Connective Models for Topological Modular Forms of Level $n$

The goal of this article is to construct and study connective versions of topological modular forms of higher level like $\mathrm{tmf}_1(n)$. In particular, we use them to realize Hirzebruch's level-$n$ genus as a map of ring spectra.


Introduction
The basic tenet of Waldhausen's philosophy of brave new algebra is to replace known notions for commutative rings by corresponding notions for E ∞ -ring spectra.These days replacing the integers by the sphere spectrum is actually no longer so brave and new, but rather a well-established principle.In extension, we might want to find and study E ∞ -analogues of other prominent rings as well.The aim of the present paper is to do this for rings of holomorphic modular forms with respect to congruence subgroups of SL 2 (Z).
Topological analogues of modular forms for SL 2 (Z) itself were already introduced about twenty years ago.Indeed, Goerss, Hopkins and Miller introduced three spectra TMF, Tmf and tmf of topological modular forms.Recall that the rings M * (SL 2 (Z); Z) and M * (SL 2 (Z); Z) of holomorphic and meromorphic integral modular forms can be defined as the global sections H 0 (M ell ; ω ⊗ * ) and H 0 (M ell ; ω ⊗ * ) of powers of a certain line bundle ω on the compactified and uncompactified moduli stack of elliptic curves, respectively. 1In analogy, TMF is defined as the global sections of a sheaf O top of E ∞ -ring spectra on M ell with π 2k O top ∼ = ω ⊗k and Tmf as the global sections of an analogous sheaf on M ell .The edge maps of the resulting descent spectral sequences take the form of homomorphisms π 2 * TMF → M * (SL 2 (Z); Z) and π 2 * Tmf → M * (SL 2 (Z); Z).
The former morphism is an isomorphism after base change to Z[ 1 6 ] (while taking higher cohomology of ω ⊗ * into account at the primes 2 and 3) and thus TMF can be really seen as the rightful analogue of M (SL 2 (Z); Z).In contrast, π * Tmf has torsionfree summands in negative degree, whereas M * (SL 2 (Z), Z) is concentrated in non-negative degrees.The solution is to define tmf simply as the connective cover τ ≥0 Tmf and one can show that indeed π 2 * tmf[ 1 6 ] is isomorphic to M * (SL 2 (Z), Z[ 1 6 ]).We mention that one of the motivations for constructing tmf was lifting the Witten genus to a map of E ∞ -ring spectra M String → tmf as achieved in [AHR10].For applications to the stable homotopy groups of spheres and exotic spheres see e.g.[HM98], [BHHM20], [WX17] and [IWX20].
In number theory, it is very common not only to consider modular forms with respect to SL 2 (Z), but also to congruence subgroups of these; the most important being Γ = Γ 0 (n), Γ 1 (n) or Γ(n).Algebro-geometrically, such modular forms can be defined as sections of the pullback of ω ⊗ * to compactifications M(Γ) of stacks classifying generalized elliptic curves with certain level structures (see e.g.[DR73], [DI95], [Con07], [Mei22a]); for example, M(Γ 1 (n)) classifies generalized elliptic curves with a chosen point of order n whose multiples intersect every irreducible component of every geometric fiber.Hill and Lawson [HL15] defined sheaves of E ∞ -ring spectra on these stacks and obtained spectra Tmf(Γ), as their global sections, and TMF(Γ), by restriction to the loci of smooth elliptic curves.The latter spectra are good topological analogues of the rings M (Γ; Z[ 1 n ]) of meromorphic modular forms in the sense that π * TMF(Γ) is isomorphic to this ring if Γ is Γ 1 (n) or Γ(n) (with n ≥ 2) and, if we invert 6, also in the case Γ = Γ 0 (n).
In contrast, neither Tmf(Γ) nor its connective cover τ ≥0 Tmf(Γ) are in general good analogues of the ring of holomorphic modular forms M (Γ; Z[ 1 n ]), even in the nice case of Γ = Γ 1 (n) and n ≥ 2. Writing Tmf 1 (n) for Tmf(Γ 1 (n)), the reason is that H 1 (M(Γ 1 (n)); ω) and thus π 1 Tmf 1 (n) is non-trivial in general (with n = 23 being the first example), while this contribution does not occur in M (Γ; Z[ 1 n ]).Following an idea of Lawson, we define a connective version tmf 1 (n) by "artificially" removing π 1 , while still retaining the E ∞structure on tmf 1 (n).The following will be proven as Theorem 2.12 and Theorem 2.22.
