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

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

which has radius of convergence \(2\pi \).

A divisor function is a multiplicative arithmetic function of the form

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

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

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.

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

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.

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.

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 \(2g\).

An algebraic curve is an algebraic variety of dimension \(1.\)

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 \(k=\mathbb {C}\) 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 \(H^1(C,\mathcal{O}_C)\).

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.

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

of distinct irreducible subvarieties of \(V\).

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

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.

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

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

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

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 \(g+1\).

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 \(F\).

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.

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.

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 \(\mathbb {Q}\), proved that the abelian group \(A(K)\) is finitely generated. Thus

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 \(A\).

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

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

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

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

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

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 \(\mathbb {A}^n\).

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

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

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 \(K[x_1,\dots ,x_n]\).

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 \(s_1*s_2\).

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

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

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 \(a\in A\),

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

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

Let \(N \ge 1\). Let \(\mu _N\) be the group of \(N\)th roots of unity in some algebraically closed field of characteristic not dividing \(N\). Let \(M\) be a free rank \(2\) \(\mathbb {Z}/N\mathbb {Z}\)-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 \(\mathbb {Z}/N\mathbb {Z}\times \mu _N\) 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 \(\mathbb {Z}/N\mathbb {Z}\)-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 \(\mathbb {Z}/N\mathbb {Z}\)-module is enriched with a Galois action to make a Galois module, or replaced by a finite étale group scheme that is \((\mathbb {Z}/N\mathbb {Z})^2\) étale locally.

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

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 \(K/\mathbb {Q}\) be a Galois extension and \(\rho :\operatorname{Gal}(K/\mathbb {Q})\to \textrm{GL}(V)\) 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 \(K_{\mathfrak {p}}/\mathbb {Q}_p\) 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 \(K_{\mathfrak {p}}/\mathbb {Q}_p\), and \(G_1\) is the wild inertia group, which is a finite \(p\)-group.

Let \(g_i = |G_i|\). Then

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 \(f(\rho ,p){\gt}0\).

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

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

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

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

If \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{GL}_n(\mathbb {C})\) 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 \(\rho \).

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.

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

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

Here for \(m\in \mathbb {Z}_{{\gt} 0}\), \(A[m]\) 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 \(\mathbb {Z}_p\)-module of rank \(2g\). It carries an action of the absolute Galois group of \(K\), and thus has an associated Galois representation.

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.

A Dirichlet character is a function \(\chi : \mathbb {Z}\to \mathbb {C}\) 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 \((n,q){\gt}1\) 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.

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 \(q\).

The Galois orbit of a Dirichlet character \(\chi \) of modulus \(q\) and order \(n\) is the set \([\chi ]:=\{ \sigma (\chi ): \sigma \in \operatorname{Gal}(\mathbb {Q}(\zeta _n)/\mathbb {Q})\} \), 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 \(\operatorname{Gal}(\mathbb {Q}(\zeta _n)/\mathbb {Q})\) on the set of Dirichlet characters of modulus \(q\) and order \(n\), each of which has \(\mathbb {Q}(\zeta _n)\) as its field of values.

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

where \(n\) is the order of \(\chi \) and \(t_i:=\mathrm{tr}_{\mathbb {Q}(\chi )/\mathbb {Q}}(\chi (i))\) is the trace of \(\chi (i)\) from the field of values of \(\chi \) to \(\mathbb {Q}\). 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 \(1\).

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.

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.

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.

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 \((n,q){\gt}1\) then \(\chi (n)=0\), whereas if \((n,q)=1\), then \(\chi (n)\) is a root of unity.

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 \(n\)th roots of unity.

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.

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.

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

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 \(X_{\bar k}\).

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

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

The discriminant of a genus 2 curve over \(\mathbb {Q}\) 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.

Every (smooth, projective, geometrically integral) curve of genus 2 can be defined by a Weierstrass equation of the form

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.

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 \(g{\gt}1\).

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.

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

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 \(\mathbb {Q}\), 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 \(2\)).

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 \(K\).

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 \(\textrm{GL}_2(\mathbb {F}_p)\) are maximal subgroups that fix a one-dimensional subspace of \(\mathbb {F}_p^2\); every such subgroup is conjugate to the subgroup of upper triangular matrices.

Subgroup labels containing the letter B identify a subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) 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

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

Let \(R\) be a commutative ring. Given a free rank \(2\) étale \(R\)-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 \(A^\times \to \operatorname{Aut}\)

An exceptional subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) does not contain \(\textrm{SL}_2(\mathbb {F}_p)\) 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 \(\textrm{PGL}_2(\mathbb {F}_p)\), 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 [ .

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 \(\widehat{\mathbb {Z}}\) or \(\mathbb {Z}_{\ell }\) and \(H\) is a subgroup of level \(N\), this is the same as the index of \(H\) in the reduction of \(G\) modulo \(N\).

The level of an open subgroup \(H\) of a matrix group \(G\) over \(\widehat{\mathbb {Z}}\) 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 \(\mathbb {Z}_{\ell }\), in which case the level is necessarily a power of \(\ell \).

