# Arithmetic Fuchsian Groups In this section, we computing the basic invariants of an arithmetic Fuchsian group $\Gamma$. ## Creation We begin by giving the basic constructors for arithmetic Fuchsian groups. ### `FuchsianGroup(O): AlgQuatOrd -> GrpPSL2` ### `FuchsianGroup(O): AlgAssVOrd -> GrpPSL2` Returns the arithmetic Fuchsian group $\Gamma$ corresponding to the group ${\cal O}^*$ of units of reduced norm $1$ in the quaternion order ${\cal O}$. ### `FuchsianGroup(A): AlgQuat -> GrpPSL2` Returns the arithmetic Fuchsian group $\Gamma$ corresponding to the group ${\cal O}^*$ of units of reduced norm $1$ in a maximal order ${\cal O}$ in the quaternion algebra $A$. ### `FuchsianGroup(A, N): AlgQuat, RngOrdIdl -> GrpPSL2` ### `FuchsianGroup(A, N): AlgQuat, RngIntElt -> GrpPSL2` Returns the arithmetic Fuchsian group $\Gamma$ corresponding to the group ${\cal O}^*$ of units of reduced norm $1$ in an order ${\cal O}$ of level ${\frak N}$ in the quaternion algebra $A$. ### `Example: Construct AFG1 (ex-3c7078)` In this example, we construct arithmetic Fuchsian groups in three ways. First, we construct the group associated to the quaternion algebra of discriminant $6$ over ${\mathbb{Q}}$. ```magma > A := QuaternionAlgebra(6); > G := FuchsianGroup(A); > G; Arithmetic Fuchsian group arising from order of Quaternion Algebra with base ring Rational Field > O := BaseRing(G); > O; Order of Quaternion Algebra with base ring Rational Field with coefficient ring Integer Ring > Discriminant(O); 6 > Algebra(O) eq A; true ``` Next, we construct a group “by hand”, associated to the quaternion algebra over the totally real subfield $F={\mathbb{Q}}(\zeta_9)^+$ of ${\mathbb{Q}}(\zeta_9)$ ramified only at two of the three real infinite places. ```magma > K := CyclotomicField(9); > F := sub; > Degree(F); 3 > Z_F := MaximalOrder(F); > Foo := InfinitePlaces(F); > A := QuaternionAlgebra(ideal, Foo[2..3]); > Discriminant(A); Principal Ideal of Z_F Generator: [1, 0, 0] [ 2nd place at infinity, 3rd place at infinity ] > O := MaximalOrder(A); > G := FuchsianGroup(O); > G; Arithmetic Fuchsian group arising from order of Quaternion Algebra with base ring Field of Fractions of Maximal Equation Order with defining polynomial x^3 - 3*x - 1 over its ground order ``` Lastly, we construct the group of level $3$ inside a quaternion algebra ramified at $2\infty_1$ over ${\mathbb{Q}}(\sqrt{5})$. ```magma > F := NumberField(Polynomial([-5,0,1])); > Z_F := MaximalOrder(F); > A := QuaternionAlgebra(ideal, InfinitePlaces(F)[1..1]); > G := FuchsianGroup(A, ideal); > O := BaseRing(G); > O; Order of Quaternion Algebra with base ring Field of Fractions of Z_F with coefficient ring Maximal Order of Equation Order with defining polynomial x^2 - 5 over its ground order > Discriminant(O); Principal Ideal of Z_F Generator: 6/1*Z_F.1 > Discriminant(A); Principal Prime Ideal of Z_F Generator: [2, 0] [ 1st place at infinity ] > Level(G); Principal Ideal of Z_F Generator: 3/1*Z_F.1 ``` ## Quaternionic Functions In this section, we provide some functions related quaternion algebras which underlie the other functions in this chapter. Let $A$ be a quaternion algebra over a totally real field $F$, represented in standard form by $\alpha,\beta \in A$ satisfying $\alpha^2=a$, $\beta^2=b$, and $\beta\alpha=-\alpha\beta$. Suppose that $A$ has a unique split real place $v$; we then take this place to be the identity and consider $F$ as a subfield of ${\mathbb{R}}$. Then at least one of $a,b>0$, and without loss of generality we may assume that $a>0$. We then define the embedding $\iota:A \to M_2({\mathbb{R}})$, given by $$ \alpha \mapsto \left(\begin{matrix}\sqrt{a} & 0 \\ 0 & -\sqrt{a}\end{matrix}\right), \quad \beta \mapsto \left(\begin{matrix}0 & \sqrt{|b|} \\ {\rm sgn}(b)\sqrt{|b|} & 0\end{matrix}\right). $$ This particular choice of embedding is governed by the following. The reduced norm ${\rm nrd}:A \to {\mathbb{R}}$ given by $$ \gamma = x+y\alpha+z\beta+w\alpha\beta \mapsto {\rm nrd}(\gamma)=x^2-ay^2-bz^2+abw^2 $$ is not a Euclidean norm, so it is natural to instead take $$ \gamma \mapsto {\rm nrd}'(\gamma)=x^2 + |a|y^2 + |b|z^2 + |ab|w^2, $$ which corresponds to $$ {\rm nrd}'(\gamma)=x^2+y^2+z^2+w^2{\rm\ if\ }\iota(\gamma)=\left(\begin{matrix}x & y \\ z & w\end{matrix}\right). $$ Combining this new norm, defined for the identity real place, with the ones corresponding to the nonidentity real places $\sigma$, we may then define the definite norm $$ N(\gamma) = {\rm nrd}'(\gamma) + \sum_{\sigma \neq {\operatorname{id}}} \sigma({\rm nrd}(\gamma)). $$ This norm (and the corresponding Gram matrix) on $A$ are used in the application to fundamental domains (see the next section). ### `QuaternionOrder(G): GrpPSL2 -> AlgQuatOrd` ### `BaseRing(G): GrpPSL2 -> AlgQuatOrd` The order in some quaternion algebra that was used to define the Fuchsian group $G$. ### `QuaternionAlgebra(G): GrpPSL2 -> AlgQuat` The quaternion algebra used to define the Fuchsian group $G$. ### `SplitRealPlace(A): AlgQuat -> PlcNum` Returns the unique real place at which $A$ is split, if it exists. ### `FuchsianMatrixRepresentation(A): AlgQuat -> Map` Returns the map $A \to M_2({\mathbb{R}})$ when $A$ has a unique split place, as defined above. ### `DefiniteNorm(gamma): AlgQuatElt -> FldReElt` Returns the definite norm of $\gamma \in A$ for $A$ a quaternion algebra with a unique split real place, as defined above. ### `DefiniteGramMatrix(B): SeqEnum[AlgQuatElt] -> FldReElt` Returns the definite Gram matrix for the basis $B$ of a quaternion algebra $A$ with a unique split real place, as defined above. ### `Example: Quaternionic Functions (ex-755a66)` We begin by defining the quaternion algebra over $F={\mathbb{Q}}(\sqrt{5})$ which is ramified at the prime ideal $2$ and the second real place $\infty_2$, as indexed by Magma. ```magma > F := QuadraticField(5); > Z_F := MaximalOrder(F); > Foo := InfinitePlaces(F); > A := QuaternionAlgebra(ideal, [Foo[2]]); > a, b := StandardForm(A); > F!a, F!b; -1 w + 2 ``` The algorithm is random; entering the preceding input again will produce a different choice of $a$ and $b$. Next we perform an independent check that $a$ and $b$ define the same algebra. ```magma > IsIsomorphic(A, QuaternionAlgebra< F | a, b >); true ``` We see that $A$ is isomorphic to the algebra represented in standard form by $$ \alpha^2=a,\ \beta^2=b,\ \beta\alpha=-\alpha\beta $$ where $a=-1$, $b=2+\sqrt{5}$. (Note that the algorithm used to construct $A$ is probabilistic, so $a$ and $b$ are not canonically given.) We now demonstrate the real embedding corresponding to the split real place $\infty_1$. ```magma > v := SplitRealPlace(A); > v; 1st place at infinity > iota := FuchsianMatrixRepresentation(A); > iota(alpha); [0.0000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000 1.000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000] [-1.000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000 0.000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000] > iota(beta); [2.0581710272714922503219810475804504212387300996778194862815423526404796231583 78925542297104146968394 0.0000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000] [0.0000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000 -2.05817102727149225032198104758045042123873009967 