Arithmetic function. (LMFDB)

An arithmetic function is a complex-valued function whose domain is the positive integers.

Bernoulli numbers. (LMFDB)

The Bernoulli numbers are the rational numbers B_n that appear as coefficients of the formal power series

\frac{T}{e^T-1}=\sum_{n\ge 0}B_n\frac{T^n}{n!},

which has radius of convergence 2\pi.

Divisor function. (LMFDB)

A divisor function is a multiplicative arithmetic function of the form

\sigma_{\tau}(n)=\sum_{d\mid n}d^\tau,

for some fixed \tau\in\mathbb{C}.

Depends on: Definition 1.1 Definition 1.4

Multiplicative arithmetic function. (LMFDB)

An arithmetic function f:\mathbb{Z}_{>0}\to \C is multiplicative if f(mn)=f(m)f(n) for all coprime integers m,n>0, and is not the zero-function (in particular, f(1)=1).

Abelian variety. (LMFDB)

An abelian variety defined over the field K is a smooth connected projective variety equipped with the structure of an algebraic group. The group law is automatically commutative.

An abelian variety of dimension 1 is the same as an elliptic curve.

Depends on: Definition 1.28 Definition 3.1

Affine space. (LMFDB)

Affine space \mathbb{A}^n(K) of dimension n over a field K is the set K^n.

If P=(x_1,\dots,x_n) is a point in \mathbb{A}^n(K), the x_i are called the *affine coordinates* of P. Thus

\mathbb{A}^n(K) = \{(x_1,\dots,x_n)\mid x_1,\dots,x_n\in K\}.

Depends on: Definition 1.133

Base change. (LMFDB)

Let V be an algebraic variety defined over a field K. If L/K is a field extension, then any set of equations that define V over K can be used to define an algebraic variety over L, the base change of V from K to L (typically denoted V_L).

An algebraic variety over a field L is said to be a base change if it is the base change of an algebraic variety defined over a proper subfield of L, equivalently, if its base field is not a minimal field of definition.

Depends on: Definition 1.28 Definition 1.133

Base field. (LMFDB)

The base field, of an algebraic variety is the field over which it is defined; it necessarily contains the coefficients of a set of defining equations for the variety, but it is not necessarily a minimal field of definition.

Depends on: Definition 1.5 Definition 1.21

Complex multiplication. (LMFDB)

A simple abelian variety of dimension g is said to have complex multiplication (CM) if its endomorphism algebra is a CM field of degree 2g, or equivalently, if its endomorphism ring is an order in a CM field of degree 2g.

Depends on: Definition 1.5 Definition 1.13 Definition 1.14 Definition 1.15 Definition 1.17 Definition 2.8 Definition 2.12 Definition 2.39

Algebraic curve. (LMFDB)

An algebraic curve is an algebraic variety of dimension 1.

Depends on: Definition 1.28

Genus of a smooth curve. (LMFDB)

The genus of a smooth projective geometrically integral curve C defined over a field k is the dimension of the k-vector space of regular differentials H^0(C, \omega_C). When k=\C this coincides with the topological genus of the corresponding Riemann surface.

The quantity defined above is sometimes also called the algebraic genus or the geometric genus of C. Because of our assumption on the smoothness of C, it coincides with the arithmetic genus H^1(C,\mathcal{O}_C).

Depends on: Definition 1.25

Smoothness of an algebraic curve. (LMFDB)

Let C be an algebraic curve over a perfect field k. Then C is called smooth if the extension of C to the algebraic closure of k is non-singular at all of its points.

Depends on: Definition 1.10 Definition 1.27

Dimension of an algebraic variety. (LMFDB)

The dimension of an algebraic variety V is the maximal length d of a chain

V_0 \subset V_1 \subset \cdots \subset V_d

of distinct irreducible subvarieties of V.

Depends on: Definition 1.19 Definition 1.28

Endomorphism algebra. (LMFDB)

The endomorphism algebra of an abelian variety A is the \Q-algebra \textrm{End}(A)\otimes\Q, where \textrm{End}(A) is the endomorphism ring of A.

Depends on: Definition 1.5 Definition 1.15

Endomorphism ring. (LMFDB)

An endomorphism of an abelian variety A over a field k is a homomorphism \varphi \colon A \to A defined over k. The set of endomorphisms of an abelian variety A can be given the structure of a ring in which addition is defined pointwise (using the group operation of A) and multiplication is composition; this ring is called the endomorphism ring of A, denoted \textrm{End}(A).

For endomorphisms defined over an extension of k, we instead speak about the geometric endomorphism ring.

Depends on: Definition 1.5

Geometric endomorphism ring. (LMFDB)

For an abelian variety A over a field F, the geometric endomorphism ring of A is \End(A_{\overline{F}}), the endomorphism ring of the base change of A to an algebraic closure \overline{F} of F.

Depends on: Definition 1.15

Geometrically simple. (LMFDB)

An abelian variety over a field k is geometrically (or absolutely) simple if it is simple when viewed as a variety over \bar k.

Depends on: Definition 1.5 Definition 1.26

Hyperelliptic curve. (LMFDB)

A hyperelliptic curve X over a field k is a smooth projective algebraic curve of genus g\ge 2 that admits a 2-to-1 map X\to \mathbb{P}^1 defined over the algebraic closure \bar k.

If X is a hyperelliptic curve over k, then the canonical map X \to \mathbb{P}^{g-1} is a 2-to-1 map onto a smooth genus 0 curve Y. The curve Y is isomorphic to \mathbb{P}^1 if and only if Y has a k-rational point.