A non-split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) is a Cartan subgroup that is not diagonalizable over \(\mathbb {F}_p\). Every non-split Cartan subgroup is a cyclic group isomorphic to \(\mathbb {F}_{p^2}^\times \).

For \(p=2\) the label Cn identifies the unique index 2 subgroup of \(\textrm{GL}_2(\mathbb {F}_2)\). For \(p{\gt}2\) the label Cn identifies the nonsplit Cartan subgroup consisting of matrices of the form

with \(x,y\in \mathbb {F}_p\) not both zero and \(\varepsilon \) the least positive integer generating \((\mathbb {Z}/p\mathbb {Z})^\times \simeq \mathbb {F}_p^\times \), corresponding to \(x+y\sqrt{\varepsilon }\in \mathbb {F}_{p^2}^\times \). 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

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

For \(p{\gt}2\) the normalizer of a Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) is a maximal subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) that contains a Cartan subgroup with index 2. It is the normalizer in \(\textrm{GL}_2(\mathbb {F}_p)\) of the Cartan subgroup it contains.

For \(p=2\) the Cartan subgroups of \(\textrm{GL}_2(\mathbb {F}_2)\) 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 \(\textrm{GL}_2(\mathbb {F}_2)\) has order 2 (which makes it conjugate to the Borel subgroup), while the normalizer of a non-split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_2)\) has order 6 (which makes it all of \(\textrm{GL}_2(\mathbb {F}_2)\)).

For \(p{\gt}2\) the normalizer of a non-split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) is a maximal subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) that contains a non-split Cartan subgroup with index 2, and it is the normalizer in \(\textrm{GL}_2(\mathbb {F}_p)\) 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 \(\textrm{GL}_2(\mathbb {F}_2)\), which contains its (already normal) non-split Cartan subgroup with index 2.

For \(p{\gt}2\) the label Nn identifies the normalizer of the nonsplit Cartan subgroup generated by the non-split Cartan subgroup Cn and the matrix

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

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

The normalizer of a split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) is a maximal subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) that contains a split Cartan subgroup with index 2. For \(p{\gt}2\) such a group is in fact the normalizer in \(\textrm{GL}_2(\mathbb {F}_p)\) of the split Cartan subgroup it contains, but for \(p=2\) this is not the case (the split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_2)\) is already normal).

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

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

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

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

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

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 \(\mathbb {Z}/n\mathbb {Z}\) (take \(R=\mathbb {Z}_{\ell }\) or \(R=\widehat{\mathbb {Z}}\), for example), we use \(G(n)\) to denote the image of \(G\) under the group homomorphism induced by the projection \(R\to \mathbb {Z}/n\mathbb {Z}\). 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 \(H\)).

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 \(\textrm{GL}_2(\mathbb {Z}/n\mathbb {Z})\) as \(n\) varies over positive integers, order them by divisibility of \(n\) and consider the inverse system equipped with reduction maps \(\textrm{GL}_2(\mathbb {Z}/n\mathbb {Z})\to \textrm{GL}_2(\mathbb {Z}/m\mathbb {Z})\) for all positive integers \(m|n\), then the inverse limit

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

A split Cartan subgroup of \(\textrm{GL}_2(\mathbb {F}_p)\) is a Cartan subgroup that is diagonalizable over \(\mathbb {F}_p\). Every split Cartan subgroup is conjugate to the subgroup of diagonal matrices, which is isomorphic to \(\mathbb {F}_p^\times \times \mathbb {F}_p^\times \).

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

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

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

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.

A group is abelian if its operation is commutative.

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

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

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

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

and similarly, a right coset of \(H\) is a set

The left cosets partition \(G\), as do the right cosets.

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 \(G\).

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

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

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 \(\textrm{SL}_2(\mathbb {Z})\).

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>

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

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

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

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 \(1/n\).

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

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

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

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 \(H=G\).

If \(G\) is a group, a subnormal series for \(G\) is a chain of subgroups

where each subgroup \(H_i\) is normal in \(H_{i+1}\) for all \(i\).

A subnormal series where \(H_i\) is normal in \(G\) for all \(i\) is a normal series.

The order of a group is its cardinality as a set.

A presentation of a group \(G\) is a description of \(G\) as the quotient \(F/R\) of a free group \(F\) generated by a specified set of generators, modulo the normal subgroup \(R\) generated by a set of words in those generators. When \(G\) is abelian we instead express \(G\) as a quotient of a free abelian group \(F\) so that we can omit commutator relations.

In what follows, we denote by \(g^h\) the conjugate \(h^{-1}gh\) and by \([g, h]\) the commutator \(ghg^{-1}h^{-1}\).

We only give presentations for finite solvable groups, where they can take a special form. A polycyclic series is a subnormal series \(G = G_1 \trianglerighteq G_2 \trianglerighteq \dots \trianglerighteq G_n \trianglerighteq G_{n+1} = \{ 1\} \) so that \(G_i/G_{i+1}\) is cyclic for each \(i\). A polycyclic sequence is a sequence of elements \((g_1, \dots , g_n)\) of \(G\) so that \(G_i/G_{i+1} = \langle g_i G_{i+1}\rangle \). The relative orders of a polycyclic series are the orders \(r_i\) of the cyclic quotients \(G_i / G_{i+1}\). The polycyclic presentation associated to a polycyclic sequence has generators \(g_1, \dots , g_n\) and relations of the following shape.

