Families of ϕ‐congruence subgroups of the modular group

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ‐congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi‐unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi‐unipotent case, we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve y2=x3−1728$y^2=x^3-1728$ defined by the commutator subgroup of the modular group.


INTRODUCTION
The modular group Γ = SL 2 () and its finite index subgroups play a fundamental role throughout mathematics and physics.The most well-known of these subgroups are the congruence subgroups, which have the property that membership in the group can be tested against a finite list of congruence conditions.Equivalently, a subgroup is congruence if it contains one of the principal congruence subgroups Γ() = { ∈ Γ |  ≡ 1 (mod )}.
The quotients of the complex upper-half plane by these congruence subgroups define the modular curves, which classify elliptic curves decorated with additional structures related to torsion points on the curves.In this paper, we study a generalization of this notion of congruence subgroup to families of noncongruence subgroups.
Most subgroups of Γ of finite index are not congruence subgroups.In recent years, it has been shown that they too define natural moduli spaces [9], and their arithmetic has been shown to impinge on problems of classical interest.For example, Chen has shown [8] that the arithmetic of certain noncongruence moduli spaces is related to the description of Markoff triples [4].Now that the unbounded denominators conjecture on noncongruence modular forms has been resolved [6], one of the most interesting arithmetic open problems in this area, aside from the Grothendieck-Teichmuller conjecture [26], is to determine the Eisenstein ideals [12] associated to noncongruence subgroups, which encode the unbounded denominators phenomenon.Determining the Eisenstein ideals arising from SL 2 (  )-Teichmuller level structures would shed light on the work of [8].
Future exploration of phenomena such as this could be aided by having at hand a healthy supply of interesting spaces and groups to study.Thinking along these lines, in this paper we generalize the congruence subgroups in a natural way.The notion depends on a group homomorphism where  is some algebraic group defined over a number field.Given such data, there exists a number field ∕ such that  takes values in (  [1∕𝑀]) for some integer  ⩾ 1, where   is the ring of integers in .Therefore, it makes sense to reduce this homomorphism  modulo all but finitely many prime ideals of .The corresponding principal -congruence subgroups of level  for gcd(, ) = 1 are defined as Γ(, ) = { ∈ Γ | () ≡ 1 (mod )}.
A subgroup of Γ is said to be -congruence if it contains some principal -congruence subgroup.These are all finite index subgroups of the modular group, and when  is the inclusion of SL 2 () into SL 2 () we recover the usual congruence subgroups.When  has Zariski dense image, these groups are related to the Malcev completion of Γ (see [16] for a nice discussion), where Γ is interpreted as the fundamental group of the moduli stack of elliptic curves.
Below we study in Section 2 basic properties of these -congruence subgroups and, in particular, introduce some tools that allow us under suitable hypotheses to lift surjective maps Γ → (  ) to continuous surjections Γ → (  ), where Γ denotes the profinite completion of Γ and   denotes the -adic integers.
In Section 3, we study a particular family of -congruence subgroups arising from an uppertriangular but not decomposable representation of Γ of rank 2. The modular curves that arise as -congruence modular curves are precisely the curves that live beneath the abelian covers of the elliptic curve  2 =  3 − 1728.These curves have quite a long history, going back at least to investigations of Poincaré, though there remain some open problems (for example, describing the modular curves and Eisenstein constants for the genus zero groups   introduced in Subsection 3.4).We describe how to compute equations for the curves defined by -congruence subgroups that lie between the commutator Γ ′ and the double-commutator Γ ′′ , as well as give a detailed discussion of the corresponding modular forms and the Eisenstein constants.This material uses classical results on division polynomials associated to elliptic curves.
Finally, in Section 4 we use the moduli space of rank 4 irreducible representations of Γ described by Tuba-Wenzl [29] to identify some -congruence subgroups related to Sp 4 .We were led to consider this case in view of the probabilistic results of [22] which show that it is rarer to find surjective maps from Γ onto Sp 4 (  ) than for other linear algebraic groups.We compute genus and dimension formulae for the corresponding modular curves and modular forms, though we say nothing about finding equations for these curves, which are very high degree covers of the moduli space of elliptic curves, nor do we say anything about the corresponding Eisenstein constants.
By combining our symplectic results with the general theory worked out in Section 2, we are able to establish the following result: Theorem 1.1.Let  ∈  × ⧵ ( × ) 2 , and let  ∶ Γ → Sp 4 () be defined by the matrices .
Let  > 7 be a prime such that  is a primitive root mod , and let Γ denote the profinite completion of Γ.Then  gives rise to a continuous surjection φ ∶ Γ → Sp 4 (  ).
See Section 4 for more technical details on this result, and for proof of the main statements in Theorem 1.1.In particular, we warn the reader that the homomorphism  of Theorem 1.1 keeps a nonstandard symplectic form invariant, though the corresponding symplectic group is equivalent to that defined by more standard conventions.Briefly, the proof strategy of Theorem 1.1 has two key steps: we first establish the theorem mod  by showing that the image of  is not contained in any of the maximal subgroups of Sp 4 (  ), using the explicit enumeration and description of these subgroups as detailed in [19].Next we use the material from Subsection 2.2 to give a Hensel lifting type argument to deduce surjectivity mod prime powers from the prime case.The difficulty of executing this strategy more generally rests on the complicated nature of the maximal subgroup structure of classes of finite groups of Lie type, cf.[1].To give the flavor of the difficulties that one encounters, even when considering symplectic groups of degree four over general finite fields, one must consider stabilizers of much more complicated symplectic spreads [2] than those that arise here over   .For other groups, the possibilities are presumably even more foreboding.
There remain many interesting open questions about -congruence subgroups, aside from constructing nice examples.For example, how does the absolute Galois group Gal(∕) act on the corresponding curves and associated dessins d'enfants?If the prime  is fixed in Theorem 1.1 but  is varied, does this describe a single orbit under the action of the Galois group?In a slightly different direction, under what conditions do these families give rise to expander graphs as in the case of modular curves [23]?We hope to return to some of these questions in future work.
-If ∕ is a number field, then   denotes the ring of integers in .
-  and   denote the -adic numbers and the -adic integers, respectively.
- denotes a coordinate on the complex upper-half plane.
-We write  =  2∕6 in Section 3. Any other occurrences of  should be interpreted as meaning  =  2 .