Theorem 1.1.There is an essentially unique connective E ∞ -ring spectrum tmf 1 (n) with an E ∞ -ring map tmf 1 (n) → Tmf 1 (n) that identifies the homotopy groups of the source with M (Γ 1 (n); Z[ 1 n ]).Moreover, the involution of M(Γ 1 (n)) sending a point of order n on the universal elliptic curve to its negative defines on tmf 1 (n) the structure of a genuine C 2 -spectrum.Its slices in the sense of [HHR16] are trivial in odd degrees and can be explicitly identified in even degrees.
The analogous theorem also works to define tmf(n), but tmf 0 (n) we define only in certain cases since in the general case it is not yet clear what the "correct" definition is.The spectrum tmf(n) has been further investigated in [HR21,Theorem 3.14], where a criterion for the non-vanishing of its Tate spectrum is proven.
One of the principal motivations for the consideration of tmf 1 (n) is its connection to the Hirzebruch level-n genera M U * → M (Γ 1 (n); Z[ 1 n ]).They specialize for n = 2 to the classic Ochanine elliptic genus and have similar rigidity properties in general [HBJ92].We will prove the following as Theorem 3.6.
Theorem 1.2.For every n ≥ 2, there is a ring map M U → tmf 1 (n) realizing on homotopy groups the Hirzebruch level-n-genus.Moreover, this map refines to a map We have two further classes of results on the spectra tmf 1 (n) and their cousins.The first is the following compactness result, contained in Theorem 4.4 and Corollary 4.6.
Theorem 1.3.The tmf[ 1 n ]-modules tmf 0 (n), tmf 1 (n) and tmf(n) are perfect, i.e. they are compact objects in the module category, in the cases they are defined.In particular, their F p -cohomologies are finitely presented over the Steenrod algebra and thus their p-completions are fp-spectra in the sense of [MR99].
By a result of Kuhn [Kuh18, Theorem 1.7] this implies for example that the Hurewicz image of π * tmf(Γ) ∼ = π * Ω ∞ tmf(Γ) in H * (Ω ∞ tmf(Γ); F p ) is finite dimensional, where tmf(Γ) denotes either tmf 0 (n), tmf 1 (n) or tmf(n).We also note that in contrast to the theorem, tmf 1 (n) will not be a perfect tmf 0 (n)-module in general.We also show that tmf 0 (n), tmf 1 (n) and tmf(n) are faithful as tmf[ The second result is a variant of the decomposition results of [Mei22b], which we state in this introduction only at the prime 2 and for tmf 1 (n), and which will be proven as Theorem 5.6.
Theorem 1.4.Let n > 1 be odd.If one can lift every weight 1-modular form for Γ 1 (n) over F 2 to a form of the same weight and level over Z (2) , we have a C 2 -equivariant splitting where ρ denotes the real regular representation of C 2 .
In [Mei22a, Appendix C], it is shown that for 1 < n < 65 odd indeed every weight-1-modular form for Γ 1 (n) over F 2 lifts to a form of the same weight and level over Z (2) , while for n = 65 it does not.See also [Mei22a,Remark 3.14] for a further discussion of this condition.
I want to thank Tyler Lawson for explaining to me the idea how to construct a connective model for TMF 1 (n) and for the sketch of an argument that tmf 1 (3) is not perfect as a tmf 0 (3)-module.It is also a pleasure to acknowledge the influence and encouragement of Mike Hill.Furthermore I want to thank Eva Höning and Birgit Richter for their interest and remarks on a preliminary version and the anonymous referee for their extensive comments.
Finally, I want to thank the Hausdorff Institute for hospitality in 2015 when part of this work was undertaken.Apologies for the subsequent delay in publication.