7819486281542352640479623158378925542297104146968394] > Sqrt(Evaluate(b, v)); 2.05817102727149225032198104758045042123873009967781948628154235264047962315837 8925542297104146968394 > DefiniteNorm(alpha); 3.00000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000 ``` We see indeed that $$ \alpha \mapsto \left(\begin{matrix}0 & 1 \\ -1 & 0\end{matrix}\right), \quad \beta \mapsto \left(\begin{matrix}\sqrt{2+\sqrt{5}} & 0 \\ 0 & -\sqrt{2+\sqrt{5}}\end{matrix}\right) $$ and $$ N(\alpha)={\rm nrd}'(\alpha)+\sigma({\rm nrd}(\alpha))=(0^2+1^2+(-1)^2+0^2)+1=3. $$ ### `MultiplicativeOrder(gamma): AlgAssVOrdElt -> SeqEnum` ### `MultiplicativeOrder(gamma): AlgQuatElt -> SeqEnum` ```magma PlusMinus: BoolElt Default: true ``` Computes the order of the element $\gamma$ of a quaternion algebra; either a finite number or $0$ if the element has infinite order. If `PlusMinus` eq `true`, then compute the order in the group of units modulo $\pm 1$. ### `Quaternion(g): GrpPSL2Elt -> AlgQuatElt` For $g$ an element of an arithmetic Fuchsian group, return the underlying element of the quaternionic order. ## Basic Invariants In this section, we compute the basic invariants associated to an arithmetic Fuchsian group $\Gamma$. The first is the hyperbolic volume ${\rm vol}(X)$ of the orbit space $X=\Gamma\backslash{\mathbb{H}}$, which can be computed via a formula of Shimizu: $$ {\rm vol}(X)=\left({1 \over 2}\right)^{d-2} \zeta_F(-1) \prod_{{\frak p}\mid {\rm disc}A} \left(N({\frak p})-1\right) $$ where $d=[F:{\mathbb{Q}}]$. This volume is normalized so that an ideal triangle (i.e. a triangle with all vertices on the boundary of ${\mathbb{H}}$) has volume $1/2$. With this normalization, ${\rm vol}(X)$ takes rational values. The next invariant is the signature of $\Gamma$. A point $z \in {\mathbb{H}}$ is an *elliptic point* of order $k \geq 2$ if the stabilizer of $z$ under $\Gamma$, which is a finite cyclic group, has order $k$. There are only finitely many $\Gamma$-conjugacy classes of elliptic points, also known as *elliptic cycles*. If $\Gamma$ has $t$ elliptic cycles of order $m_1,\dots,m_t$, and $X$ has genus $g$, then we say that $\Gamma$ has *signature* $(g;m_1,\dots,m_t)$. We compute the number of elliptic cycles of order $k$ by a formula (too complicated to state here) which requires the computation of the relative class group of a CM extension of number fields. The genus $g$ can then be calculated by the Riemann-Hurwitz formula: $$ {\rm vol}(X) = 2g-2+\sum_{i=1}^t \left(1-{1 \over m_i}\right). $$ ### `ArithmeticVolume(G): GrpPSL2 -> FldRatElt` Returns the hyperbolic volume of the quotient of the upper half-plane by $G$ for an arithmetic Fuchsian group $G$. The volume is normalized arithmetic volume, so the “usual” volume is divided by $2\pi$; this gives an ideal triangle volume $1/2$. ### `EllipticInvariants(G): GrpPSL2 -> SeqEnum` Returns the number of elliptic cycles of the arithmetic Fuchsian group $G$ as a sequence of elliptic orders and their multiplicities. ### `Signature(G): GrpPSL2 -> SeqEnum` Returns the signature of the arithmetic Fuchsian group $G$. ## Group Structure We now compute a finite presentation for an arithmetic Fuchsian group $\Gamma$. If $\Gamma$ has signature $(g;m_1,\dots,m_t)$, then there exists a standard presentation for $\Gamma$, generated (freely) by $\alpha_1,\beta_1,\dots,\alpha_g,\beta_g,\gamma_1,\dots,\gamma_t$ with the relations $$ \gamma_1^{m_1}=\dots=\gamma_t^{m_t}=[\alpha_1,\beta_1] \cdots [\alpha_g,\beta_g] \gamma_1 \cdots \gamma_t = 1 $$ where $[\alpha_i,\beta_i]=\alpha_i\beta_i\alpha_i^{-1}\beta_i^{-1}$ is the commutator. In