Definitions
Let Γ = SL 2 (), let  be a finite group, and suppose given a homomorphism The kernel of  is necessarily of finite index, and we are interested in two basic questions.
As most subgroups of finite index in Γ are noncongruence, the answer to question (b) is "almost always," and in practice it can be tested easily using Wohlfahrt's criterion.For example, one has the following well-known result whose proof demonstrates this principle: Lemma 2.1.If  ⩾ 5 and  ∶ Γ →   is surjective, then ker  is noncongruence.
Proof.Assume that Γ() ⊆ ker  for some  ⩾ 2. This yields an isomorphism where  is the image of ker  in the quotient Γ∕Γ() ≅ ∏   || SL 2 (∕  ).As   has a unique nontrivial normal subgroup, we must have that  =   is a prime power.Let  be the kernel of the reduction map SL 2 (∕  ) → SL 2 (  ) and let  ∶ SL 2 (∕  ) →   denote the surjective homomorphism obtained from our congruence hypothesis.Then () is a normal subgroup of   , and so must be {1},   or   .But  is a -group, and so the only possibility is () = {1}.Hence,  ⊆  and we obtain a surjection SL 2 (  ) →   , which is absurd, as PSL 2 (  ) and   belong to members of different families of finite simple groups.□ The answer to question (a) is more delicate as, for example, the maximal subgroup structure of  can be quite complicated, cf.[1].Nevertheless, the results of papers such as [22] suggest again that question (a) should frequently have a positive answer.Our aim in this paper is to describe some methods for using global homomorphisms  into linear algebraic groups to identify and compute with families of finite-index subgroups of the modular group.We begin with some notation and basic properties.
Let  be a linear algebraic group defined over a number field and let  ∶ Γ → () denote a homomorphism.As Γ is finitely generated, there exists a number field ∕ with ring of integers   such that  takes values in (  [1∕]) for some integer  ⩾ 1.That is, we have a map It makes sense to reduce  mod integers  coprime to , and the kernels of such maps are our analogues of principal congruence subgroups: Definition 2.2.For  as above, we define the corresponding principal -congruence subgroups of level  ⩾ 1 for gcd(, ) = 1 as Definition 2.3.More generally, a subgroup Λ ⊆ Γ is said to be a -congruence subgroup if there exists  ⩾ 1 with Γ(, ) ⊆ Λ.
Remark 2.4.As isomorphism of representations corresponds to matrix conjugation, it is clear that the -congruence subgroups of Γ only depend on  up to isomorphism.Remark 2.5.Notice that Γ(, 1) = Γ for all representations .Therefore, one does not have, for example, Γ(, 1) ∩ Γ() = Γ(, ).Remark 2.6.Suppose that Λ ⊆ Γ is of finite index, and let  be a representation of Λ defined over .One can define -congruence subgroups Λ(, ) analogously to above.However, by considering the block description of matrices defining Ind Γ Λ , one sees that Therefore, every -congruence subgroup of Λ is an Ind Γ Λ -congruence subgroup of Γ and so, from this perspective, it suffices to consider -congruence subgroups of the full modular group Γ as defined above.
(1) Each Γ(, ) is a normal subgroup of finite index in Γ, and so each -congruence subgroup of Γ is of finite index in Γ.
Lemma 2.8.With  as above, Proof.By definition where Γ denotes the profinite completion of Γ.In the next subsection, we introduce some basic tools, likely known to experts, that will be used in Section 4 to describe explicit surjections for certain (conjecturally) infinite sets of primes .See Corollary 4.4 for more details.
Remark 2.11.All modular curves obtained from the -congruence groups Γ(, ) have moduli interpretations in the sense of [9].One expects the finite groups underlying the Teichmuller level structures of these moduli interpretations to be closely related to the image of , though the relationship is not always clear.The proof of [5, Theorem 5.1] can be used to make this relationship explicit in concrete examples.The possible discrepancy between the image of  and the corresponding Teichmuller level structures should be borne in mind while reading Section 3, where we study a family of noncongruence groups arising from a homomorphism  with metabelian image.In [10], it is shown that the subgroups of Γ defined by metabelian Teichmuller level structures are all congruence subgroups of Γ.Thus, in the examples of Section 3, the groups underlying the Teichmuller level structures certainly differ from the images of the reductions of .