If X admits a 2-to-1 map to \mathbb{P}^1 that is defined over k, then X has a Weierstrass model of the form y^2+h(x)y=f(x); when the characteristic of k is not 2 one can complete the square to put this model in the form y^2=f(x).

In general, there is always a model for X in \mathbb{P}^3 of the form

h(x,y,z)=0\qquad w^2=f(x,y,z)

where h(x,y,z) is a homogeneous polynomial of degree 2 (a conic) and f(x,y,z) is a homogeneous polynomial of degree g+1.

Depends on: Definition 1.10 Definition 1.11

Irreducible variety. (LMFDB)

A variety defined over a field F is irreducible if it is nonempty and cannot be decomposed as the union of two strictly smaller varieties over F. It is geometrically irreducible if it remains irreducible when seen as a variety over the algebraic closure of F.

Depends on: Definition 1.28

Jacobian of a curve. (LMFDB)

The Jacobian of a (smooth, projective, geometrically integral) curve X of genus g over a field k is a g-dimensional principally polarized abelian variety J that is canonically associated to X.

If X has a k-rational point, then J(k) is isomorphic to the group of degree zero divisors on X modulo linear equivalence. A choice of rational point on X determines a morphism X \to J called an Abel-Jacobi map; it is an embedding if and only if g \ge 1, and an isomorphism if and only if g=1.

The Torelli theorem states that if X and Y are curves whose Jacobians are isomorphic as *principally polarized* abelian varieties, then X and Y are isomorphic. It is possible, however, for non-isomorphic curves to have Jacobians that are isomorphic as unpolarized abelian varieties.

Depends on: Definition 1.5

Minimal field of definition. (LMFDB)

Let V/k be an algebraic variety defined over a field k and let S be the set of subfields k_0\subseteq k for which there exists an algebraic variety V_0/k_0 whose base change to k is isomorphic to V.

Any field k_0\in S that contains no other elements of S is a minimal field of definition for V.

In general, an algebraic variety may have more than one minimal field of definition; this does not occur for elliptic curves but it does occur for curves of genus 2.

Depends on: Definition 1.7 Definition 1.11 Definition 1.28 Definition 1.133

Mordell-Weil group of an abelian variety. (LMFDB)

The Mordell-Weil group of an abelian variety A over a number field K is its group of K-rational points A(K).

Weil, building on Mordell's theorem for elliptic curves over \Q, proved that the abelian group A(K) is finitely generated. Thus

A(K)\simeq \Z^r \oplus T,

where r is a nonnegative integer called the Mordell-Weil rank of A, and T is a finite abelian group called the torsion subgroup.

The torsion subgroup T is the product of at most 2g cyclic groups, where g is the dimension of A.

Depends on: Definition 1.5 Definition 1.13

Projective space. (LMFDB)

Projective space \mathbb{P}^n(K) of dimension n over a field K is the set (K^{n+1}\setminus\{0\})/{}\sim{}, where

(x_0,x_1,\dots,x_n) \sim (y_0,y_1,\dots,y_n) \iff x_0=\lambda y_0, \dots, x_n=\lambda y_n\quad\text{for some}\ \lambda\in K^*.

The equivalence class of (x_0,x_1,\dots,x_n) in \mathbb{P}^n(K) is denoted by (x_0:x_1:\dots:x_n), and the x_i are called homogeneous coordinates. Thus

\mathbb{P}^n(K) = \{(x_0:\dots:x_n)\mid x_0,\dots,x_n\in K,\ \text{not all zero}\}.

Depends on: Definition 1.133

Quotient curve. (LMFDB)

Let X be an algebraic curve and let H be a finite subgroup of its automorphism group.

The quotient curve X/H is the algebraic curve obtained by identifying points of X that lie in the same H-orbit (equations defining X/H as an algebraic variety of dimension one can be constructed from the equations defining X and the automorphisms in H).

The natural projection X\to X/H that sends each point on X to its H-orbit is a surjective morphism

Depends on: Definition 1.10 Definition 1.28 Definition 1.57

Riemann surface. (LMFDB)

A Riemann surface is a connected complex manifold of dimension one. Compact Riemann surfaces can be identified with smooth projective curves over \C.

Simple. (LMFDB)

An abelian variety is simple if it is nonzero and not isogenous to a product of abelian varieties of lower dimension.

Depends on: Definition 1.5

Non-singular point (definition). (LMFDB)

Let V be a variety over a perfect field F. A point P of V is non-singular if the module of differentials of V is locally free at P. According to the Jacobian criterion, if V is defined in a neighborhood of P by affine polynomial equations f_1(X_1, \ldots, X_n) = \ldots =f_r(X_1, \ldots, X_n)=0, then V is non-singular at P if the Jacobian matrix \left( \frac{\partial f_i}{\partial X_j} \right)_{ij} has the same rank as the codimension of V in \mathbb{A}^n.

Algebraic variety. (LMFDB)

There are two main kinds of algebraic varieties, *affine varieties* and *projective varieties*. Both are defined as the set of common zeros of a collection of polynomials. Let K be a field with algebraic closure \overline{K}.

An affine algebraic set is a subset of affine space \mathbb{A}^n(\overline{K}) of the form

V(I) = \{P \in \mathbb{A}^n(\overline{K}) : f(P) = 0\text{ for all }f \in I\}

where I \subseteq \overline{K}[x_1,\dots,x_n] is an ideal. Given an affine algebraic set V, its defining ideal is

I(V) = \{ f \in \overline{K}[x_1,\dots,x_n] : f(P)=0\text{ for all }P \in V\}.