the special case where $g=0$, there exists an algorithm to reconstruct the standard presentation; if $g>0$, we only obtain a finite presentation, and it is possible that the construction of a standard presentation may be algorithmically impractical. Our algorithm uses a fundamental domain $F$ for $\Gamma$ (see section [Fundamental Domains](fundamental-domains.md#grppsl2fds)). In brief, we compute the set of elements which pair the sides of $F$; these elements are provably generators for the group $\Gamma$, and the relations between them are given as the set of minimal loops in a graph which records the side pairings. ### `Group(G): GrpPSL2 -> GrpFP, Map, Map` Returns a presentation $U$ for the arithmetic Fuchsian group $G$, a map $U \to G$, and a map $U \to {\cal O}$ where ${\cal O}$ is the quaternion order corresponding to $G$. ### `Example: Basic Invariants (ex-5d9bc5)` In this example, we compute the basic invariants of the arithmetic Fuchsian group associated to a maximal order in the quaternion algebra of discriminant $6$ over ${\mathbb{Q}}$. ```magma > A := QuaternionAlgebra(6); > O := MaximalOrder(A); > G := FuchsianGroup(O); > ArithmeticVolume(G); 1/3 > EllipticInvariants(G); [ <2, 2>, <3, 2> ] > Genus(G); 0 > Signature(G); <0, [2, 2, 3, 3]> ``` We verify the Riemann-Hurwitz formula $1/3 = 2\cdot 0-2+2(1-{1 \over 2})+2(1-{1 \over 3})$. We can also compute a finite presentation of this group. ```magma > U, m := Group(G); > U; Finitely presented group U on 3 generators Relations U.1^2 = Id(U) U.2^3 = Id(U) U.3^3 = Id(U) (U.2 * U.1 * U.3)^2 = Id(U) > [Matrix(m(U.i)) : i in [1..2]]; [ [-0.44721359549995793928183473374634344441856698198020262470211905349191088 22850343946302640398390408845 -1.09544511501033222691393956560182030636 1751136512834480425665136619270191207412304725356516468009374] [1.095445115010332226913939565601820306361751136512834480425665136619270191 207412304725356516468009374 0.44721359549995793928183473374634344441856 69819802026247021190534919108822850343946302640398390408845], [0.170820393249936908922752100619735659953958748115047636673027591262383066 3212351288138489551667724006 0.4184228011239092912722673110503248597003 073307330865863518420141891341904873780366272041685003642481] [-2.86791254390708738946955138575621625166625481138975671478453458144009456 7944693063167063601605004483 -1.170820393249936908922752100619735659953 958748115047636673027591262383066321235128813848955166772401] ] > [A!Quaternion(m(U.i)) : i in [1..2]]; [ -1/5*j + 1/5*k, -1/2 - 1/2*i + 3/10*j - 3/10*k ] ``` Next we compute a more complicated example, exhibiting an elliptic cycle of order $11$. The group $\Gamma$ corresponds to $F={\mathbb{Q}}(\zeta_{11})^+$ and $A$ the quaternion algebra only at real places. (See also the section on triangle groups.) ```magma > K := CyclotomicField(11); > F := sub; > Foo := InfinitePlaces(F); > Z_F := MaximalOrder(F); > A := QuaternionAlgebra(ideal, Foo[1..4]); > G := FuchsianGroup(A); > Signature(G); <0, [ 2, 3, 11 ]> > U, m := Group(G); > U; Finitely presented group U on 2 generators Relations U.1^2 = Id(U) U.2^3 = Id(U) (U.1 * U.2^-1)^11 = Id(U) ``` As a final example, we show that the finite presentation of $G$ may indeed be quite complicated. Here, we examine the (torsion-free) group associated to the quaternion algebra over ${\mathbb{Q}}$ with discriminant $35$. ```magma > G := FuchsianGroup(QuaternionOrder(35)); > Signature(G); <3, []>; > time U := Group(G); Time: 78.870 > U; Finitely presented group U on 6 generators Relations U.5 * U.4^-1 * U.5^-1 * U.1 * U.3 * U.2^-1 * U.3^-1 * U.6 * U.4 * U.2 * U.1^-1 * U.6^-1 = Id(U) ```