Conventions and notation
All notions are to be understood suitably derived or ∞-categorical.This means that pushout means either a pushout in the respective ∞-category or a homotopy pushout in the underlying model category.We will use ⊗ for the (derived) smash product.Note that this coincides with the coproduct in the ∞-category CAlg of E ∞ -ring spectra.
When we use G-spectra, we will always mean genuine G-spectra.The notations τ ≤k and τ ≥k denote the k-(co)connective cover of a spectrum and we use the same notation for the slice-(co)connective covers of a G-spectrum.Furthermore, we denote by S the sphere (G-)spectrum.In some parts of this article, we have the opportunity to use RO(C 2 )-graded homotopy groups of C 2 -spectra.We will use the notation σ for the sign representation and ρ or C for the regular representation of C 2 .
We will use the notations TMF 1 (n) and TMF(Γ 1 (n)) interchangeably and similarly in related contexts.

The construction of connective topological modular forms
The aim of this section is to construct connective spectra tmf(Γ) of topological modular forms and thereby prove Theorem 1.1.Here Γ denotes a congruence subgroup Γ in the following sense, which is a bit more restrictive than the standard definition.
As explained in [HL15] and [Mei22b, Section 2.1], we can associate with every such Γ a (non-connective and non-periodic) E ∞ -ring spectrum Tmf(Γ).(See also [Sto12,Theorem 5.2] for the case of Γ(n).)These arise as global sections of sheaves of E ∞ -ring spectra O top on stacks M(Γ) classifying generalized elliptic curves with certain level structures; the details will not be important for the purposes of this article, but see e.g.[DR73], [Con07], [Čes17], [Mei22a].Our goal in this section is to construct a nice connective version tmf(Γ) for Tmf(Γ).For this, we will fix a localization Z S of the integers and restrict mostly to tame congruence subgroups.Definition 2.2.We say that a congruence subgroup Γ of level n is tame with respect to The definition ensures that the order of every automorphism of a point in M(Γ) is invertible and thus the stack is of cohomological dimension 1.As explained in [Mei22b, Section 2.1], in this case π * τ ≥0 Tmf(Γ) is concentrated in even degrees except for π 1 Tmf(Γ), which might be nonzero.(The smallest n for which this happens is 23.)Moreover, the even homotopy groups of Tmf(Γ) are precisely isomorphic to the ring of holomorphic modular forms M (Γ; Z[ 1 n ]).Following the lead of [Law15, Proposition 11.1] (and additional explanations by its author), we will first describe a general procedure to kill π 1 for E ∞ -rings that applies to τ ≥0 Tmf(Γ) for Γ tame.We will then present a C 2 -equivariant refinement that helps to define a nice version of tmf(Γ) also in some non-tame case (see Construction 2.24).We note that our techniques are only necessary if π 1 Tmf(Γ) is non-trivial as else the usual connective cover defines a perfectly good version of tmf(Γ).

The non-equivariant argument
Let R be a connective E ∞ -ring spectrum with π 0 R an étale extension of Z S , a localization of Z, and η • 1 = 0; here, η ∈ π 1 S is the Hopf element and 1 ∈ π 0 R the unit.(The relevant example for us is ring spectra, which is injective on π * and with cokernel π 1 R.In the following, we localize everything implicitly at the set S. In particular, Z really means Z S etc.
Let A first be a general E ∞ -ring spectrum.For an A-module M , we denote by the free unital E ∞ -A-algebra on M (cf.[Lur12, 3.1.3.14]).
Definition 2.3.Let x : Σ k A → A be an A-linear map.We define its E ∞ -cone C A (x) as the pushout A ⊗ P A (Σ k A) A of E ∞ -ring spectra.Here, the first map P A (Σ k A) → A is the free E ∞ -map on x, while the second arises from applying P A to the unique map of the first two summands and the identity id A .
Lemma 2.4.If x = 0, the canonical map C(x) → C A (x) is split as a map of A-modules.

Proof. The pushout square
arises from the pushout square via the functor Mod A → CAlg A of square-zero extension.In particular, it is a diagram of E ∞ -A-algebras.As the E ∞ -pushout square (P) defining C A (0) arises from (2.6) as well, but via P A , we see that the square (2.5) receives a map from the square (P).The resulting map C A (0) → C(0) defines a splitting of C(0) → C A (0) by the universal property of the pushout square (2.5).