• \(g_i^{r_i} = \prod _{k=i+1}^n g_k^{a_{i,k}}\) for all \(i\);

• \(g_i^{g_j} = \prod _{k=j+1}^n g_k^{b_{i,j,k}}\) for \(j {\lt} i\).

Any finite solvable group has a polycyclic presentation. When the size of \(G\) is not too large, we choose a presentation with the following properties:

• it has a minimal number of generators;

• among such, it has a maximal number of \(i\) so that all \(a_{i,k} = 0\);

• among such, it has a maximal number of commuting \(g_i\);

• among such, aim for an increasing sequence of relative orders;

• among such, minimize the sum of the \(b_{i,j,k}\) for noncommuting generators \(g_i\) and \(g_j\).

The rank of a finite group \(G\) is the minimal number of elements required to generate it, which is often smaller than the number of generators in a polycyclic presentation. For \(p\)-groups, the rank can be computed by taking the \(\mathbb {F}_p\)-dimension of the quotient by the Frattini subgroup.

The modular group is the group of \(2\times 2\) matrices with integer coefficients and determinant \(1\); it is denoted by \( \mathrm{SL}(2,\mathbb {Z}) \) or \(\mathrm{SL}_2(\mathbb {Z})\).

A standard set of generators for the modular group are the matrices:

If \(G\) is a group, a subset \(H\subseteq G\) is a subgroup of \(G\) if the binary operation of \(G\) restricts to a binary operation on \(H\), and \(H\) is a group for this induced operation.

Equivalently, the subset \(H\) must satisfy the following conditions:

1. for all \(a,b\in H\), \(a*b\in H\) 2. the identity of \(G\) is an element of \(H\) 3. for every \(a\in H\), the inverse of \(a\) in \(G\) is also in \(H\).

The index of a subgroup \(G'\) of a group \(G\), denoted \([G:G']\), is the order of the set of left cosets of \(G'\) in \(G\).

If \(H\) is a subgroup of a group \(G\), then \(H\) is normal if any of the following equivalent conditions hold:

1. \(gHg^{-1}=H\) for all \(g\in G\) 2. \(gHg^{-1}\subseteq H\) for all \(g\in G\) 3. \(gH=Hg\) for all \(g\in G\) 4. \((aH)*(bH)=(ab)H\) is a well-defined binary operation on the set of left cosets of \(H\)

If \(H\) is a normal subgroup, we write \(H \lhd G\), and the set of left cosets \(G/H\) form a group under the operation given in (4) above.

If \(p\) is a prime and \(G\) is a finite group of order \(p^nm\) where \(p\nmid m\), then a \(p\)-Sylow subgroup of \(G\) is any subgroup of order \(p^n\).

Sylow subgroups exist for every finite group and prime \(p\).

A torsion group is a group in which every element has finite order.

The elements of finite order in an abelian group \(A\) form a torsion group called the torsion subgroup of \(A\).

If \(K/F\) is an extension of fields, its automorphism group is

Note, a finite extension is Galois if and only if \(|\textrm{Aut}(K/F)| = [K:F]\).

Let

• \(K\) be a \(p\)-adic field.

• \(L\) a finite Galois extension of \(K\).

• \(\mathcal{O}_K\), \(\mathcal{O}_L\) the rings of integers for \(K\), \(L\),

• \(P_K\), \(P_L\) the unique maximal ideals of \(\mathcal{O}_K\), \(\mathcal{O}_L\), and

• \(\kappa =\mathcal{O}_K/P_K\), \(\lambda =\mathcal{O}_L/P_L\) the

residue fields of \(K\), \(L\).

Then each \(\sigma \in \operatorname{Gal}(L/K)\) induces a element of \(\operatorname{Gal}(\lambda /\kappa )\). The kernel of the resulting homomorphism

is the inertia group of \(L/K\).

A local field is a field \(K\) with a non-trivial absolute value \(|\ |\) that is locally compact in the topology induced by the distance metric \(d(x,y):=|x-y|\).

An archimedean local field is a local field whose absolute value is archimedean; such a field is isomorphic to \(\mathbb {R}\) or \(\mathbb {C}\).

A nonarchimedean local field is a local field whose absolute value is nonarchimedean. Such a field is either isomorphic to a finite extension of \(\mathbb {Q}_p\) when \(K\) has characteristic zero (in which case it is a \(p\)-adic field), or to a finite extension of \(\mathbb {F}_p((t))\) when \(K\) has characteristic \(p\). In both cases \(p\) is the characteristic of the residue field of \(K\)..

The maximal ideal of a nonarchimedean local field \(K\) is the unique maximal ideal of its ring of integers \(\mathcal O_K\).

It consists of all elements of \(\mathcal O_K\) that are not units, equivalently, all elements of \(K\) whose absolute value is strictly less than 1.