An affine variety over \overline{K} is an affine algebraic set whose defining ideal I \subseteq \overline{K}[x_1,\dots,x_n] is a prime ideal. An affine variety over K is an affine variety over \overline{K} whose defining ideal can be generated by polynomials in K[x_1,\dots,x_n].

We define projective notions similarly. A projective algebraic set is a subset of projective space \mathbb{P}^n(\overline{K}) defined by a *homogeneous* ideal I \subseteq \overline{K}[x_1,\dots,x_n]. A projective variety over \overline{K} is a projective algebraic set whose defining ideal is a homogeneous prime ideal. A projective variety over K is a projective variety over \overline{K} whose defining ideal can be generated by homogeneous polynomials in K[x_1,\dots,x_n].

Depends on: Definition 1.6 Definition 1.23 Definition 1.133 Definition 1.143

Binary operation. (LMFDB)

A binary operation on a set S is a function S\times S\to S.

If the operation is denoted by *, then the output of this function applied to (s_1,s_2) is typically denoted s_1*s_2.

Associative binary operation. (LMFDB)

If * is a binary operation on a set A, then * is associative on A if for all a,b,c\in A,

a*(b*c)=(a*b)*c. /div>

Depends on: Definition 1.29

Commutative binary operation. (LMFDB)

If * is a binary operation on a set A, then * is commutative on A if for all a,b\in A,

a*b=b*a.

Depends on: Definition 1.29

Identity for a binary operation. (LMFDB)

If * is a binary operation on a set A, then A has an identity element with respect to * if there exists e\in A such that for all a\in A,

a*e = e*a = a.

Such an identity element e, if it exists, is unique and is thus called the identity element of A with respect to *.

Depends on: Definition 1.29

Inverse for a binary operation. (LMFDB)

If * is a binary operation on a set A having identity element e\in A, then an element a\in A has an inverse in A with respect to * if there exists a'\in A such that

a*a' = a'*a = e.

Depends on: Definition 1.29 Definition 1.32

Symplectic isomorphism. (LMFDB)

Let N \ge 1. Let \mu_N be the group of Nth roots of unity in some algebraically closed field of characteristic not dividing N. Let M be a free rank 2 \Z/N\Z-module together with an isomorphism \alpha \colon \bigwedge^2 M \stackrel{\sim}\to \mu_N, or equivalently with a nondegenerate alternating pairing M \times M \to \mu_N. For example, M could be E[N] for an elliptic curve E, together with the Weil pairing. Or M could be \Z/N\Z \times \mu_N with the "determinant" pairing (a,\gamma),(b,\delta) \mapsto \delta^a/\gamma^b.

A symplectic isomorphism from M to another such structure M' is a \Z/N\Z-module isomorphism M \to M' such that the induced isomorphism \bigwedge^2 M \to \bigwedge^2 M' gets identified via \alpha and \alpha' with the identity \mu_N \to \mu_N.

The same definition makes sense in a context in which each free rank 2 \Z/N\Z-module is enriched with a Galois action to make a Galois module, or replaced by a finite etale group scheme that is (\Z/N\Z)^2 etale locally.

Artin representation (definition). (LMFDB)

An Artin representation is a continuous homomorphism \rho:\mathrm{Gal}(\overline{\Q}/\Q)\to\GL(V) from the absolute Galois group of \Q to the automorphism group of a finite-dimensional \C-vector space V. Here continuity means that \rho factors through the Galois group of some finite extension K/\Q. The smallest such K is called the Artin field of \rho.

Depends on: Definition 1.84 Definition 2.20

Conductor of an Artin representation. (LMFDB)

The conductor of an Artin representation is a positive integer that measures its ramification. It can be expressed as a product of local conductors.

Let K/\Q be a Galois extension and \rho:\Gal(K/\Q)\to\GL(V) an Artin representation. Then the conductor of \rho is \prod_p p^{f(\rho,p)} for non-negative integers f(\rho,p), where the product is taken over prime numbers p.

To define the exponents f(\rho,p), fix a prime \mathfrak{p} of K above p and consider the corresponding extension of local fields K_{\mathfrak{p}}/\Q_p with Galois group G. Then G has a filtration of higher ramification groups in lower numbering G_i, as defined in Chapter IV of Serre's Local Fields . In particular, G_{-1}=G, G_0 is the inertia group of K_{\mathfrak{p}}/\Q_p, and G_1 is the wild inertia group, which is a finite p-group.

Let g_i = |G_i|. Then

f(\rho, p) = \sum_{i\geq 0} \frac{g_i}{g_0} (\dim(V) - \dim(V^{G_i}))

where V^{G_i} is the subspace of V fixed by G_i.

Note that if p is unramified in K, then f(\rho,p)=0 and conversely, if \rho is faithful and p is ramified in K, then f(\rho,p)>0.

Depends on: Definition 1.35 Definition 1.40 Definition 1.62 Definition 1.101 Definition 1.104 Definition 1.107 Definition 2.29 Definition 2.44

Number field associated to an Artin representation. (LMFDB)

The Artin field is a number field associated to an Artin representation \rho:\mathrm{Gal}(\overline{\Q}/\Q)\to\GL(V) by being the smallest Galois extension K/\mathbb{Q} such that \rho factors through \mathrm{Gal}(K/\Q).

Depends on: Definition 1.35 Definition 2.1

Parity of a representation. (LMFDB)

An Artin representation \rho:\Gal(\overline{\Q}/\Q)\to \GL(V) is even or odd if \det(\rho(c)) equals 1 or -1, respectively, where c is a complex conjugation.