We will apply our general consideration to the connective E ∞ -ring spectrum R we have fixed.As η is zero in π * R, we obtain an Lemma 2.7.The 1-coconnective cover τ ≤1 C S (η) is equivalent to HZ.
Proof.We claim that the canonical map C(η) → C S (η) is 2-connected.By the Hurewicz theorem, we can test this after tensoring with HZ and thus it suffices to show that the resulting map C(η ⊗ HZ) → C HZ (η ⊗ HZ) is 2-connected.But η ⊗ HZ agrees with the 0-map ΣHZ → HZ.Thus, we have to show that As noted above, the map is split injective and thus must be indeed an isomorphism on π i even for i ≤ 3.
By [Lur12, Theorem 7.5.0.6], we can extend the E ∞ -map This construction provides the existence part of the following proposition.
Proposition 2.9.Let R be a connective E ∞ -ring spectrum such that π 0 R is an étale extension of a localization Z S of the integers and η • 1 = 0 in π 1 R. Then there exists a morphism R ′ → R of E ∞ -ring spectra inducing an isomorphism on π i for i = 1 and satisfying π 1 R ′ = 0.Moreover, for every other R ′′ → R with these properties, there is an equivalence R ′′ → R ′ of E ∞ -ring spectra over R.
Proof.It remains to show uniqueness.We localize again everything implicitly at S. We first note that the map HZ → τ ≤1 R constructed above is actually the unique E ∞ -map with this source and target.Indeed: For connectivity reasons, we have an equivalence of mapping spaces Map CAlg (HZ, τ ≤1 R) ≃ Map CAlg (C S (η), τ ≤1 R).The latter is equivalent to the space of nullhomotopies of η in τ ≤1 R, i.e.Map Sp (Σ 2 S, τ ≤1 R) ≃ * .Using that thus τ ≤1 R has an essentially unique structure of an HZ-E ∞ -algebra, we deduce again from [Lur12, Theorem 7.5.0.6] that the space of E ∞ -maps from Hπ 0 R to τ ≤1 R is equivalent to the set of ring homomorphisms We see that R ′′ arises as a pullback of a diagram of the same shape as (2.8), but possibly with a map Hπ 0 R → τ ≤1 R inducing a different isomorphism f on π 0 than the identity.The paragraph above implies that using the map f on Hπ 0 R we obtain an equivalence between the cospans constructing R ′ and R ′′ and thus between R ′ and R ′′ over R.
To apply this to topological modular forms, we need the following two lemmas.
Lemma 2.10.Let Γ be a tame congruence subgroup with respect to a localization Z S .Then η is zero in π 1 Tmf(Γ) S .
Proof.According to [Mei22b, Proposition 2.5], the descent spectral sequence For the reminder of the proof, assume that we are in this case and set G = Γ/Γ 1 (n).
We will argue that the map is isomorphic to the inclusion of G-invariants.As η vanishes in the target, this will imply the vanishing of η in the source.The map is concentrated in the zero-line since the order of G is invertible in Z (2) by the tameness of Γ.Thus, is indeed the inclusion of G-invariants.
Lemma 2.11.Let Γ be a tame congruence subgroup with respect to a localization Z S .Then Proof.As recalled above, we have This allows us to use Proposition 2.9 to define tmf(Γ) S in the tame case by killing π 1 from τ ≥0 Tmf(Γ) S .Summarizing we obtain: Theorem 2.12.For every set of primes S and every congruence subgroup Γ that is tame with respect to Z S , there is up to equivalence a unique connective E ∞ -ring spectrum tmf(Γ) S with an E ∞ -ring map tmf(Γ) S → Tmf(Γ) S that identifies the homotopy groups of the source with the ring of holomorphic modular forms M (Γ; Z S ).
Formally, we could also apply this procedure in some non-tame cases (e.g. if we localize away from 2), but the author knows of no reason to regard these constructions in these cases as "correct".Notation 2.13.We will use the abbreviations when these make sense.