A \(p\)-adic field (or local number field) is a finite extension of \(\mathbb {Q}_p\), equivalently, a nonarchimedean local field of characteristic zero.

The residue field of a nonarchimedean local field is the quotient of its ring of integers by its unique maximal ideal.

The residue field is finite and its characteristic \(p\) is the residue field characteristic. Finite extensions of \(\mathbb {Q}_p\) have residue field characteristic \(p\).

The ring of integers of a local field \(K\) with absolute value \(|\ |\) is the subring \(\mathcal O_K := \{ x\in K:|x|\le 1\} \); it is a discrete valuation ring.

The wild inertia group of a Galois extension \(K/\mathbb {Q}_p\) is the unique \(p\)-Sylow subgroup of its inertia group.

An (analytic) L-function is a Dirichlet series that has an Euler product and satisfies a certain type of functional equation.

It is expected that all L-functions satisfy the Riemann Hypothesis, that all of the zeros in the critical strip are on the critical line. Selberg has defined a class \(\mathcal S\) of Dirichlet series that satisfy the Selberg axioms. It is conjectured (but far from proven) that \(\mathcal S\) is precisely the set of all L-functions. Selberg’s axioms have not been verified for all of the L-functions in this database but are known to hold for many of them.

It is also conjectured that a precise form of the functional equation holds for every element of \(\mathcal S\). Under this assumption the functional equation is determined by a quadruple known as the Selberg data, consisting of the degree, conductor, spectral parameters, and sign.

The analytic rank of an L-function \(L(s)\) is its order of vanishing at its central point.

When the analytic rank \(r\) is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be \(r\) zeroes located very near to the central point).

An L-function \(L(s) = \sum _{n=1}^{\infty } a_n n^{-s}\) is called arithmetic if its Dirichlet coefficients \(a_n\) are algebraic numbers.

The central point of an L-function is the point on the real axis of the critical line. Equivalently, it is the fixed point of the functional equation.

In the analytic normalization, the central point is \(s=1/2\), in the arithmetic normalization, it is \(s=\frac{w+1}{2}\), where \(w\) is the weight of the L-function.

The critical line of an L-function is the line of symmetry of its functional equation.

In the analytic normalization, the functional equation relates \(s\) to \(1-s\) and the critical line is the line \(\Re (s) = \frac12\).

In the arithmetic normalization, the functional equation relates \(s\) to \(1 + w - s\), where \(w\) is the motivic weight. In that normalization the critical line is \(\Re (s) = \frac{1+w}2\).

A Dirichlet series is a formal series of the form \(F(s) = {\displaystyle \sum _{n=1}^{\infty } \frac{a_n}{ n^{s}}}\), where \(a_n \in {\mathbb {C}}\).

The dual of an L-function \(L(s) = \sum _{n=1}^{\infty } \frac{a_n}{n^s}\) is the complex conjugate \(\bar{L}(s) = \sum _{n=1}^{\infty } \frac{\bar{a_n}}{n^s}\).

It is expected that the Euler product of an L-function of degree \(d\) and conductor \(N\) can be written as

where for \(p\nmid N\)

and for \(p\mid N\),

The functions \(L_p(s)\) are called Euler factors (or local factors).

All known analytic L-functions have a functional equation that can be written in the form

where \(N\) is an integer, \(\Gamma _{\mathbb {R}}\) and \(\Gamma _{\mathbb {C}}\) are defined in terms of the \(\Gamma \)-function, \(\mathrm{Re}(\mu _j) = 0 \ \mathrm{or} \ 1\) (assuming Selberg’s eigenvalue conjecture), and \(\mathrm{Re}(\nu _k)\) is a positive integer or half-integer,

and \(\varepsilon \) is the sign of the functional equation. With those restrictions on the spectral parameters, the data in the functional equation is specified uniquely. The integer \(d = J + 2 K\) is the degree of the L-function. The integer \(N\) is the conductor (or level) of the L-function. The pair \([J,K]\) is the signature of the L-function. The parameters in the functional equation can be used to make up the 4-tuple called the Selberg data.

The axioms of the Selberg class are less restrictive than given above.

Note that the functional equation above has the central point at \(s=1/2\), and relates \(s\leftrightarrow 1-s\).

For many L-functions there is another normalization which is natural. The corresponding functional equation relates \(s\leftrightarrow w+1-s\) for some positive integer \(w\), called the motivic weight of the L-function. The central point is at \(s=(w+1)/2\), and the arithmetically normalized Dirichlet coefficients \(a_n n^{w/2}\) are algebraic integers.

The complex functions

that appear in the functional equation of an L-function are known as gamma factors. Here \(\Gamma (s):=\int _0^\infty e^{-t}t^{s-1}dt\) is Euler’s gamma function.

The gamma factors satisfy \( \Gamma _{\mathbb {C}}(s) = \Gamma _{\mathbb {R}}(s) \Gamma _{\mathbb {R}}(s + 1) \) and can also be viewed as “missing” factors of the Euler product of an L-function corresponding to (real or complex) archimedean places.

The leading coefficient of an arithmetic L-function is the first nonzero coefficient of its Laurent series expansion at the central point.