Depends on: Definition 1.35

Ramified prime of an Artin representation. (LMFDB)

If \rho:\Gal(\overline{\Q}/\Q)\to\GL_n(\C) is an Artin representation with Artin field K, then a prime p is ramified if it is ramified in K/\Q.

Equivalently, a prime is ramified if the inertia subgroup for a prime above p is not contained in the kernel of \rho.

Depends on: Definition 1.35 Definition 1.37 Definition 2.45

Unramified prime of an Artin representation. (LMFDB)

If \rho:\Gal(\overline{\Q}/\Q)\to\GL_n(\C) is an Artin representation, a prime p is unramified if it is not ramified.

Equivalently, a prime is unramified if the inertia subgroup for a prime above p in the Artin field of \rho is contained in the kernel of \rho.

Depends on: Definition 1.35 Definition 1.37 Definition 1.39

Isogeny of abelian varieties. (LMFDB)

An isogeny of abelian varieties is a surjective algebraic group homomorphism with finite kernel.

Two abelian varieties are isogenous if there is an isogeny between them. This defines an equivalence relation on the set of isomorphism classes. Equivalence classes are called isogeny classes.

Depends on: Definition 1.5

Simple abelian variety. (LMFDB)

An abelian variety is simple if it is nonzero and not isogenous to a product of abelian varieties of lower dimension.

Depends on: Definition 1.5

Tate module of an abelian variety. (LMFDB)

Let p \in \mathbb{Z}_{\geq 0} be a prime and A an abelian variety of dimension g defined over a field K. The p-adic Tate module of A is the inverse limit

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

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

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

Depends on: Definition 1.41 Definition 1.84

Twist of an abelian variety. (LMFDB)

A twist of an abelian variety A is an abelian variety A' over the same field that becomes isomorphic to A upon base change to an algebraic closure.

Depends on: Definition 1.5 Definition 1.7

Dirichlet character. (LMFDB)

A Dirichlet character is a function \chi: \Z\to \C together with a positive integer q called the modulus such that \chi is completely multiplicative, i.e. \chi(mn)=\chi(m)\chi(n) for all integers m and n, and \chi is periodic modulo q, i.e. \chi(n+q)=\chi(n) for all n. If (n,q)>1 then \chi(n)=0, whereas if (n,q)=1, then \chi(n) is a root of unity. The character \chi is primitive if its conductor is equal to its modulus.

Conductor of a Dirichlet character. (LMFDB)

The conductor of a Dirichlet character \chi modulo q is the least positive integer q_1 dividing q for which \chi(n+kq_1)=\chi(n) for all n and n+kq_1 coprime to q.

Depends on: Definition 1.45

Galois orbit of a Dirichlet character. (LMFDB)

The Galois orbit of a Dirichlet character \chi of modulus q and order n is the set [\chi]:=\{\sigma(\chi): \sigma\in \Gal(\Q(\zeta_n)/\Q)\}, where \sigma(\chi) denotes the Dirichlet character of modulus q defined by k \mapsto \sigma(\chi(k)). The map \chi\to \sigma(\chi) defines a faithful action of the Galois group \Gal(\Q(\zeta_n)/\Q) on the set of Dirichlet characters of modulus q and order n, each of which has \Q(\zeta_n) as its field of values.

Depends on: Definition 1.45 Definition 1.52 Definition 1.53 Definition 1.56 Definition 2.20

Orbit index of a Dirichlet character. (LMFDB)

The Galois orbits of Dirichlet characters of modulus q are ordered as follows. Let \chi be any character in the Galois orbit [\chi] and define the N-tuple of integers

t([\chi]) := (n,t_1,t_2,\ldots,t_{q-1}) \in \Z^q,

where n is the order of \chi and t_i:=\mathrm{tr}_{\Q(\chi)/\Q}(\chi(i)) is the trace of \chi(i) from the field of values of \chi to \Q. The q-tuple t([\chi]) is independent of the choice of representative \chi and uniquely identifies the Galois orbit [\chi].

The orbit index of \chi is the index of t([\chi]) in the lexicographic ordering of all such tuples arising for Dirichlet characters of modulus q; indexing begins at 1, which is always the index of the Galois orbit of the principal character of modulus 1.

Depends on: Definition 1.45 Definition 1.47 Definition 1.52 Definition 1.53 Definition 1.55 Definition 1.56

Label of a Galois orbit of a Dirichlet character. (LMFDB)

The label of a Galois orbit of a Dirichlet character \chi of modulus N takes the form N.a, where a is a letter or string of letters representing the index of the Galois orbit. The index 1 is written as a, the index 2 is written as b, the index 27 is written as ba, and so on.

Depends on: Definition 1.45 Definition 1.47 Definition 1.48

Induced Dirichlet character. (LMFDB)

A Dirichlet character \chi_1 of modulus q_1 is said to be induced by a Dirichlet character \chi_2 of modulus q_2 dividing q_1 if \chi_1(m)=\chi_2(m) for all m coprime to q_1.

A Dirichlet character is primitive if it is not induced by any character other than itself; every Dirichlet character is induced by a uniquely determined primitive Dirichlet character.

Depends on: Definition 1.45 Definition 1.52

Minimal Dirichlet character. (LMFDB)

A Dirichlet character \chi of prime power modulus N is minimal if the following conditions both hold:

1. The conductor of \chi does not lie in the open interval (\sqrt{N},N), and if N is a square divisible by 16 then {\rm cond}(\chi)\in \{\sqrt{N},N\}.