Lifting surjectivity mod prime powers
When considering families as above, it is natural to ask when the image of a map is surjective.It is automatic, by the Chinese remainder theorem, that it suffices to check that    is surjective for each   ||.In this section, we show that typically, it would suffice to check each   where |.
For the remainder of this section, we fix a reductive group  defined over , the ring of integers of some number field, and a prime | of  that is unramified.We shall denote by  = |∕| and   = ∕.We shall also fix a subgroup  ⊂ ().
We will denote by   = (∕  ) and for  <  by  , the kernel of the map   →   .Similarly we shall denote by   the image of  in   and  , the kernel of the map   →   .Lemma 2.12.For all but finitely many , we have for each  ⩾ 1 that  +1, is isomorphic to the Lie algebra, Lie(   ), over   and the conjugation action of  +1 on the normal subgroup  +1, induces an action of  1 on  +1, given by the usual adjoint action on Lie(   ).
Proof.We recall that Lie() can be identified with the kernel of the map So with  embedded in GL  we may identify this with the  by  matrices  for which Similarly, we have that  +1, is precisely the  by  integer matrices  for which The map []∕ 2 → ∕ +1 given by  ↦   then induces the map Lie() ⊗   ↪  +1, , which is clearly equivariant for the action of conjugation by (  ).
One can readily verify by case analysis that for almost all  the map is bijective for all semisimple groups.In particular, in the applications in this paper we only require this for Sp 4 , and so this concludes the proof in that case.
Alternatively, for the general case, notice that for almost all  we have that the implicit function theorem allows us to lift elements of  +1, to  +2, , so that the map  +2, →  +2, given by  ↦   , so: (1 +   )  = 1 +  +1  (mod  +2 ) induces an injection  +1, ↪  +2,+1 that is seen to be a group homomorphism.Whenever the implicit function theorem applies we have that is indeed an isomorphism, as are the maps  +1, →  +2,+1 .
Proof.This is a simple application of Burnside's irreducibility criterion [17,Theorem 3.10].□ Remark 2.14.Note that the adjoint action would typically be irreducible for simple groups, but not for instance for GL  , where the adjoint action has a summand being the trivial representation.
Proof.Without loss of generality we may suppose that the element g ∈  is an element whose image in  1 is nilpotent.Then g has the form 1 + , will satisfy g  = 1 (mod ) and has nilpotency degree  ′ ⩽  so that for  = 1, … ,  ′ − 1 we have   are linearly independent over   , for  =  ′ , … ,  − 1 we have   = 0 (mod ) and finally   = 0 (mod  2 ).Then we have . □ Remark 2.16.The matrix ( 4 1 0 4 ) is nilpotent modulo 3 but its cube is the identity modulo 9.
Combining these we obtain: Theorem 2.17.Suppose  is at least twice the maximum nilpotency degree of  and | is such that  2,1 ≅ Lie(   ).Suppose also that the adjoint action of () on the lie algebra, Lie(   ), is irreducible (and nontrivial).Then if  1 =  1 we have for all  ⩾ 0 that   =   .

Generalities on the commutator subgroup
Let Γ = SL 2 (), and  = () where  is a primitive 12th root of unity, so that  4 −  2 + 1 = 0.If  ≡ 1 (mod 4) then   2 contains all of the roots of  4 −  2 + 1, so at times below we will restrict to such primes.On the other hand, we will sometimes want the image of   to not be contained in SL 2 (  ), thus we will insist that  ≡ 5 (mod 12).This ensures that  4 −  2 + 1 factors into two irreducible quadratics over   .
This representation is not irreducible, but it is also not reducible into a direct sum of two onedimensional subrepresentations.Notice that  12 ∈ ker , so that this representation is certainly not injective!Lemma 3.1.One has Proof.First observe the following identities: ) .
This, combined with the description of (), shows that the set on the right of the equality is contained inside (Γ).The reverse inclusion is obvious: notice that set on the right is a subgroup, containing () and (), which generate (Γ).
Proof.As Γ ′ is freely generated by  and , which are of infinite order, it follows that If  ≡ 5 (mod 12), then the intermediate identity is the reduction mod  map.It follows that Γ ′′ = ⋂ ≡5 (mod 12) ker   .As we have seen that Γ ′′ ⊆ ker  ⊆ ker   for all , the lemma follows.
□ Summarizing all of this, we have shown that if then there is an exact sequence The next lemma illustrates that the notion of -congruence subgroups is natural and useful.
Lemma 3.4.The discrete group  satisfies the congruence subgroup property.Therefore, the finite index subgroups  ⊆ Γ containing Γ ′′ are exactly the -congruence subgroups for this indecomposable representation .
Proof.Let  ′ ⊆  be a finite index subgroup, and let  ′ 0 =  ′ ∩ , which is finite index in  because an intersection of finite index subgroups is of finite index.Now  ′ 0 ⊆  is abelian, and as  ≅  ⊕  3 visibly satisfies the congruence subgroup property, we find that  ′ 0 contains the congruence subgroup   for some  ⩾ 1.But then  ′ also contains   , so that it is a congruence subgroup.By the exact sequence (1), the inverse images under  of the finite index subgroups of  are exactly the finite index subgroups between Γ ′′ and Γ. □ We can describe Γ ′′ more explicitly.Define Δ() − to be the smallest normal subgroup of Γ containing −  .
Proof.One checks that In particular, the induced conjugation action of  and  2 on Γ ′ ∕Γ ′′ ≅  2 is fixed-point free.We have ∕Γ ′′ ≅ 1,  or  2 .In the first case  = Γ ′′ , while in the latter we have  of finite index in Γ ′ , hence it is -congruence by Lemma 3.4.To rule out the case ∕Γ ′′ ≅  entirely, assume it is so, and notice that the action of the order 6 element  would be given by multiplication by ±1.In particular,  2 acts trivially on ∕Γ ′′ in this case, contradicting its fixed-point freeness.Therefore, we cannot have ∕Γ ′′ ≅ , and this concludes the proof.□ Proposition 3.8.All but finitely many of the finite-index normal subgroups of SL 2 () of genus one that contain −1 are -congruence subgroups for the indecomposable representation  introduced above.