In its arithmetic normalization, an L-function \(L(s)\) of weight \(w\) has its central value at \(s=\frac{w+1}{2}\) and the functional equation relates \(s\) to \(1 + w - s\). For L-functions defined by an Euler product \(\prod _p L_p(s)^{-1}\) where the coefficients of \(L_p\) are algebraic integers, this is the usual normalization implied by the definition.

The analytic normalization of an L-function is defined by \(L_{an}(s):=L(s+w/2)\), where \(L(s)\) is the L-function in its arithmetic normalization. This moves the central value to \(s=1/2\), and the functional equation of \(L_{an}(s)\) relates \(s\) to \(1-s\).

The Riemann hypothesis is the assertion that if \(\rho \) is a zero of an analytic L-function then \(\mathrm{Re}(\rho ){\gt}0\) implies that \(\mathrm{Re}(\rho )=1/2.\)

An L-function \(L(s) = \sum _{n=1}^{\infty } \frac{a_n}{n^s}\) is called self-dual if its Dirichlet coefficients \(a_n\) are real.

The sign of the functional equation of an analytic L-function, also called the root number, is the complex number \(\varepsilon \) that appears in the functional equation of \(\Lambda (s)=\varepsilon \overline{\Lambda }(1-s)\). The sign appears as the 4th entry in the quadruple known as the Selberg data.

We define the Dedekind eta function \(\eta (z)\) by the formula

where \(q=e^{2\pi iz}\).

It is related to the Discriminant modular form via the formula

The upper half-plane \(\mathcal{H}\) is the set of complex numbers whose imaginary part is positive, endowed with the topology induced from \(\mathbb {C}\).

The completed upper half-plane \(\mathcal{H}^*\) is

endowed with the topology such that the disks tangent to the real line at \(r \in \mathbb {Q}\) form a fundamental system of neighbourhoods of \(r\), and strips \(\{ z \in \mathcal{H} \mid \operatorname {Im} z {\gt} y \} \cup \{ \infty \} \), \(y{\gt}0\), form a fundamental system of neighbourhoods of \(\infty \), which should therefore be thought of as \(i \infty \).

The modular group \(\textrm{SL}_2(\mathbb {Z})\) acts properly discontinuously on \(\mathcal{H}\) and \(\mathcal{H}^*\) by the formula

with the obvious conventions regarding \(\infty \).

For each open subgroup \(H \le \textrm{GL}_2(\widehat{\mathbb {Z}})\), there is a modular curve \(X_H\), defined as a quotient of the full modular curve \(X_{\text{full}}(N)\), where \(N\) is the level of \(H\). More precisely, \(H\) is the inverse image of a subgroup \(H_N \le \textrm{GL}_2(\mathbb {Z}/N\mathbb {Z})\), which acts on \(X_{\text{full}}(N)\) over \(\mathbb {Q}\), and \(X_H\) is the quotient curve \(H_N \backslash X_{\text{full}}(N)\), also defined over \(\mathbb {Q}\).

Like \(X_{\text{full}}(N)\), the curve \(X_H\) is smooth, projective, and integral, and when \(\det (H)=\widehat{\mathbb {Z}}\) it is also geometrically integral, but in general it may have several geometric components, as is the case for \(X_{\text{full}}(N)\) when \(N{\gt}2\).

Rational points: When \(-1\in H\) the rational points of \(X_H\) consist of cusps and \(\operatorname{Gal}_{\mathbb {Q}}\)-stable isomorphism classes of pairs \((E,[\iota ]_H)\), where \(E\) is an elliptic curve over \(\mathbb {Q}\), and \([\iota ]_H\) is an \(H\)-level structure on \(E\). Such points exist precisely when the image of the adelic Galois representation \(\rho _E\colon \operatorname{Gal}_{\mathbb {Q}}\to \textrm{GL}_2(\widehat{\mathbb {Z}})\) is conjugate to a subgroup of \(H\).

Complex points: The congruence subgroup \(\Gamma _H:= H\cap \textrm{SL}_2(\mathbb {Z})\) acts on the completed upper half-plane \(\overline{\mathfrak {h}}\); one connected component of \(X_H(\mathbb {C})\) is biholomorphic to the quotient \(\Gamma _H \backslash \overline{\mathfrak {h}}\).

The curve \(X_H\) can alternatively be constructed as the coarse moduli space of the stack \(\mathcal X_H\) over \(\mathbb {Q}\) defined in Deligne-Rapoport [ . Both constructions of \(X_H\) can be carried out over any field of characteristic not dividing \(N\), or even over \(\mathbb {Z}[1/N]\).

The cusps on \(X_H\) are the points whose image under the canonical morphism \(j\colon X_H\to X(1)\simeq \mathbb {P}^1\) is \(\infty \). It is only the noncuspidal points that parametrize elliptic curves (with level structure).

The cusps of a modular curve \(X_H\) correspond to the complement of \(Y_H\) in \(X_H\), where \(Y_H\) is the coarse moduli stack \(\mathcal M_H^0\) defined in [ .

The rational cusps (also called \(\mathbb {Q}\)-cusps) are the cusps fixed by \(\operatorname{Gal}_{\mathbb {Q}}\).