2. Both the order and conductor of \chi are minimal among the set of all Dirichlet character \chi\psi^2 for which {\rm cond}(\psi){\rm cond}(\chi\psi) | N.

This includes all primitive Dirichlet characters of prime power modulus, but not every minimal Dirichlet character of prime power modulus is primitive.

For a composite modulus N with prime power factorization N=p_1^{e_1}\cdots p_n^{e_n}, a Dirichlet character \chi of modulus N is minimal if and only if every character in its unique factorization into Dirichlet characters of modulus p_1^{e_1},\cdots,p_n^{e_n} is minimal. The trivial Dirichlet character is minimal.

Depends on: Definition 1.45 Definition 1.46 Definition 1.52 Definition 1.53 Definition 1.54 Definition 1.55

Modulus of a Dirichlet character. (LMFDB)

A Dirichlet character is a function \chi: \mathbb{Z}\to \mathbb{C} together with a positive integer q, called the modulus of the character, such that \chi is completely multiplicative, i.e. \chi(mn)=\chi(m)\chi(n) for all integers m and n, and \chi is periodic modulo q, i.e. \chi(n+q)=\chi(n) for all n. If (n,q)>1 then \chi(n)=0, whereas if (n,q)=1, then \chi(n) is a root of unity.

Order of a Dirichlet character. (LMFDB)

The order of a Dirichlet character \chi is the least positive integer n such that \chi^n is the trivial character of the same modulus as \chi. Equivalently, it is the order n of the image of \chi in \mathbb{C}^\times, the group of nth roots of unity.

Depends on: Definition 1.45 Definition 1.52 Definition 1.55

Primitive Dirichlet character. (LMFDB)

A Dirichlet character \chi is primitive if its conductor is equal to its modulus; equivalently, \chi is not induced by a Dirichlet character of smaller modulus.

Depends on: Definition 1.45 Definition 1.46 Definition 1.50 Definition 1.52

Principal Dirichlet character. (LMFDB)

A Dirichlet character is principal (or trivial) if it has order 1, equivalently, if it is induced by the unique Dirichlet character of modulus 1.

The value of the principal Dirichlet character of modulus q at an integer n is 1 if n is coprime to q and 0 otherwise.

Depends on: Definition 1.45 Definition 1.50 Definition 1.52

Field of values of a Dirichlet character. (LMFDB)

The field of values of a Dirichlet character \chi\colon\Z\to\C is the field \Q(\chi(\Z)) generated by its values; it is equal to the cyclotomic field \Q(\zeta_n), where n is the order of \chi.

Depends on: Definition 1.53

Automorphism group of an algebraic curve. (LMFDB)

An automorphism of an algebraic curve is an isomorphism from the curve to itself. The set of automorphisms of a curve X form a group \mathrm{Aut}(X) under composition; this is the automorphism group of the curve.

The automorphism group of a genus 2 curve necessarily includes the hyperelliptic involution (x,y)\mapsto(x,-y), which is an automorphism of order 2; this means that the automorphism group of a genus 2 curve is never trivial.

The geometric automorphism group of a curve X/k is the automorphism group of X_{\bar k}.

Depends on: Definition 1.10 Definition 1.59

Discriminant of a genus 2 curve. (LMFDB)

The discriminant \Delta of a Weierstrass equation y^2+h(x)y=f(x) can be computed as

\Delta := \begin{cases} 2^8\text{lc}(f)^2\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has odd degree},\\ 2^8\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has even degree}, \end{cases}

where \text{lc}(f) denotes the leading coefficient of f and \text{disc}(f) its discriminant.

The discriminant of a genus 2 curve over \Q is the discriminant of a minimal equation for the curve; it is an invariant of the curve that does not depend on the choice of minimal equation.

Genus 2 curve. (LMFDB)

Every (smooth, projective, geometrically integral) curve of genus 2 can be defined by a Weierstrass equation of the form y^2+h(x)y=f(x)

with nonzero discriminant and \deg h \le 3 and \deg f \le 6; in order to have genus 2 we must have \deg h = 3 or \deg f =5,6. Over a field whose characteristic is not 2 one can complete the square to make h(x) zero, but this will yield a model with bad reduction at 2 that is typically not a minimal equation for the curve.

This equation can be viewed as defining the function field of the curve, or as a smooth model of the curve in the weighted projective plane. Every curve of genus 2 admits a degree 2 cover of the projective line (consider the function x) and is therefore a hyperelliptic curve.

Depends on: Definition 1.18 Definition 1.58 Definition 1.61

Primes of good reduction. (LMFDB)

A variety X over \mathbb{Q} is said to have good reduction at a prime p if it has an integral model whose reduction modulo p defines a smooth variety of the same dimension; otherwise, p is said to be a prime of bad reduction.

When X is a curve, any prime of good reduction for X is also a prime of good reduction for its Jacobian, but the converse need not hold when X has genus g>1.

For all of the genus 2 curves currently in the LMFDB, every prime of good reduction for the curve is also a prime of good reduction for the Jacobian of the curve.

Depends on: Definition 1.20 Definition 1.28

Minimal equation of a hyperelliptic curve. (LMFDB)

Every (smooth, projective, geometrically integral) hyperelliptic curve X over \Q of genus g can be defined by an integral Weierstrass equation

y^2+h(x)y=f(x),

where h(x) and f(x) are integral polynomials of degree at most g+1 and 2g+2, respectively. Each such equation has a discriminant \Delta. A minimal equation is one for which |\Delta| is minimal among all integral Weierstrass equations for the same curve. Over \Q, every hyperelliptic curve has a minimal equation. The prime divisors of \Delta are the primes of bad reduction for X.