3.2
The tower of isogenies

The modular curve 𝑋(Γ ′ )
To simplify discussions with modular forms in this section, we now let Γ = PSL 2 (), which we maintain for the remainder of the section.The modular forms we consider transform under the preimage in SL 2 ().In particular, −1 is contained in the preimage and there are no nonzero modular forms of odd weight.Note that Γ∕Γ ′ is cyclic of order 6.Recall the following facts about the modular curve (Γ ′ ).
(1) There are no elliptic points.
(2) There is a unique cusp, which has width 6.
For explanation of the terminology above, see [11, chapter 2].
Taking the cusp to be the distinguished point at infinity, it is seen to be an elliptic curve with complex multiplication by [

Étale covers
If  ∶  → (Γ ′ ) is any isogeny of elliptic curves, then as the map  is étale, we must have  ≅ (), for some finite-index subgroup Γ ′′ ⊂  ⊂ Γ ′ .Moreover,  has no elliptic points and the cusps, which are the kernel of the isogeny, have width 6.From the Weierstrass uniformization we see that such isogenies are naturally in bijection with lattices in [ then () has no elliptic points and, because the image of Δ(6) − inside PSL 2 () is contained in , all cusps have width 6.It follows both that () is genus 1, and the cover () → (Γ ′ ) is étale.In fact, it is simply an isogeny of elliptic curves.
We thus have a natural bijection between finite index lattices in [ √ −3] and finite index subgroups satisfying Γ ′′ ⊂  ⊂ Γ ′ .The simplest part of this correspondence is We see that the index of Γ ′ () in PSL 2 () is 6 2 .
As Γ ′ () is normal in Γ ′ , it follows that Γ ′ acts on (Γ ′ ()).By our description of these curves and the corresponding tower of isogenies, there exists a map where (Γ ′ ())[] denotes the -torsion of this elliptic curve, such that the natural action of Γ ′ on (Γ ′ ()) is given by  ↦  +   (g).
These expressions can be used to solve for -expansions of x and ỹ.
Noting that  2  and   are actually polynomials in x of degrees, respectively,  2 − 1 and  2 , of the form , we observe that the equation gives a polynomial satisfied by x over [[]], where  =  6 =  2∕6 .These equations can be used to solve for the -series of x and ỹ, see below.Note that x and ỹ have poles of order 2 and 3, respectively, at the point at infinity, equivalently  = 0, so that the solutions to these equations are given by Laurent series.
Remark 3.12.By comparing divisors, we can see that We next use the preceding material to describe the primes occurring in unbounded denominators of the modular forms for the groups Γ ′ ().Of course, such existence results follow much more generally by [6], though in the very explicit case considered here one can be more precise about the primes that arise in denominators.Note also that as the forms that arise here can be expressed in terms of forms that transform under a character of the congruence group Γ ′ , this case was considered even earlier by Kurth-Long in [21].Nevertheless, we believe that our approach via isogenies of elliptic curves is of independent interest.Proposition 3.13.The  series of x has coefficients in .Moreover, suppose that  is a prime not dividing .Then the -series of x is -integral.
Proof.Set x =  2 x and η4 =  −1  4 , so that x and η4 are both in [[]].We have that x satisfies the equation over whose reduction modulo  is where the variable  above should not be confused with the modular curves (Γ ′ ()).From this it follows, using that the constant term of x is nonvanishing, that x =  2 (mod ).
To apply Hensel's lemma to solve Equation ( 2) and thereby find the -series of x, we shall need to consider the derivative The sources of any irrational quantities that might arise while applying Hensel's lemma are the coefficients of the polynomial occurring in Equation (2), as well as from the solution of the first step of Hensel's lemma (the solution modulo ).The sources of denominators that arise from this procedure are likewise the coefficients, and inverting the constant term, as a series in , of M′ ( x).We conclude that the coefficients of x are rational and, in fact, integral away from primes dividing .□ The following result is a standard part of the theory of isogenies of elliptic curves: Lemma 3.14.Suppose  > 3 is prime and write   () = ∑      .Then if  is an ordinary prime for (Γ ′ ), we have  |   for  > ( 2 − )∕2, whereas if  is a supersingular prime, we have  |   for all  > 0. When  = 2 or  = 3 we have that   () = 0 (mod ).
Proof.For supersingular primes there are no (nontrivial) -torsion points modulo , hence   () is a constant modulo .
For ordinary primes there are  − 1 (nontrivial) -torsion points modulo , so the degree of   () modulo  must be divisible by ( − 1)∕2.As the lead coefficient is divisible by , so two are the first ( − 1)∕2.
The case of  = 2 and  = 3 is a direct check.But note that in contrast to the case of other primes, these do not give division polynomials modulo 2 and 3, as the discriminant of our curve's model is divisible by both 2 and 3. □ Lemma 3.15.Suppose  > 3 is prime and write    () = ∑   (  )    .Then we have that Proof.We note, by comparing divisors and lead terms, that we have the recurrence ) , from which we find −   () 2+1   () − where for convenience  =  2(−1) −1 2 . Comparing the degree of   (), which is  2 , to that of   (), which is ( 2 − 1)∕2, we see that the highest order terms have -adic valuations coming Proof.We note, by comparing divisors and lead terms, that we have the recurrence As above, we simply look at the source of the high-order terms.□ Remark 3.17.The preceding lower bounds on valuations in the two lemmas above are not tight in most cases.
Next, we note that implies that in either case the divisibility claim we are trying to establish, as well as the claim about reduction for ψ , follow from the analogous claims for ψ  and φ  .Similarly, using ) we see that the claims about the reduction reduce to those for the case of    , and using the divisibility claim about    .
The lemma thus holds in these cases.
For the inductive step, consider the recursive identity and divide both sides by  2 .The claim concerning ψ follows by induction.Next, in considering the recursion ) , we see that the claim concerning φ follows by induction.This concludes the proof of the lemma.□ Proposition 3.19.Suppose  > 3 is an odd prime.For any  |  the -series for x has unbounded denominators at .
Proof.Beginning as in Proposition 3.13, we now set and note x = 1 (mod q).Next set Observe that M() ∈ [[ q]][].Note also that, as a series in q, we have Consequently: We note that if x has unbounded denominators at , then we are done, as then x must also have unbounded denominators at .We next consider the case that x has integral coefficients.Then x is a nonzero solution to the nonzero polynomial: But then x∕ q2 is a nonzero solution to the nonzero polynomial   () 2 .However, as x∕ q2 = 1∕ q2 + (1∕ q) this is not possible.Now, suppose for the purpose of contradiction that the -denominators of x as a power series in q are bounded.Then consider the minimal power  ⩾ 1 so that x =   x clears them.Now consider the minimal power  so that M() =   M(∕  ) has integral coefficients.By Lemma 3.16, we can see that  =  2 − 2  () as the minimal value of  must come from the coefficient of   2 in   ().It follows that x is a nonzero solution to   2 = 0 (mod ), which is a contradiction.□ Observe that M() ∈ [[ q]][] and as now it follows that we have x ∈  [[ q]].This shows that the -part of the Eisenstein constant in this case is bounded above by  2  () where   () is the -adic valuation of .From the proof of Proposition one obtains a lower bound of    () .These bounds can be observed in Table 1.
Example 3.22.The computations above can be used to compute the -expansions of x for various groups Γ ′ () (see Table 1).

