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 1.134

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 1.134

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 2.42

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 1.134