliptic regulator of E. It is a symmetric positive definite bilinear form on E(\Q)\otimes\R.

For a number field K, the canonical height of P\in E(K) is given by \hat{h}(P)=\lim_{n\to\infty} n^{-2}h\bigl(x(nP)\bigr), where h is the Weil height.

Depends on: Definition 3.1 Definition 3.84 Definition 2.66

Conductor of an elliptic curve over \Q. (LMFDB)

The conductor N of an elliptic curve E defined over \Q is a positive integer divisible by the primes of bad reduction and no others. It has the form N=\prod p^{e_p}, where the exponent e_p is

• e_p=1 if E has multiplicative reduction at p,

• e_p=2 if E has additive reduction at p and p\ge5,

• 2\leq e_p\leq 5 if E has additive reduction and p=3, and

• 2\leq e_p\leq 8 if E has additive reduction and p=2.

For all primes p, there is an algorithm of Tate that simultaneously creates a local 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>

Depends on: Definition 3.1 Definition 3.2 Definition 3.39