Eisenstein series
Suppose that the group  is normal in Γ ′ and that g 1 , … , g  are coset representatives for Γ ′ ∕.Then representatives for the cusps are given by g 1 ∞, … , g  ∞.The stabilizer of the cusp g  ∞ is precisely the weight 2 Eisenstein series for the cusp g  ∞.
Proposition 3.23.Let  be a character of Γ ′ ∕ and define Then unless  = 1 and  = 1, the function is a holomorphic modular form of weight 2 and character .When  = 1 and  = 1 this is simply a multiple of  2 .
For  ∈  2 (, ), the divisor of  is precisely Proof.For  ≠ 1 we use the standard results used to construct the Weil pairing.That is, there exists a function g on () that has divisor and this function transforms with character ⟨,   (g)⟩.The function  4 g is then a basis for the one-dimensional space  2 (, ).
Proof.One readily verifies that the divisor of , so that the collection of forms given are indeed all holomorphic.A dimension count then verifies that this gives the whole space.□ Proposition 3.27.The Laurent series for 1   ( x, ỹ) has bounded denominators for all primes  with (, ) = 1.
Proof.From the proof of Proposition 3.13, we verify that the Laurent series of  2   is of the form ) from which if follows that inverting   does not introduce denominators for primes not dividing .□ Proposition 3.28.The unbounded denominators for modular forms on Γ ′ () are precisely at odd primes dividing , and if 4|, also at 2. Specifically, there are modular forms with unbounded denominators at each of these primes, and these are the only primes that can contribute unbounded denominators.
Proof.That no other primes appear in denominators follows from Propositions 3.13, 3.26, and 3.27, and using that from the equation ỹ2 = x3 − 1728 the denominators for ỹ cannot be worse than those for x, except hypothetically at  = 2.However, in the case of 4 ∤  then x generates the function field as an extension of a congruence curve, hence ỹ can be expressed as a polynomial in x with coefficients modular functions with bounded denominators.That unbounded denominators exist follows immediately from 3.21, 3.20, 3.26, and 3.27.□

Dimension formulae
Proposition 3.29.Suppose Γ ′′ ⊂  ⊂ Γ ′ .Denote by   and   , respectively, the space of cusp forms of weight , and the space generated by Eisenstein series.Then the following facts hold.
(1) The dimension for the space of modular forms  2 () is precisely [ ∶ Γ ′ ], and, for  > 1, we have that  2 () has dimension precisely The dimension for the space of modular forms  2 (, ) is precisely .
The claims all follow from general principles, see for example [27].
Proof.For  > 1 we have  2 (, ) =  4  2(−1) (, ) which, using the dimension formulae, reduces the problem to finding at least one element of  2 (, ) ⧵  2 (, ) coming from lower weights.Letting  =  1 +  2 we see that is precisely such an element.□ Remark 3.32.Notice that the condition on the group of characters in Proposition 3.31 is necessary, as indeed (Γ ′ ) is not generated in weight 2.