The equation y^2+h(x)y=f(x) uniquely determines a homogeneous equation of weighted degree 6 in variables x,y,z, where y has weight g+1, while x and z both have weight 1: one homogenizes h(x) to obtain a homogeneous polynomial h(x,z) of degree g+1 and homogenizes f(x) to obtain a homogeneous polynomial f(x,z) of degree 2g+2. This yields a smooth projective model y^2+h(x,z)y=f(x,z) for the curve X.

One can always transform the minimal equation into a simplified equation y^2 = g(x) = 4f(x)+h(x)^2, but this equation need not have minimal discriminant and may have bad reduction at primes that do not divide the minimal discriminant (it will always have bad reduction at the prime 2).

Depends on: Definition 1.11 Definition 1.12 Definition 1.18 Definition 1.58 Definition 1.60

Galois group. (LMFDB)

The Galois group of an irreducible separable polynomial of degree n can be embedded in S_n through its action on the roots of the polynomial, with the image being well-defined up to labeling of the roots. Different labelings lead to conjugate subgroups. The subgroup acts transitively on \{1,\ldots,n\}. Conversely, for every transitive subgroup G of S_n with n\in\mathbb{Z}^+, there is a field K such that G is the Galois group of some polynomial over K.

Borel subgroup. (LMFDB)

A Borel subgroup of a general linear group is a subgroup that is conjugate to the group of upper triangular matrices.

The Borel subgroups of \GL_2(\F_p) are maximal subgroups that fix a one-dimensional subspace of \F_p^2; every such subgroup is conjugate to the subgroup of upper triangular matrices.

Subgroup labels containing the letter B identify a subgroup of \GL_2(\F_p) that lies in the Borel subgroup of upper triangular matrices but is not contained in the subgroup of diagonal matrices; these are precisely the subgroups of a Borel subgroup that contain an element of order p.

The label B is used for the full Borel subgroup of upper triangular matrices

The label B.a.b denotes the proper subgroup of B generated by the matrices

\begin{pmatrix}a&0\\0&1/a\end{pmatrix},\ \begin{pmatrix}b&0\\0&r/b\end{pmatrix},\ \begin{pmatrix}1&1\\0&1\end{pmatrix},

where a and b are minimally chosen positive integers and r is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, as defined in .

Cartan subgroup. (LMFDB)

Let R be a commutative ring. Given a free rank 2 etale R-algebra A equipped with a basis, any a \in A^\times defines an R-linear multiplication-by-a map A \to A, so we get an injective homomorphism A^\times \to \Aut_{\text{R-module}}(A) \simeq \GL_2(R), and the image is called a Cartan subgroup of \GL_2(R). The canonical involution of the R-algebra A gives another element of \Aut_{\text{R-module}}(A); we call the group generated by it and the Cartan subgroup A^\times the extended Cartan subgroup. The Cartan subgroup has index 2 in the extended Cartan subgroup.

If R=\F_p, there are two possibilities for A: the split algebra \F_p \times \F_p and the nonsplit algebra \F_{p^2}; the resulting Cartan subgroups are called split and nonsplit. The extended Cartan subgroup equals the normalizer of the Cartan subgroup in \GL_2(\F_p) except when p=2 and A is split. In the split case, if we use the standard basis of \F_p \times \F_p, the Cartan subgroup is the subgroup of diagonal matrices in \GL_2(\F_p), and the extended Cartan subgroup is this together with the coset of antidiagonal matrices in \GL_2(\F_p).

If R=\Z/p^e\Z, again there are two possibilities for A: the split algebra R \times R, or the nonsplit algebra. The nonsplit algebra can be described as \mathcal{O}/p^e \mathcal{O} where \mathcal{O} is either the degree 2 unramified extension of \Z_p or a quadratic order in which p is inert. The nonsplit algebra can also be described as the ring of length e Witt vectors W_e(\F_{p^2}).

If R=\Z/N\Z for some N \ge 1, then A can be split or nonsplit independently at each prime dividing N.

Exceptional subgroup. (LMFDB)

An exceptional subgroup of \GL_2(\F_p) does not contain \SL_2(\F_p) and is not contained in a Borel subgroup or in the normalizer of a Cartan subgroup.

Exceptional subgroups are classified according to their image in \PGL_2(\F_p), which must be isomorphic to one of the alternating groups A_4 or A_5, or to the symmetric group S_4. These groups are labelled using identifiers containing one of the strings A4, A5, S4, as described in .

Depends on: Definition 1.63 Definition 1.69

Index of an open subgroup. (LMFDB)

The index of an open subgroup H of a profinite group G is the positive integer [G:H].

When G is a matrix group over \widehat{\Z} or \Z_{\ell} and H is a subgroup of level N, this is the same as the index of H in the reduction of G modulo N.

Depends on: Definition 1.67 Definition 1.72 Definition 1.73

Level of an open subgroup. (LMFDB)

The level of an open subgroup H of a matrix group G over \widehat{\Z} is the least positive integer N for which H is equal to the inverse image of its projection to the reduction of G modulo N.

This also applies to open subgroups of matrix groups over \Z_{\ell}, in which case the level is necessarily a power of \ell.

Depends on: Definition 1.72

Non-split Cartan subgroup. (LMFDB)

A non-split Cartan subgroup of \GL_2(\F_p) is a Cartan subgroup that is not diagonalizable over \F_p. Every non-split Cartan subgroup is a cyclic group isomorphic to \F_{p^2}^\times.