Remark 2.14.For every ring spectrum R, we can consider the stack X R associated to the graded Hopf algebroid In [MO20, Definition 5.5] we introduced cubic versions M 1 (n) cub and M 0 (n) cub of the moduli stacks M(Γ 1 (n)) and M(Γ 0 (n)).These come with a finite morphism to the moduli stack M cub of cubic curves, where we allow arbitrary Weierstraß equations.We showed in [MO20,Theorem 5.19 In combination, we see that In the case n = 1, the corresponding equivalence X tmf ≃ M cub has a quite different character and was shown in [Mat16].Whether there are equivalences X tmf 0 (n) ≃ M 0 (n) cub for a suitable definition of tmf 0 (n) remains open to the knowledge of the author, even for n = 3.

The C 2 -equivariant argument
All the stacks M(Γ) come with an involution induced from postcomposing the level structure with the [−1]-automorphism of the elliptic curve.As explained in Remark 2.15, this induces a C 2 -action on Tmf(Γ).Our goal in this subsection is to define suitable C 2 -spectra tmf(Γ) in the tame case.This will allow us to construct an E ∞ -ring spectrum tmf(Γ) also if there is just a tame subgroup Γ ′ ⊂ Γ of index 2 (see Construction 2.24).
Remark 2.15.The goal of this remark is to clarify the construction of the C 2 -action on Tmf(Γ) sketched above.
Denote the automorphism of M(Γ) described above by t.As t commutes with the forgetful map pr : M(Γ) → M ell , this defines a C 2 -action inside the slice category (Stacks/M ell ) ét,op of stacks étale over M ell .We will use a lax commutative triangle As explained in [Mei22b, Example 6.12], the C 2 -action induced by t on TMF(Γ) is equivalent to the one induced by the C 2 -action in (Stacks/M ell ) ét,op given by id M(Γ) on M(Γ), but choosing the [−1]-isomorphism between the elliptic curves classified by pr and pr id M(Γ) .This C 2 -action induces multiplication by −1 on the pullback of ω to M(Γ): Indeed, the [−1]-automorphism of an elliptic curve induces multiplication by −1 on the sheaf of differentials.Moreover, pr classifies precisely the pullback of the universal elliptic curve E uni and ω is the restriction of Ω 1 E uni /M ell to M ell along the zero section.Thus, if Γ is tame, this implies that C 2 acts by (−1) k on π 2k TMF(Γ) ∼ = H 0 (M(Γ); ω ⊗k ).Since π 2k Tmf(Γ) injects in the tame case into π 2k TMF(Γ), the same is true for π 2k Tmf(Γ).
Note that the action t can be trivial, e.g. for Γ = Γ 0 (n) or Γ(2).This forces π 2k Tmf(Γ) = 0 for k odd in these cases (as t acts both by 1 and −1 and the groups are torsionfree).This corresponds to the classical fact that there are no modular forms of odd weight if − id is in Γ.
In the following we will use standard notation from equivariant homotopy theory.In particular, we denote for an inner product space V with G-action by S(V ) the unit sphere and by S V the 1-point compactification as G-spaces.We denote by a = a σ : S 0 → S σ the inclusion for σ the real sign representation of C 2 .
The Hopf map defines a C 2 -map η : S(C 2 ) → S C , where C 2 acts on C via complex conjugation.This stabilizes to an element in π C 2 σ S, which restricts to η ∈ π e 1 S.
Lemma 2.16.The homotopy groups π C 2 σ (S) and π C 2 −σ S are infinite cyclic and generated by η and a, respectively.
Proof.For π C 2 σ (S), this is proven as formula (8.1) in [AI82].(Note that they use the notation π s p,q for our π C 2 pσ+q (S).)Proposition 7.0 in op.cit.implies that the homomorphism π C 2 −σ S → π 0 S, taking a map S → Σ σ S to its geometric fixed points, is an isomorphism.Taking fixed points of the map a clearly gives the identity map S 0 → S 0 , which yields the result.
In the following, we denote by τ ≤i the slice coconnective cover, by τ ≥i the slice connective cover and by τ i = τ ≥i τ ≤i the i-th slice for C 2 -spectra.We refer to [HHR16] for background about the slice filtration.We denote by HZ the C 2 -Eilenberg-MacLane spectrum for the constant Mackey functor Z. Lemma 2.17.We have an equivalence τ ≤1 Cη ≃ HZ.