The groups 𝑮 𝒑
There are many intermediate -congruence subgroups that are not contained in Γ ′ , and so are not accounted for in the preceding discussion.One way to study modular forms coming from such groups  is to relate forms on  to forms on  ∩ Γ ′ .For example, by noting that the associated curves are quotients of an elliptic curve by an action of the group ∕( ∩ Γ ′ ), we can already see via the Hurwitz genus formula that these curves would virtually all have genus 0. We will not pursue a detailed study of all these groups here, but instead we close this discussion by identifying a family of these genus zero -congruence subgroups that are maximal subgroups of Γ.
To see that this is not so, notice that ) .
As  3 −  is a primitive 12th root of unity, our hypothesis  ≡ 5 (mod 12) ensures that  3 −  does not reduce mod  to an element of   .We deduce that  ∉   , which shows that Γ(3) is not contained in   .□ Lemma 3.34.Let  ≡ 5 (mod 12).The congruence closure of   is the unique normal subgroup Γ (3) of Γ of index 3.
Proof.For this, let This is a normal subgroup of   (Γ) of index 3.Therefore,  −1  (  ) must be the unique normal subgroup Γ (3) of Γ of index 3, which is congruence of level 3.This shows that   ⊆ Γ (3) .As the index of   inside Γ is 3, and  is prime, we see that the congruence closure of   must be Γ (3) as claimed. □ We now briefly illustrate how to study modular forms on these intermediate groups by passing to Γ ′ and using the preceding material.Let G = Γ ′ ∩   then ( G ) is a genus 1 elliptic curve with a -isogeny to (Γ ′ ).So, ( G ) has  cusps, the kernel of the isogeny, and no elliptic points.The group G is index 2 in   .The group   ∕ G acts on ( G ) by the hyper-elliptic involution and hence (  ) is the quotient by this involution.
We conclude that (  ) has ( + 1)∕2 cusps, the origin being the only -torsion point fixed by the involution.Additionally, (  ) has 3 elliptic points of order 2, these are the images of the nontrivial 2-torsion points on ( G ).Finally, (  ) has no elliptic points of order 3.The Hurwitz genus formula allows us to see that the genus is 0.
The dimension formula for the spaces of modular forms of weight 2 on ( G ) is , whereas on (  ) it is Comparing with the dimension formula for (Γ (3) ), where recall that Γ (3) is the unique normal subgroup of Γ of index 3, we see that the dimension for the space of forms on (  ) is essentially half of that on ( G ) with rounding issues relating to forms on ( G ) with trivial character, that is, forms in the subring [ 8 ,  4 ,  6 ].As the modular forms on (  ) are those on ( G ) that are fixed by the hyperelliptic involution, we have in the notation of Proposition 3.26, and either using Proposition 3.30, or adapting to ( G ) rather than (Γ ′ ()), the forms on (  ) are of the form Finally, noting that the hyperelliptic involution acts on characters by  ↦  −1 , a natural basis for forms on (  ) comes from adding together forms from   ( G , ) and   ( G ,  −1 ) for each character  of Γ ′ ∕ G .Though we focused on the specific case of   above, the analysis is similar if the image of  in Γ∕Γ ′ has order 2 and Γ ′ () ⊂  with  odd (when  is even the group  need not include the hyperelliptic involution).When the image has elements of order 3, there is also a quotient by a degree three automorphism of the curve.In this case the analysis is simpler if 3 ∤ , in which case the fixed points will be three of the 3-torsion points on the elliptic curve, which leads to two elliptic points.
We conclude this section by illustrating in Figure 1 the Belyi map corresponding to the smallest genus zero example   for  = 5, which is of index 15 inside Γ. Descriptions on how to illustrate Belyi maps via dessins d'enfants are found in a variety of sources, such as [18].

SOME SYMPLECTIC FAMILIES
The results of [22] imply that surjective homomorphisms Γ → Sp 4 (  ) are somewhat rarer than for other classical groups.In this section, we study families of subgroups arising from such homomorphisms obtained via the moduli results of [29] on irreducible representations of Γ of rank 4. Throughout this section, we take Γ = SL 2 (), and we write  in place of  for our homomorphism, to distinguish this case from the previous one.

Moduli of representations of rank 4
In [29], equations for the moduli space of four-dimensional irreducible representations of Γ were determined.The corresponding vector-valued modular forms were studied in [14].Our goal now is to study which representations in this moduli space admit a symplectic structure.Let us recall from [29, Proposition 2.6] that the moduli space of irreducible representations of Γ has irreducible components determined by matrices , for nonzero complex numbers , , , , and where  = √ ∕ denotes a choice of square-root.Moreover, the Corollary above 2.10 of [29] shows that necessarily () 3 = 1, so that the space of irreducible representations has 6 irreducible components, corresponding to the third root of unity  and the choice of sign in .See [29, section 2.10] for polynomial conditions that the eigenvalues of () must satisfy in order to ensure that  is irreducible, as we will be interested in generic specializations of , we will not be overly concerned with these conditions.Notice that  is defined over a number field if and only if the eigenvalues of () are algebraic.
To simplify some of the commutative algebra to follow, we write  =  2 ,  =  2 ,  =  2 ,  =  2 , so that  = ±   .As we do not intend to be exhaustive below, we consider only the case  =   and omit discussion of the case  = −   .To begin, in terms of our new variables, we have , where () 6 = 1.Initially we look for an antisymmetric matrix such that ()  () =  and ()  () = .Using that  ≠ 0, this already reduces us to considering a matrix of the form:

𝐸
Again, as we do not intend to be exhaustive, we shall concentrate on the case  = 0.In this case, we can extract some polynomial equations that encode the condition that  defines a symplectic form for .These equations describe a variety with eight components, four that are one-dimensional, and four that are two-dimensional.The two-dimensional components are given by the ideals: In what follows, we consider only the case  4 , where  = 0, and we set  = 1, which we are free to do as we may rescale .Under these hypotheses, we can write things easily in terms of our original variables and we find that we are considering the following surface inside of the Tuba-Wenzl moduli space: This family of representations preserves the symplectic form

Combinatorics of some 𝐒𝐩 𝟒 families
We now have a generous supply of homomorphisms  ∶ Γ → Sp 4 ().
Reducing modulo  for all but finitely many primes  (we must avoid 3 where  has bad reducibility properties, as well as the divisors of ) we obtain homomorphisms As we will see in Theorem 4.3, by choosing  and  carefully, this homomorphism is surjective for (conjecturally † ) infinitely many primes .We thereby obtain noncongruence subgroups ker   of Γ of finite index: which grows like ( 10 ).This index is too large and poses a problem for explicit study.To circumvent this problem, we compose   with a primitive action of Sp 4 (  ) on the set of Lagrangian subspaces in  4  , which will pick out a maximal noncongruence subgroup of Γ as a point stabilizer.These symplectic point stabilizers are the analogues in our setting of the classical congruence groups Γ 0 ().
Assume  ⩾ 5 (later we will require  ⩾ 11) and let our symplectic form be given by  as above.Let  denote the corresponding symplectic Grassmanian.If  > 3 is prime, then either 3 or −3 is a quadratic residue mod .Using this, one can show that  is isomorphic mod  to the usual symplectic Grassmanian, and so which grows instead like ( 3 ).Point stabilizers under this action are known to be isomorphic with GL 2 (  ) semidirect producted with the additive group Sym 2 (  ) of symmetric 2 × 2-matrices.An explicit computation with the symplectic form defined by  gives: Let (, , ), (, ), () and  denote these Lagrangian subspaces.We wish to describe how (), () and () act on .