Let \(H\) be an open subgroup of \(\textrm{GL}_2(\widehat{\mathbb {Z}})\) of level \(N\), let \(\pi _N\colon \textrm{GL}_2(\widehat{\mathbb {Z}})\to \textrm{GL}_2(\mathbb {Z}/N\mathbb {Z})\) be the natural projection, and let \(E\) be an elliptic curve over a number field \(K\).

An \(H\)-level structure on \(E\) is the \(H\)-orbit \([\iota ]_H:=\{ h\circ \iota \colon h\in \pi _N(H)\} \) of an isomorphism \(\iota \colon E[N]\overset {\sim }{\rightarrow }(\mathbb {Z}/N\mathbb {Z})^2\).

An \(H\)-level structure on \(E\) is rational if it lies in a \(\operatorname{Gal}_K\)-stable isomorphism class of pairs \((E,[\iota ]_H)\), where \(\sigma \in \operatorname{Gal}_K\) acts via \((E,[\iota ]_H)\mapsto (E^\sigma ,[\iota \circ \sigma ^{-1}]_H)\). Two pairs \((E,[\iota ]_H)\) and \((E',[\iota ']_H)\) are isomorphic if there is an isomorphism \(\phi \colon E\to E'\) that induces an isomorphism \(\phi _N\colon E[N]\to E'[N]\) for which \(\phi _N^*([\iota ']_H) = [\iota ]_H\).

If \(E\) admits a rational \(H\)-level structure \([\iota ]_H\) then image of its adelic Galois representation \(\rho _E\colon \operatorname{Gal}_K\to \textrm{GL}_2(\widehat{\mathbb {Z}})\) is conjugate to a subgroup of \(H\) and the isomorphism class of \((E,[\iota ]_H)\) is a non-cuspidal \(K\)-rational point on the modular curve \(X_H\).

When \(-1\in H\) every non-cuspidal \(K\)-rational point on \(X_H\) arises in this way. When \(-1\not\in H\) this is almost true, but there may be exceptions at points with \(j(E)=0,1728\).

Invariants of a rational \(H\)-level structure include:

• Cyclic \(\boldsymbol {N}\)-isogeny field degree: the minimal degree of an extension \(L/K\) over which the base change \(E_L\) admits a rational cyclic isogeny of degree \(N\); equivalently, the index of the largest subgroup of \(H\) fixing a subgroup of \((\mathbb {Z}/N\mathbb {Z})^2\) isomorphic to \(\mathbb {Z}/N\mathbb {Z}\).

• Cyclic \(\boldsymbol {N}\)-torsion field degree: the minimal degree of an extension \(L/K\) for which \(E_L\) has a rational point of order \(N\); equivalently, the index of the largest subgroup of \(H\) that fixes a point of order \(N\) in \((\mathbb {Z}/N\mathbb {Z})^2\).

• N-torsion field degree the minimal degree of an extension \(L/K\) for which \(E[N]\subseteq E(L)\); this is simply the cardinality of the reduction of \(H\) to \(\textrm{GL}_2(\mathbb {Z}/N\mathbb {Z})\).

There are three variants of the modular curve \(Y(N)\):

1. There is a functor sending each \(\mathbb {Z}[1/N]\)-algebra \(R\) to the set of (isomorphism classes of) pairs \((E,\alpha )\) such that \(E\) is an elliptic curve over \(R\) and \(\alpha \colon E[N] \to (\mathbb {Z}/N\mathbb {Z})^2\) is an isomorphism of group schemes. Suppose that \(N \ge 3\); then this functor is represented by a smooth affine \(\mathbb {Z}[1/N]\)-scheme \(Y_{\mathrm{full}}(N)\), called the full modular curve of level \(N\). (If \(N{\lt}3\), it is representable only by an algebraic stack, and one must take the coarse moduli space to get a scheme.) For any field \(k\) with \(\operatorname {char} k \nmid N\), the set \(Y_{\mathrm{full}}(N)(k)\) is the set of isomorphism classes of triples \((E,P,Q)\) , where \(E\) is an elliptic curve over \(k\) and \(P,Q \in E(k)\) form a \((\mathbb {Z}/N\mathbb {Z})\)-basis of \(E[N]\). The curve \(Y_{\mathrm{full}}(N)_{\mathbb {Q}}\) is integral but typically has several geometric components.

2. Let \(\zeta _N \in \overline{\mathbb {Q}}\) be a primitive \(N\)th root of unity. There is a functor sending each \(\mathbb {Z}[1/N,\zeta _N]\)-algebra \(R\) to the set of pairs \((E,\alpha )\) such that \(E\) is an elliptic curve over \(R\) and \(\alpha \colon E[N] \to (\mathbb {Z}/N\mathbb {Z})^2\) is an isomorphism of group schemes such that the resulting elements \(P,Q \in E[N](R)\) satisfy \(e_N(P,Q)=\zeta _N\). For \(N \ge 3\), this functor is represented by a smooth affine \(\mathbb {Z}[1/N,\zeta _N]\)-scheme \(Y(N)\), called the classical modular curve of level \(N\). Over any \(\mathbb {Z}[1/N,\zeta _N]\)-field \(k\), the curve \(Y(N)_k\) is geometrically integral.