Proof.It suffices to show that the first slice of Cη is null and the zeroth slice is HZ.As shown in [HHR16] and summarized in [HM17, Section 2.4], this is implied by the calculations π 0 Cη ∼ = Z and π σ Cη = 0.These follows easily by the long exact sequence arising from the cofiber sequence Spelled out, the latter condition is equivalent to As τ ≥k+1 X and its suspension are k-slice connected, the direction discused above shows H W (τ ≥k+1 X; Z) = H W (Στ ≥k+1 X; Z) = 0. Thus, is an isomorphism.As summarized in [HM17, Section 2.4], the slice τ k X is of the form S W ⊗HM for some Mackey functor M and we deduce that H 0 (HM ; We know that τ 0 S = HZ.As HM is (slice) connective, a similar argument to before shows that M ∼ = π 0 (S ⊗ HM ) ∼ = π 0 (HZ ⊗ HM ) = H 0 (HM ; Z) = 0.
Thus, τ k X = 0 as was to be shown.
as in the non-equivariant situation in the preceding subsection.The arguments for the following two results are quite analogous to those of the preceding section, so we allow ourselves to be brief.Proof.By Lemma 2.18 it suffices to check that C(η) ⊗ HZ → C S (η) ⊗ HZ is slice-2connected.Since π σ HZ = 0 and thus η becomes zero in HZ, this agrees with Analogously to Lemma 2.4, the map is split injective and thus indeed slice-2-connected (even slice-3-connected).
Together with Lemma 2.17 this implies that τ ≤1 C S (η) ≃ HZ.To deduce the analogue of Proposition 2.9, we will need one more categorical result.
Lemma 2.20.Let G be a finite group and Sp G be the ∞-category of G-spectra.Denote by Sp ≥0 G the full subcategory of connective G-spectra and by Sp inherits the structure of a symmetric monoidal ∞-category from Sp ≥0 G and τ ≤k is strong symmetric monoidal, while the inclusion Sp G is lax symmetric monoidal.The same proposition gives that the resulting maps Sp σ R. Then there is an E ∞ -ring C 2 -spectrum R ′ with an E ∞ -map R ′ → R inducing an equivalence on slices in degree 0 and degrees at least 2 and such that τ 1 R ′ = 0.Moreover, for every other R ′′ → R with these properties, there is an equivalence The proof of uniqueness is analogous to Proposition 2.9.
To formulate the consequences for tmf(Γ), we want to recall from [HM17] that a C 2spectrum E is strongly even if its odd slices vanish and its even slices are of the form S kρ ⊗ HA or, equivalently, if π kρ E is constant and π kρ−1 E = 0. Theorem 2.22.For every set of primes S and every congruence subgroup Γ 1 (n) ⊂ Γ ⊂ Γ 0 (n) that is tame with respect to Z S , we can define a strongly even connective E ∞ -ring C 2 -spectrum tmf(Γ) S with an E ∞ -ring C 2 -map tmf(Γ) S → Tmf(Γ) S that identifies the underlying homotopy groups of the source with M (Γ; Z S ).
Given these claims, applying Proposition 2.21 to R = τ ≥0 Tmf(Γ) yields the C 2 -spectrum R ′ = tmf(Γ) with the required properties: the first claim implies that we can apply Proposition 2.21, while the other two ensure that tmf(Γ) is strongly even.
For proving the claims, we will distinguish the (overlapping) cases that 1 2 ∈ Z S and that Γ is tame for Z (2) .
For the first claim, note that π this follows from the homotopy fixed points spectral sequence; else, use the line after (6.15) in [Mei22b].Since η restricts to η ∈ π 1 Tmf(Γ), Lemma 2.10 implies thus the vanishing of η.
