# Dirichlet Characters Let $R$ be a ring. Then a *Dirichlet character over* $R$ *of modulus* $N$ is a homomorphism $$ \varepsilon : ({\mathbb{Z}}/N{\mathbb{Z}})^* \rightarrow R^*, $$ where $R^*$ is the group of invertible elements of $R$. We extend $\varepsilon$ to a set theoretic map on the whole of ${\mathbb{Z}}$ by defining $\varepsilon(x) = 0$ if $\gcd(x,N)\neq 1$. The *conductor* of $\varepsilon$ is the smallest positive integer $M$ such that the homomorphism $({\mathbb{Z}}/N{\mathbb{Z}})^* \rightarrow R^*$ factors through $({\mathbb{Z}}/M{\mathbb{Z}})^*$ via the natural map $({\mathbb{Z}}/N{\mathbb{Z}})^*\rightarrow ({\mathbb{Z}}/M{\mathbb{Z}})^*$. ## Creation ### `DirichletGroup(N): RngIntElt -> GrpDrch` The group of Dirichlet characters modulo $N$ with values in `RationalField()`. (Note that this is a group of exponent at most $2$.) ### `FullDirichletGroup(N): RngIntElt -> GrpDrch` The group of Dirichlet characters modulo $N$ taking values in the $m$th cyclotomic field, where $m$ is the exponent of the unit group modulo $N$. ### `DirichletGroup(N, R): RngIntElt, Rng -> GrpDrch` The group of Dirichlet characters modulo $N$ with values in the ring $R$. Here $R$ can be the integers, rationals, a number field or a finite field. (Note that this group may be smaller than the full Dirichlet group.) ### `DirichletGroup(N, R, z, r): RngIntElt, Rng, RngElt, RngIntElt -> GrpDrch` The group of Dirichlet characters mod $N$ with values in the cyclic subgroup generated by the root of unity $z$ in the ring $R$. Here $z$ must be an element of $R$ of exact order $r$ (where $r$ may be smaller than the exponent of the full Dirichlet group). ### `BaseExtend(G, R): GrpDrch, Rng -> GrpDrch` ### `BaseExtend(G, R, z): GrpDrch, Rng, RngElt -> GrpDrch` The group of Dirichlet characters corresponding to $G$ with values in the ring $R$. In the second form, the distinguished root of unity of the base ring of $G$ is identified with the given element $z$. ### `AssignNames(~G, S): GrpDrch, [MonStgElt]` Assign names to the generators of the Dirichlet group $G$. ## Element Creation ### `Elements(G): GrpDrch -> [GrpDrchElt]` A sequence containing all Dirichlet characters in the Dirichlet group $G$. ### `Random(G): GrpDrch -> GrpDrchElt` A random element of the Dirichlet group $G$. ### `G . i: GrpDrch, RngIntElt -> GrpDrchElt` The $i$th generator of the group $G$. ### `G ! x: GrpDrch, Any -> GrpDrchElt` This coerces the given element $x$ into the Dirichlet group $G$. Here $x$ may be a Dirichlet character belonging to a different group, or a sequence of integers specifying an element of the `AbelianGroup` of $G$. ### `KroneckerCharacter(D): RngIntElt -> GrpDrchElt` ### `KroneckerCharacter(D, R): RngIntElt, Rng -> GrpDrchElt` The Kronecker character $n\mapsto (d/n)$, where $d$ is the fundamental discriminant associated to the integer $D$. When a ring $R$ is given, this is returned as a character with values in $R$. ## Attributes of Dirichlet Groups ### `BaseRing(G): GrpDrch -> Rng` The ring in which characters in $G$ take values. ### `Modulus(G): GrpDrch -> RngIntElt` The integer $N$ such that G is a group of Dirichlet characters on ${\mathbb{Z}}/N$. ### `Order(G): GrpDrch -> RngIntElt` The order of the Dirichlet group $G$. ### `Exponent(G): GrpDrch -> RngIntElt` The exponent of the Dirichlet group $G$. ### `NumberOfGenerators(G): GrpDrch -> RngIntElt` The number of generators of the Dirichlet group $G$. ### `Generators(G): GrpDrch -> [GrpDrchElt]` A sequence containing generators for the Dirichlet