3. There is a functor sending each \(\mathbb {Z}[1/N]\)-algebra \(R\) to the set of pairs \((E,\alpha )\) consisting of an elliptic curve \(E\) over \(R\) and a symplectic isomorphism \(\alpha \colon E[N] \to \mathbb {Z}/N\mathbb {Z}\times \mu _N\). For \(N \ge 3\), this functor is represented by a smooth affine \(\mathbb {Z}[1/N]\)-scheme \(Y_{\mathrm{arith}}(N)\). Over any field \(k\) with \(\operatorname {char} k \nmid N\), the curve \(Y_{\mathrm{arith}}(N)_k\) is geometrically integral.

• Relationships*: Over any \(\mathbb {Z}[1/N,\zeta ]\)-field \(k\), the curve \(Y_{\mathrm{arith}}(N)_k\) is isomorphic to \(Y(N)_k\) and to a connected component of \(Y_{\mathrm{full}}(N)_k\).

• Complex points*: The group \(\Gamma (N)\) acts on the upper half-plane \(\mathfrak {h}\), and the quotient \(\Gamma (N) \backslash \mathfrak {h}\) is biholomorphic to \(Y(N)(\mathbb {C}) \simeq Y_{\textup{arith}}(\mathbb {C})\) (choosing \(\zeta _N \in \mathbb {C}\)).

• Compactifications*: For each variant, there is a corresponding smooth projective model, denoted \(X_{\mathrm{full}}(N)\), \(X(N)\), or \(X_{\mathrm{arith}}(N)\).

• Quotients*: For each open subgroup \(H \le \textrm{GL}_2(\widehat{\mathbb {Z}})\), there is a quotient \(X_H\) of \(X_{\mathrm{full}}(N)\).

A ring is a set \(R\) with two binary operations \(+\) and \(\cdot \) such that

1. \(R\) is an abelian group with respect to \(+\) 2. \(\cdot \) is associative on \(R\) 3. the distributive laws hold, i.e., for all \(a,b,c\in R\),

4. there is an identity element with respect to the operation \(\cdot \), typically denoted by \(1_R\) or, more simply, by \(1\).

The identity element of \(R\) as a group with respect to \(+\) is typically denoted by \(0_R\) or, more simply, by \(0\).

The ring \(R\) is a commutative ring if \(R\) is a ring such that the operation \(\cdot \) is commutative on \(R\).

We say that \(R\) is a rng (also called ring without identity) if conditions 1-3 (but not necessarily 4) are satisfied.

Let \(A\) be a commutative ring. An \(A\)-field is an \(A\)-algebra that is a field.

The characteristic of a ring is the least positive integer \(n\) for which

if such an \(n\) exists, and \(0\) otherwise. Equivalently, it is the exponent of the additive group of the ring.

The characteristic of a field \(k\) is either \(0\) or a prime number \(p\), depending on whether the prime field of \(k\) is isomorphic to \(\mathbb {Q}\) or \(\mathbb {F}_p\).

A Dedekind domain \(D\) is a integral domain which is not a field such that

1. \(D\) is Noetherian; 2. every non-zero prime ideal is maximal; 3. \(D\) is integrally closed.

The ring of integers of a number field is always a Dedekind domain, as is every discrete valuation ring.

In a Dedekind domain, every non-zero ideal \(I\) can be written as a product of non-zero prime ideals,

and the product is unique up to the order of the factors. Repeated factors are often grouped, so we write \(I=Q_1^{e_1}\cdots Q_g^{e_g}\) where the \(Q_i\) are non-zero prime ideals of \(D\).

In addition, every fractional ideal \(I\) is invertible in the sense that there exists a fractional ideal \(J\) such that \(IJ=D\).

A field is a commutative ring \(R\) such that \(0_R\neq 1_R\) and every nonzero element of \(R\) has an inverse in \(R\) with respect to multiplication.

If \(R\) is an integral domain, then its field of fractions \(F\) is the smallest field containing \(R\).

It can be constructed by mimicking the set of fractions \(a/b\) where \(a,b\in R\) with \(b\neq 0\) following the usual rules for fraction arithmetic. It is unique, up to unique isomorphism.

If \(R\) is an integral domain with field of fractions \(K\), then a fractional ideal \(I\) of \(R\) is an \(R\)-submodule of \(K\) such that there exists \(d\in R-\{ 0\} \) with

If \(R\) is a ring, a subset \(I\subseteq R\) is an ideal of \(R\) if \(I\) is a subgroup of \(R\) for \(+\) and for all \(a\in I\) and all \(r\in R\),

In a polynomial ring \(R[X_1,\dots ,X_n]\), an ideal is homogeneous if it can be generated by homogeneous polynomials.

If \(R\subseteq S\) are commutative rings, an element \(s\in S\) is integral over \(R\) if there exists \(n\in \mathbb {Z}^+\) and \(a_i\in R\) such that

The integral closure of \(R\) in \(S\) is \(\{ s\in S\mid s \text{ is integral over } R\} \).