Remark 2.23.The case that Γ = Γ 0 (n) is not excluded in the previous theorem, but one easily checks that Γ 0 (n) can only be tame if 1 2 ∈ Z S .In this case, we obtain simply the cofree C 2 -spectrum of tmf 0 (n) S with the trivial action.Construction 2.24.Given Γ ′ ⊂ Γ ⊂ Γ 0 (n) with Γ ′ tame with respect to Z S and Γ/Γ ′ ∼ = C 2 , we can extend our previous definition by defining tmf(Γ) S as tmf(Γ ′ ) C 2 S (so e.g.tmf 0 (3) = tmf 1 (3) C 2 as in [HM17]).If Γ itself is already tame, then 1 2 ∈ Z S .One then easily computes (e.g. with the slice spectral sequence) that π * tmf(Γ ′ ) C 2 S ∼ = π * tmf(Γ) S and one can use the uniqueness part of Theorem 2.12 to identify our new definition with the previous one.

Realization of Hirzebruch's level-n genus
In the previous section we have defined ring spectra tmf 1 (n) = tmf(Γ 1 (n)).The spectra tmf 1 (n) are even for n ≥ 2 and thus complex orientable.We want to show that there is a complex orientation for tmf 1 (n) such that the corresponding map agrees with the level-n genus introduced by Hirzebruch [Hir88] and Witten [Wit88] and studied e.g. in [Kri90], [Fra92], [Her07] and [WWY21].We we recall its definition below.For this purpose it will be convenient to use algebro-geometric language, for which we recall first the following set of definitions.
Definition 3.1.A formal group over a base scheme S is a Zariski sheaf F : Sch op S → Ab that Zariski locally on an affine open U = Spec R ⊂ S is isomorphic to Spf R t .The R-modules R t glue to the structure sheaf O F on S and the R-modules (R t /t) • dt glue to the line bundle ω F/S .4An invariant differential of a formal group F is a trivialization of Remark 3.2.There are different ways to state the definition of a formal group, e.g. as an abelian group object in one-dimensional formal Lie varieties (see [Goe08, Definitions 1.29 and 2.2]).To compare them, note that our formal groups are automatically fpqc sheaves since Spf R t is an fpqc sheaf.On the other hand, a trivialization of the sheaf of differentials of a one-dimensional formal Lie variety over Spec R determines an equivalence to Spf R t and such trivializations exist Zariski locally.
We note that the differential ds of a coordinate s of a formal group F is an invariant differential of F , sending a 0 t + a 1 t 2 + • • • to a 0 dt locally.If S = Spec R, a coordinate of F is equivalent datum to an isomorphism F ∼ = Spf R s .
Recall that given an arbitrary even ring spectrum E, a complex orientation is an element in E 2 (CP ∞ ) restricting to 1 ∈ E 2 (CP 1 ) after a homeomorphism CP 1 ∼ = S 2 is chosen.The formal spectrum Spf E 2 * (CP ∞ ) is a formal group over Spec E 2 * (pt) and the line bundle ω corresponds to E * (CP 1 ); it thus comes with a canonical invariant differential corresponding to 1 ∈ E 2 (CP 1 ).A complex orientation is thus a coordinate of Spf E 2 * (CP ∞ ) in degree * = 1 whose differential is the canonical invariant differential.
We want to apply this to E = tmf 1 (n) for n ≥ 2. Essentially by construction, the maps n) , which is the total space of the G m -torsor associated with ω C/M 1 (n) , i.e. classifies generalized elliptic curves with a point of exact order n and an invariant differential.The resulting morphism is an open immersion whose image is covered by the non-vanishing loci of c 4 and ∆ [MO20, Proposition 3.5].We denote by C the pullback of ] are elliptic cohomology theories, their formal groups are identified with the restrictions of C to the non-vanishing loci of c 4 and ∆, respectively, and as a result C becomes identified with the restriction of Spf tmf an isomorphism on global sections of the structure sheaf, coordinates on Spf tmf 1 (n) 2 * (CP ∞ ) are in bijection with those on C and one checks that the canonical invariant differential on the former corresponds to the canonical invariant differential on the latter.Summarizing we obtain: Lemma 3.3.Complex orientations M U → tmf 1 (n) are in bijection with coordinates of C, which are homogeneous of degree 1 and have the canonical invariant differential as differential.