group $G$. ### `UnitGenerators(G): GrpDrch -> [RngIntElt]` This returns an ordered sequence of integers that reduce to “canonical” generators of the unit group of $Z/N$, where $N$ is the modulus of G. ### `GaloisConjugacyRepresentatives(G): GrpDrch -> [GrpDrchElt]` ### `GaloisConjugacyRepresentatives(seq): [GrpDrchElt] -> [GrpDrchElt]` This returns a sequence containing one representative from each Galois conjugacy class (over ${\mathbb{Q}}$) of characters corresponding to a character in the given group or the given sequence. ### `AbelianGroup(G): GrpDrch -> GrpAb, Map` This returns a finite abelian group isomorphic to the given group $G$ of Dirichlet characters (as an abstract group), and secondly returns a map from the abstract group to $G$. It is necessary to use this function in order to make group theoretic constructions involving $G$. ## Attributes of Elements ### `BaseRing(chi): GrpDrchElt -> Rng` The ring in which the Dirichlet character $\chi$ takes values. ### `Modulus(chi): GrpDrchElt -> RngIntElt` The modulus of the group of Dirichlet characters that contains $\chi$. ### `Conductor(chi): GrpDrchElt -> RngIntElt` The minimal conductor of the Dirichlet character $\chi$. (That is, the smallest integer $M$ such that `chi` is well-defined on the unit group of $Z/M$.) ### `ElementToSequence(chi): GrpDrchElt -> SeqEnum` A sequence of integers specifying the Dirichlet character $\chi$ (in terms of generators of the group containing $\chi$). ### `x eq y: GrpDrchElt, GrpDrchElt -> BoolElt` Return `true` iff the given characters have the same modulus and values. ### `Order(chi): GrpDrchElt -> RngIntElt` The order of the given element $\chi$ in a group of Dirichlet characters. ### `IsTrivial(chi): GrpDrchElt -> BoolElt` Returns `true` if and only if the Dirichlet character $\chi$ has order $1$. ### `IsPrimitive(chi): GrpDrchElt -> BoolElt` Returns `true` iff the Dirichlet character $\chi$ is primitive (equivalently, if its conductor equals its modulus). ### `AssociatedPrimitiveCharacter(chi): GrpDrchElt -> GrpDrchElt` The primitive character modulo the conductor of $\chi$ which takes the same values (on units) as $\chi$. ### `IsEven(chi): GrpDrchElt -> BoolElt` Returns `true` if and only if `Evaluate(chi,-1)` is equal to $1$. Note that in characteristic $0$, the space of modular forms of weight $k$ and character $\chi$ is zero if $\chi$ is even and $k$ is odd. ### `IsOdd(chi): GrpDrchElt -> BoolElt` Returns `true` if and only if `Evaluate(chi,-1)` is equal to $-1$. Note that in characteristic $0$, the space of modular forms of weight $k$ and character $\chi$ is zero if $\chi$ is odd and $k$ is even. ### `IsTotallyEven(chi): GrpDrchElt -> BoolElt` For a Dirichlet character $\chi$, this is `true` if and only if every character in the `Decomposition` of $\chi$ (into prime power components) is even. ### `Decomposition(chi): GrpDrchElt -> List` This decomposes the Dirichlet character $\chi$ as a product of characters with prime power moduli. The function returns a list (not a sequence) containing these characters (which do not belong to the same group). ### `MinimalBaseRingCharacter(chi): GrpDrchElt -> GrpDrchElt` This returns a character which is the same as $\chi$, except which takes values in the smallest possible subring of the base ring of $\chi$. ## Evaluation ### `Evaluate(chi, n): GrpDrchElt, RngIntElt -> RngElt` ### `chi(n): GrpDrchElt, RngIntElt -> RngElt` The value of the Dirichlet character $\chi$ at the integer $n$. ### `ValueList(chi): GrpDrchElt -> [RngElt]` A