Action of 𝑆
In this case, one easily checks that )  =  = 0,  ≠ 0,   =  =  = 0. † Obtaining surjectivity for infinitely many primes using our approach depends on the validity of Artin's primitive root conjecture, which is currently only known in full generality as a consequence of the Riemann hypothesis for Dirichlet -functions.
one easily checks: One could use these explicit formulae to work out the number of elliptic points of order 3 as in the case of elliptic points of order 2, but it is somewhat messier than in the preceding case.Therefore, we defer this computation until Lemma 4.8, when we are prepared to give a simpler representation theoretic proof.

Action of 𝑇
Recall that the cusps of the point stabilizers correspond to the cycles in the action of  on (  ).However, unlike with  and , the order and cycle types of  depend on  and , and the explicit calculations are not as easy or insightful.However, we can still extract useful information as follows.Let  be the character of the permutation representation above, so that (g) is the number of fixed points of (g) acting on (  ).The number of cusps of width  equals the number cycles of length  in the cycle structure of (  ).In particular,  1 = (), and by induction for  > 1, we get the formula .
Equivalently, (  ) = ∑ |   .Performing Mobius inversion yields The following result is well-known, but we include the proof for completeness.Thus, to compute the number of cusps, and therefore also the genus, it remains to understand the order of  mod , as well as the character .We start this by imposing constraints on ,  and , which forces   to be surjective, as follows.Therefore, as  ⩾ 11 and  generates  ×  , it follows that det() is invertible mod .Set  1 =  −1  and observe that . This shows that the reduction of  1 mod  has order ( − 1), and hence the same holds for its conjugate  = ().Now, as −1 = ( 2 ), it will suffice to show that the image of our subgroup in PSp 4 (  ) is the entire group.Let  ⊆ PSp 4 (  ) be this image.
The maximal subgroups of PSp 4 (  ) were worked out by Mitchell, see [19,Theorem 2.8] for a description.In the notation of that theorem, we can ignore cases (a) and (b).For the remaining cases, notice that ⟨⟩ ∩ ⟨ −1 ⟩ = {±1}.Therefore, inside of PSp 4 , both  and  −1 generate subgroups of size ( − 1)∕2 with trivial intersection.It follows that  has order satisfying || ⩾  2 ( − 1) 2 ∕4.This is enough to eliminate the cases (i) and (j) just by order considerations alone.
Next, we consider cases (g) and (h), where the maximal subgroups have order ( − 1) 2 ( + 1), and, respectively, ( − 1)( + 1) 2 , and the -Sylow subgroups have order .We eliminate the possibility that  is a subgroup of one of these two maximal subgroups in two steps: first, we show that in each case there are at most ( + 1) -Sylow subgroups, and second, we find ( + 2) conjugate subgroups of order  inside , and thus reach a contradiction.
Next, we explicitly construct  + 2 conjugate subgroups of  of order .Let  = ⟨ −1 ⟩,  = ⟨ −1  −1 ⟩, and   =  (−1)  −(−1) .Then  acts on the set of {  }  =1 .By the orbit-stabilizer Hence, to be fixed, we deduce that  = 1 and  2 = −1.This contradicts the fact that  is primitive mod , and so again we see that no (, , ) is simultaneously fixed by  and .This verifies that  is not contained in any conjugate of a Siegel parabolic and eliminates case (d) from consideration.
Next we treat the case of (e), which is the stabilizer of a pair of skew polar lines in  3 (  ).Such a pair corresponds to a pair of Lagrangian subspaces with trivial intersection.If  fixes such a pair then, as it is of order 3, it fixes each Lagrangian subspace in the pair individually.As we have already seen that  and  do not simultaneously fix a Lagrangian plane, to eliminate case (e), it suffices to show that  does not exchange any pair of Lagrangian planes with trivial intersection that are each fixed by .Again, to ensure that  has fixed points, we may suppose that  ≡ 1 (mod 4).
It remains to treat case (f), which is the case of a regular spread.By [13], this corresponds to the semidirect product PSL 2 (  2 ).⟨⟩,where  is an order two automorphism.We will examine the orders of elements in this group; as  ≠ 2, we can safely work with SL 2 (  2 ).By [3, table 1.1], the elements in SL 2 (  2 ) with order divisible by  must either have order  or 2.However, the order of  =   () is ( − 1).As  is large enough,  is not contained in any subgroup of this group, and we have eliminated the last case.Therefore, the only possibility that remains is that ⟨  (),   ()⟩ = Sp 4 (  ).□ Corollary 4.4.Let  > 7 be a prime, and let  ∈  be a primitive root mod .Then the reduction   induces a surjective map Γ → Sp 4 (  ).
Proof.This is a direct application of Theorem 2.17.The symplectic form we are using behaves poorly if reduced modulo 3, however, for  > 7 the group is isomorphic with the usual symplectic group Sp 4 (  ), and the necessary hypotheses required to apply Theorem 2.17 hold by the standard theory of linear algebraic groups.□ Next, we examine the character .Let  ≠ 2, 3. To bridge the results in [28] with our explicit computations for , consider the isomorphism of symplectic spaces  ∶  →  ′ , where (, ) is our symplectic space with invariant form , ( ′ ,  ′ ) is the symplectic space in [28] with form ) , and  has matrix with respect to the standard basis.Let   be characters as defined in [28].Next, we note that the degree of the permutation representation is |(  )| =  3 +  2 +  + 1.Using the degrees of the characters   described in [28],  decomposes as either 1 +  9 +  11 or 1 +  9 +  12 .
Consider the matrix  = Then one easily check that the fixed points of  are (1∕3, 0, ), (−1∕3, 0, ) for all  ∈   , and (0, 0).Therefore,  has 2 + 1 fixed points.One can also check that via the isomorphism  defined in Equation ( 4),  is conjugate to the class  31 in [28].Examining the tables in [28], the value of (1 +  9 +  11 ) on  31 is 2 + 1, and the value of (1 +  9 +  12 ) is  + 1, so  must decompose as 1 +  9 +  11 .□ Based on this decomposition and the tables in [28], we summarize in Table 2 the values of  on certain conjugacy classes in Sp 4 () that will be useful for cusp calculations later on.Proof.In light of Lemma 4.2, we need to consider   for |( − 1).In particular, either  is a divisor of  − 1, or  =  ⋅  ′ where  ′ is a divisor of  − 1.As in the proof of Theorem 4.3, note that  is conjugate by  to  1 .We calculate that For 1 ⩽  < −1 2 , we have that   ≠  − .In these cases, the minimal polynomial (in variable ) of   is ( −   ) 2 ( −  − ) 2 , and therefore   belongs to the conjugacy class  9 () as in [28].By Table 2, in this case (  ) = 3.