For p=2 the label Cn identifies the unique index 2 subgroup of \GL_2(\F_2). For p>2 the label Cn identifies the nonsplit Cartan subgroup consisting of matrices of the form

\begin{pmatrix}x&\varepsilon y\\y&x\end{pmatrix},

with x,y\in \F_p not both zero and \varepsilon the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, corresponding to x+y\sqrt{\varepsilon}\in\F_{p^2}^\times. Every non-split Cartan subgroup is conjugate to the group Cn.

Labels of the form Cn.a.b identify the proper subgroup of Cn generated by the matrix

\begin{pmatrix}a&\varepsilon b\\b&a\end{pmatrix},

where a and b are minimally chosen positive integers and \varepsilon is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, as defined in .

Depends on: Definition 1.64

Normalizer of a Cartan subgroup. (LMFDB)

For p>2 the normalizer of a Cartan subgroup of \GL_2(\F_p) is a maximal subgroup of \GL_2(\F_p) that contains a Cartan subgroup with index 2. It is the normalizer in \GL_2(\F_p) of the Cartan subgroup it contains.

For p=2 the Cartan subgroups of \GL_2(\F_2) are already normal and we instead define the normalizer of a Cartan subgroup to be a group that contains a Cartan subgroup with index 2. This means that the normalizer of a split Cartan subgroup of \GL_2(\F_2) has order 2 (which makes it conjugate to the Borel subgroup), while the normalizer of a non-split Cartan subgroup of \GL_2(\F_2) has order 6 (which makes it all of \GL_2(\F_2)).

Depends on: Definition 1.64

Normalizer of a non-split Cartan subgroup. (LMFDB)

For p>2 the normalizer of a non-split Cartan subgroup of \GL_2(\F_p) is a maximal subgroup of \GL_2(\F_p) that contains a non-split Cartan subgroup with index 2, and it is the normalizer in \GL_2(\F_p) of the non-split Cartan subgroup it contains. For p=2 the normalizer of a non-split Cartan subgroup is defined to be all of \GL_2(\F_2), which contains its (already normal) non-split Cartan subgroup with index 2.

For p>2 the label Nn identifies the normalizer of the nonsplit Cartan subgroup generated by the non-split Cartan subgroup Cn and the matrix

\begin{pmatrix}1&0\\0&-1\end{pmatrix},

and every normalizer of a non-split Cartan subgroup is conjugate to the group Nn.

The label Nn.a.b denotes the proper subgroup of the normalizer of the nonsplit Cartan subgroup Nn generated by the matrices

\begin{pmatrix}a&\varepsilon b\\b&a\end{pmatrix}, \begin{pmatrix}1&0\\0&-1\end{pmatrix}.

where a and b are minimally chosen positive integers and \varepsilon is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, as defined in .

Depends on: Definition 1.68

Normalizer of a split Cartan subgroup. (LMFDB)

The normalizer of a split Cartan subgroup of \GL_2(\F_p) is a maximal subgroup of \GL_2(\F_p) that contains a split Cartan subgroup with index 2. For p>2 such a group is in fact the normalizer in \GL_2(\F_p) of the split Cartan subgroup it contains, but for p=2 this is not the case (the split Cartan subgroup of \GL_2(\F_2) is already normal).

The label Ns identifies the subgroup generated by the split Cartan subgroup Cs of diagonal matrices and the matrix

\begin{pmatrix}0&1\\1&0\end{pmatrix}.

Every normalizer of a split Cartan subgroup is conjugate to the group Ns.

The label Ns.a.b identifies the proper subgroup of Ns generated by the matrices

\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-r/b&0\end{pmatrix},

where a and b are minimally chosen positive integers and r is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times.

The label Ns.a.b.c identifies the proper subgroup of the normalizer of the split Cartan subgroup generated by the matrices

\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-1/b&0\end{pmatrix}, \begin{pmatrix}0&c\\-r/c&0\end{pmatrix}

where a and b are minimally chosen positive integers and r is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, as defined in .

Depends on: Definition 1.74

Open subgroup. (LMFDB)

An open subgroup H of a profinite group G is a subgroup that is open in the topology of G, which implies that it is equal to the inverse image of its projection to a a finite quotient of G.

Open subgroups of G necessarily have finite index (since G is compact), but not every finite index subgroup of G is necessarily open.

When the profinite group G is a matrix group over a ring R that is equipped with canonical projections to finite rings of the form \Z/n\Z (take R=\Z_{\ell} or R=\widehat{\Z}, for example), we use G(n) to denote the image of G under the group homomorphism induced by the projection R\to\Z/n\Z. In this situation we may identify H with its projection to G(N), where N is the least positive integer for which H is the inverse image of its projection to G(N) (this N is the level of H).

Depends on: Definition 1.73

Profinite group. (LMFDB)

A profinite group is a compact totally disconnected topological group. Equivalently, it is the inverse limit of a system of finite groups equipped with the discrete topology.

For example, if we take the finite groups \GL_2(\Z/n\Z) as n varies over positive integers, order them by divisibility of n and consider the inverse system equipped with reduction maps \GL_2(\Z/n\Z)\to \GL_2(\Z/m\Z) for all positive integers m|n, then the inverse limit

\lim_{\overset{\longleftarrow}{n}} \GL_2(\Z/n\Z) \simeq \GL_2(\widehat{\Z})

is a profinite group which is isomorphic to the group of invertible 2\times 2 matrices over the topological ring \widehat{\Z}, which is the inverse limit of the finite rings \Z/n\Z equipped with the discrete topology.