sequence containing the values $[\chi(1),..,\chi(N)]$ of the given character $\chi$, where $N$ is the modulus of $\chi$. The list of values is stored; then in later calls to `Evaluate`, the stored value is returned. ### `ValuesOnUnitGenerators(chi): GrpDrchElt -> [RngElt]` A sequence containing the values of $\chi$ on the ordered sequence of elements of ${\mathbb{Z}}/m$ given by `UnitGenerators(Parent(chi))`, where $m$ is the modulus of $\chi$. ### `OrderOfRootOfUnity(r, n): RngElt, RngIntElt -> RngIntElt` Given an element $r$ of some ring which is *assumed* to satisfy $r^n = 1$, this returns the smallest integer $m$ such that $r^m = 1$. (This provides a convenient way to calculate the order of values of non-real characters.) ## Arithmetic ### `x * y: GrpDrchElt, GrpDrchElt -> GrpDrchElt` ### `x / y: GrpDrchElt, GrpDrchElt -> GrpDrchElt` The product or quotient (respectively) of the Dirichlet characters $x$ and $y$. This is a Dirichlet character of modulus equal to the least common multiple of the moduli of $x$ and $y$. The base rings and chosen roots of unity of the parents of $x$ and $y$ are equal. ### `x ^ n: GrpDrchElt, RngIntElt -> GrpDrchElt` The Dirichlet character $x$ raised to the power of $n$, where $n$ is any integer. ### `x ^ phi: GrpDrchElt, Map -> GrpDrchElt` The image of the Dirichlet character $x$ under the automorphism $\phi$. ### `Sqrt(x): GrpDrchElt -> GrpDrchElt` Given a Dirichlet character $x$ of odd order, this returns a square root of $x$ (in the same group). ## Example ### `Example: Dirichlet (ex-19f11f)` We begin by constructing the group of characters $({\mathbb{Z}}/5{\mathbb{Z}})^*\rightarrow{\mathbb{Q}}^*$. ```magma > G := DirichletGroup(5); G; // The default base field is Q. Group of Dirichlet characters of modulus 5 over Rational Field > #G; 2 > [Evaluate(a, n) : n in [1..5]]; [ 1, -1, -1, 1, 0 ] > Eltseq(a); [ 2 ] > a eq G![2]; true > IsEven(a); true > IsOdd(a); false > IsTrivial(a); false ``` Next we create a character by building it up “locally”. ```magma > G1 := DirichletGroup(4); > Conductor(a4); 4 > G2 := DirichletGroup(25); > Conductor(a5); 5 > eps := a4*a5; > Modulus(eps); 100 > Conductor(eps); 20 > Evaluate(eps,7) eq Evaluate(a4,7)*Evaluate(a5,7); true ``` Characters can be constructed over various fields. ```magma > G := DirichletGroup(7,GF(7)); > #G; 6 > Evaluate(a,2); 2 > > G := DirichletGroup(15,CyclotomicField(EulerPhi(15))); > G; Group of Dirichlet characters of modulus 15 over Cyclotomic Field of order 8 and degree 4 > #G; 8 > Conductor(a3); 3 > Conductor(a5); 5 > Order(a5); 4 > Evaluate(a5,2); zeta_8^2 ``` If $D$ is a fundamental discriminant, then `KroneckerCharacter(D)` is the quadratic Dirichlet character corresponding to the quadratic field ${\mathbb{Q}}(\sqrt{D})$. The following code verifies that `KroneckerCharacter` and `KroneckerSymbol` agree in the case $D=209$. ```magma > chi := KroneckerCharacter(209); > for n in [1..209] do > assert Evaluate(chi,n) eq KroneckerSymbol(209,n); > end for; ``` If $E$ is an elliptic curve with newform $f_E$, then the twist $E_D$ corresponds to $f_E$ twisted by this character, as illustrated below. ```magma > E := EllipticCurve(CremonaDatabase(),"11A"); > f := qEigenform(E,8); f; q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8) > chi := KroneckerCharacter(-7); > qEigenform(QuadraticTwist(E,-7),8); q - 2*q^2 + q^3 + 2*q^4 - q^5 - 2*q^6 + O(q^8) > R := Parent(f); > &+[Evaluate(chi,n)*Coefficient(f,n)*q^n : n in [1..7]] + O(q^8); q - 2*q^2 + q^3 + 2*q^4 - q^5 - 2*q^6 + O(q^8) ```