An integral domain is a commutative ring \(R\) such that \(1_R\neq 0_R\) and \(R\) contains no zero divisors.

Let \(R\) be an integral domain and \(F\) its field of fractions. Then \(R\) is integrally closed if \(R\) equals the integral closure of \(R\) in \(F\).

An element \(x \ne 0\) of a commutative ring \(R\) is irreducible if it is not a unit and has the property that whenever \(x=yz\) for some \(y,z \in R\), either \(y\) or \(z\) is a unit.

In a ring \(R\), an ideal \(M\) is maximal if \(M\neq R\) and for all ideals \(I\) of \(R\),

A commutative ring \(R\) is Noetherian if for every ascending chain of ideals

there exists \(N\) such that for all \(n\geq N\), \(I_n=I_N\).

If \(R\) is a commutative ring \(R\), an ideal \(I\) is prime if for all \(a,b\in R\),

Let \(R\) be an integral domain with field of fractions \(K\). If \(a\in K^\times \), then the principal fractional ideal generated by \(a\) is the set

A unit in a commutative ring \(R\) is an element \(x \in R\) so that there exists \(y \in R\) with \(xy = 1\). The set of units in \(R\) is denoted \(R^*\) or \(R^\times \) and forms a group under multiplication.

An element \(a\) in a ring \(R\) is a zero divisor if \(a\neq 0_R\) and there exists an element \(b\in R-\{ 0_R\} \) such that

The (Euler) gamma function \(\Gamma (z)\) is defined by the integral

for Re\((z) {\gt} 0\). It satisfies the functional equation

and can thus be continued into a meromorphic function on the complex plane, whose poles are at the non-positive integers \(\{ 0,-1,-2,\ldots \} \).

The Sato-Tate group of a motive \(X\) is a compact Lie group \(G\) containing (as a dense subset) the image of a representation that maps Frobenius elements to conjugacy classes. When \(X\) is an Artin motive, \(G\) corresponds to the image of the Artin representation; when \(X\) is an abelian variety over a number field, one can define \(G\) in terms of an \(\ell \)-adic Galois representation attached to \(X\).

For motives of even weight \(w\) and degree \(d\), the Sato-Tate group is a compact subgroup of the orthogonal group \(\mathrm{O}(d)\). For motives of odd weight \(w\) and even degree \(d\), the Sato-Tate group is a compact subgroup of the unitary symplectic group \(\mathrm{USp}(d)\). For motives \(X\) arising as abelian varieties, the weight is always \(w=1\) and the the degree is \(d=2g\), where \(g\) is the dimension of the variety.

The simplest case is when \(X\) is an elliptic curve \(E/\mathbb {Q}\), in which case \(G\) is either \(\mathrm{SU}(2)=\mathrm{USp}(2)\) (the generic case), or \(G\) is \(N(\mathrm{U}(1))\), the normalizer of the subgroup \(\mathrm{U}(1)\) of diagonal matrices in \(\mathrm{SU}(2)\), which contains \(\mathrm{U}(1)\) with index 2.

The generalized Sato-Tate conjecture states that when ordered by norm, the sequence of images of Frobenius elements under this representation is equidistributed with respect to the pushforward of the Haar measure of \(G\) onto its set of conjugacy classes. This is known for all elliptic curves over totally real number fields (including \(\mathbb {Q}\)) or CM fields.

A symplectic form on a vector space \(V\) over a field \(k\) is a non-degenerate alternating bilinear form \(\omega \colon V\times V\to k\). This means that

• if \(\omega (u,v)=0\) for all \(v\in V\) then \(u=0\) (non-degenerate);

• \(\omega (v,v)=0\) for all \(v\in V\) (alternating);

• \(\omega (\lambda u+v,w)=\lambda \omega (u,v)+\omega (v,w)\) and \(\omega (u,\lambda v+w)=\omega (u)+\lambda \omega (v,w)\) for all \(\lambda \in k\), \(u,v,w\in V\) (bilinear).

A finite dimensional vector space admitting a symplectic form \(\omega \) necessarily has even dimension \(2n\), and in this case \(\omega \) can be represented by a matrix \(\Omega \in k^{2n\times 2n}\) that satisfies \(u^\intercal \Omega v=\omega (u,w)\) for all \(u,v\in V\). One can always choose a basis for \(V\) so that

where \(I_n\) denotes the \(n\times n\) identity matrix.

For a positive even integer \(d\) the unitary symplectic group \(\mathrm{USp}(d)\) is the group of unitary transformations of a \(d\)-dimensional \(\mathbb {C}\)-vector space equipped with a symplectic form \(\Omega \). In other words, the subgroup of \(\textrm{GL}_d(\mathbb {C})\) whose elements \(A\) satisfy:

• \(A^{-1} = \bar A^{\intercal }\) (unitary);

• \(A^\intercal \Omega A=\Omega \) (symplectic).

It is a compact real Lie group that can also be viewed as the intersection of \(\mathrm{U}(d)\) and \(\mathrm{Sp}(d,\mathbb {C})\) in \(\textrm{GL}_d(\mathbb {C})\).