□
Before we can calculate the genus of the curve corresponding to our noncongruence group, we need one last result on the number of elliptic points of order 3.In particular,  3 is independent of  and .
Note that the minimal polynomials of classes  1 (),  ′ 1 (),  3 (), and  ′ 3 () have the general form where  is either a generator for   2 in case of conjugacy class  1 , or a generator for   in case of class  3 .Clearly,  must belong to one of these conjugacy classes, and the minimal polynomial of at least one class must be  3 + 1.As  ≠ 2, 3, we have two cases.If  ≡ 1 (mod 3), the polynomial  2 ±  + 1 has a root in   , and for no value of  as a generator of   2 and  as specified in [28] is  2 ±   + 1 = 0. Therefore,  must be in some class  3 () or  ′ 3 (), for which ( 3 ()) = ( ′ 3 ()) =

𝑝
))) , and obtain the result by matching up with the four possible cases for .□ The genus of the curves discussed in Theorem 4.9 are listed in Table 3 for the first several primes.
Remark 4.10.From the genus formula in Theorem 4.9, we can see that for weights  ⩾ 2, the dimension of the corresponding spaces of modular forms grows on the order ( 3 ).
Finally, let us conclude this discussion by confirming that the groups considered above are indeed noncongruence.

Proposition 3 . 20 .
Suppose that either  = 2 and 4 | , or  = 3 and 3 | .Then the -series for x has unbounded denominators at . Proof.Proceed as in Proposition 3.21 except set q

F I G U R E 1
The dessin for  5 with index [Γ ∶  5 ] = 15.

Lemma 4 . 2 .
Let  denote the number of cusps, and let  denote the order of the mod  reduction of ().)(  ), where  denotes Euler's totient function.Proof.As  is the least common multiple of the cycle lengths of () acting on , and these cycle lengths are the widths of the cusps, we have  = ∑ |   and therefore
the same time, the image  of { (

Lemma 4 . 8 .
Let  > 3, and let  3 denote the number of fixed points of () in its action on the symplectic Grassmanian (  ).Then  3 =  + 1 + ( + 1) Remark 2.9.As   =  ⊗ , the preceding lemma illustrates that the inclusion in part (4) of Lemma 2.7 is typically a proper inclusion.Though, if a subgroup is both  1 -congruence and  2 -congruence, then Lemma 2.7 shows that it is  1 ⊗  2 -congruence.Suppose that  satisfies strong approximation.For each prime  of   , let  , denote the -adic completion of   .Then  induces canonical maps with , , and  in [, ] division polynomials, whose definitions can be found in a variety of classical sources, such as[30, section 3.2].It follows that, where (Γ ′ ) is understood in coordinates  and , we have [12]emma 3.18, we have that1 2 M() has integral coefficients, and by Hensel's lemma we find that x will have a q-series with coefficients in [ ) from Lemma 3.18 we conclude the series has infinitely many nonzero coefficients.This yields unbounded denominators, where the Eisenstein constant of[12]has a growth rate on the order of 2 −1 , if 2  || , and 3 −1∕2 if 3  || .
□Remark 3.21.We may obtain a bound on the unboundedness of the denominators  > 3 by considering the alternative rescaling Values of  on certain conjugacy classes in Sp 4 () TA B L E 2 Lemma 4.5.Let  ≠ 2, 3.If  is the character of the permutation representation of Sp 4 (  ) on (  ), then one has  = 1 +  9 +  11 .Proof.Recall that the Siegel parabolic subgroups of Sp 4 (  ) stabilize the maximal isotropic subspaces; fixing such a Siegel parabolic , we have that the permutation representation is the induced representation Ind By[15,table 2.8], given the Borel  ⊆  ⊆ Sp 4 (  ), the character of Ind Sp 4  1 decomposes as 1 + 2 9 +  11 +  12 +  13 .By Frobenius reciprocity and transitivity of induction,  decomposes as the trivial representation, and some linear combination of  9 ,  11 ,  12 and  13 .
Genus values for small primes.If  ≡ 2 (mod 3), the root of the polynomial  2 ±  + 1 must instead lie in   2 , so  must be in some class  1 () or  ′ 1 (), for which ( 1 ()) = ( ′ 1 ()) = 0.One can easily verify that these formulae agree with  + 1 + ( + 1)( −3  ) when  > 3. □ Theorem 4.9.Suppose that ,  and  satisfy the hypotheses of Theorem 4.3 and let g be the genus of the curve defined by a point stabilizer for Γ in its action on the symplectic Grassmanian (  ) for a prime  > 7. Then Proof.As the group has index ( 2 + 1)( + 1) in SL 2 (), we use the formula Putting together the formulae for  2 ,  3 and  obtained in Lemmas 4.1, 4.2, and 4.8, we get TA B L E 3