Split Cartan subgroup. (LMFDB)

A split Cartan subgroup of \GL_2(\F_p) is a Cartan subgroup that is diagonalizable over \F_p. Every split Cartan subgroup is conjugate to the subgroup of diagonal matrices, which is isomorphic to \F_p^\times\times\F_p^\times.

The label Cs identifies the split Cartan subgroup of diagonal matrices.

The label Cs.a.b identifies the proper subgroup of Cs generated by

\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}b&0\\0&r/b\end{pmatrix},

where a and b are minimally chosen positive integers and r is the least positive integer generating (\Z/p\Z)^\times\simeq \F_p^\times, as defined in .

Depends on: Definition 1.64

Definition of group. (LMFDB)

A group \langle G, *\rangle is a set G with a binary operation * such that

1. * is associative

2. * has an identity element

3. every element g\in G has an inverse.

Depends on: Definition 1.29 Definition 1.30 Definition 1.32 Definition 1.33

Abelian group. (LMFDB)

A group is abelian if its operation is commutative.

Depends on: Definition 1.29 Definition 1.31 Definition 1.75

Automorphisms of a group. (LMFDB)

If G is a group, an automorphism of G is a group isomorphism f:G\to G.

The set of automorphisms of G, \Aut(G), is a group under composition.

Depends on: Definition 1.75 Definition 1.88

Characteristic subgroup. (LMFDB)

A subgroup H of a group G is a characteristic subgroup if \phi(H)=H for all automorphisms \phi\in \Aut(G).

Depends on: Definition 1.75 Definition 1.77 Definition 1.95

Coset of a subgroup. (LMFDB)

If G is a group and H is a subgroup of G, then a left coset of H is a set

gH = \{ gh \mid h \in H\}

and similarly, a right coset of H is a set

Hg = \{ hg \mid h \in H\}.

The left cosets partition G, as do the right cosets.

Depends on: Definition 1.75 Definition 1.95

Frattini subgroup of a group. (LMFDB)

If G is a group, then the Frattini subgroup of G, denoted \Phi(G), is the intersection of all maximal subgroups of G. If there are no maximal subgroups of G, then \Phi(G)=G.

The Frattini subgroup is always a characteristic subgroup, hence a normal subgroup, of G.

Depends on: Definition 1.78 Definition 1.89 Definition 1.97

Cusps of a subgroup of the modular group. (LMFDB)

The cusps of a subgroup \Gamma of the modular group are equivalence classes of points in \mathbb{Q}\cup\infty under the action of \Gamma by linear fractional transformation, where for

\gamma=\left(\begin{array}{ll}a&b\\c&d \end{array}\right)\in\Gamma,

we define \gamma\infty = \frac{a}{c} when c\neq 0, and \gamma\infty = \infty when c=0.

Depends on: Definition 1.94

Width of a cusp. (LMFDB)

The width of the cusp \infty for the group \Gamma is the smallest number w such that T^w=\left(\begin{matrix}1&w\\0&1\end{matrix}\right)\in\Gamma. Furthermore, for a general x\in\mathbb{P}^1(\mathbb{Q}) and \gamma\in\Gamma such that \gamma\infty=x, we define the width of x for \Gamma to be the width of \infty for \gamma^{-1}\Gamma\gamma.

Note that T=\left(\begin{matrix}1&1\\0&1\end{matrix}\right) is one of the generators of the modular group \textrm{SL}_2(\mathbb{Z}).

Depends on: Definition 1.81 Definition 1.94

Fundamental domain. (LMFDB)

If G\subseteq\Gamma is a subgroup of the modular group, then a closed set F\in\mathcal{H}\cup\mathbb{Q}\cup\{\infty\} is said to be a fundamental domain for G if: <ol> <li> For any point z\in\mathcal{H} there is a g\in G such that gz\in F.</li> <li> If z\not=z'\in F are equivalent with respect to the action of G, that is, if z'=gz for some g\in G, then z and z' belong to \partial F, the boundary of F.</li> </ol>

Depends on: Definition 1.94

Absolute Galois group. (LMFDB)

The absolute Galois group of a field K is the group of all automorphisms of the algebraic closure of K that fix the field K.

Depends on: Definition 1.133

Generators of a group. (LMFDB)

If G is a group and S is a subset of G, then S is a set of generators if the smallest subgroup of G containing S equals G.

Equivalently, S generates G if

G=\bigcap_{S\subseteq H\leq G} H \,.

The automorphism group of G acts on such S, and we say S and S' are equivalent if they are related by this action.

Depends on: Definition 1.75 Definition 1.77

Haar measure of a topological group. (LMFDB)

For G a locally compact topological group, a Haar measure on G is a nonnegative, countably additive, real-valued measure on G which is invariant under left translation on G. Any such measure is also invariant under right translation on G.

A Haar measure always exists and is unique up to multiplication by a positive scalar. If G is compact, then the normalized Haar measure on G is the unique Haar measure on G under which G has total measure 1.

As a special case, if G is finite of order n, then the normalized Haar measure is the uniform measure that assigns to each element the measure 1/n.

Depends on: Definition 1.75 Definition 1.91

Group homomorphism. (LMFDB)

If G and H are groups, then a group homomorphism from G to H is a function

f:G\to H

such that for all a,b\in G, f(a*b)=f(a)*f(b).

Depends on: Definition 1.75

Group isomorphism. (LMFDB)

A group isomorphism is a group homomorphism f:G\to H which is bijective.

Depends on: Definition 1.87

Maximal subgroup of a group. (LMFDB)

If G is a group, a subgroup M is a maximal subgroup if for every subgroup H such that M\subseteq H \subseteq G, either H=M or H=G.