The Hirzebruch genus relies on a specific such coordinate, which we will construct momentarily.Basically we will follow [HBJ92, Chapter 7], but present a more algebro-geometric approach and give an independent treatment.The key point is the existence of a certain meromorphic function on a cover of a given generalized elliptic curve.Recall to the purpose of constructing this function that every section P into the smooth part of a generalized elliptic curve C → S is an effective Cartier divisor [KM85, Lemma 1.  is an isomorphism.Thus ), where the isomorphism can be identified with the pullback along e.Thus, there is a unique section h of O C (n(P ) − n(e)) whose image is λ n .
For part (c), consider the µ n -torsor q : C ′ → C associated with the problem of extracting an n-th root out of q * h as a section of q * O C ((P ) − (e)) (i.e. the µ n -torsor associated with the pair (h, O C ((P ) − (e))) in the sense of [Mil80,p. 125]).By construction, the required root f exists on C ′ .By [DR73, Proposition II.1.17],C ′ has the structure of a generalized elliptic curve provided that we can lift e to C ′ and C ′ → S has geometrically connected fibers.For the first point, it suffices to provide a section of C ′ × C S → S, i.e. to provide an n-th root of e * h.Under the identification of part (a), this is provided by λ.For the second point, we assume that S = Spec K with K algebraically closed of characteristic not dividing n and that C ′ is not connected.The stabilizer of a component C ′ 0 must be of the form µ m with m < n and thus C ′ ∼ = C ′ 0 × µm µ n .The µ m -torsor C ′ 0 is hence associated with a pair (g, O C ((P ) − (e))) with g n/m = h.The section g provides a trivialization of O C (m(P ) − m(e)).This implies m • P = e on C ′ [DR73, Corollaire II.2.4], in contradiction with P being of exact order n.Construction 3.5.Let C be the universal generalized elliptic curve with a point of exact order n over M 1 1 (n).It comes by definition with a canonical invariant differential λ.From the preceding lemma, we obtain an n-fold étale cover q : C ′ → C together with a meromorphic function f on C ′ whose pullback along a lift of e agrees with λ.This function f provides a coordinate for C ′ ∼ = C. Moreover note that f is uniquely determined by the requirements in the lemma because C ′ is irreducible (since M 1 1 (n) is irreducible and the locus of smoothness of C ′ in it is dense) and thus every other n-th root of h would have to differ by a root of unity, resulting in a different pullback to M 1 1 (n).Pulling the orientation induced from f back along a map Spec C → M 1 (n) classifying (C/Λ, 1 n , dz) results exactly in the coordinate and orientation chosen in [HBJ92].
Theorem 3.6.For every n ≥ 2, there is a unique complex orientation of M U → tmf 1 (n) realizing on homotopy groups the Hirzebruch genus.Moreover, this can be uniquely refined to a morphism Proof.The first part follows from Lemma 3.3 as the Hirzebruch genus is given by a coordinate on the formal group associated with the universal generalized elliptic curve on For the second point, we recall from [HK01, Theorem 2.25] that C 2 -ring morphisms M U R → tmf 1 (n) are in bijection with Real orientations of tmf 1 (n), i.e. a lift of a complex orientation to a class tmf 1 (n) ρ C 2 (CP ∞ ).As CP ∞ can be built by cells in dimensions kρ, the strong-evenness of tmf 1 (n) from Theorem 2.22 implies that the forgetful map is an isomorphism; thus every complex orientation of tmf 1 (n) refines to a unique Real orientation.
Remark 3.7.We remark that in [Fra92], Franke already gave a related but different algebrogeometric treatment of the Hirzebruch genus.
Remark 3.8.After the first version of this article became available, Senger has shown in [Sen22] that the map M U → tmf 1 (n) actually refines to one of E ∞ -ring spectra.He also gives a reformulation of our treatment above in terms of Θ 1 -structures.
4 Compactness, formality and faithfulness of tmf(Γ) Given a (tame) congruence subgroup of level n, we will show that tmf(Γ) is a faithful and perfect tmf[ 1 n ]-module.In contrast, for example tmf 1 (3) will not be a perfect tmf 0 (3)module, even rationally.The latter result relies on tmf 0 (3) Q being formal (i.e.multiplicatively a graded Eilenberg-MacLane spectrum), a result we prove in greater generality in a subsection on its own.

All tmf(Γ) are perfect
Recall that for an A ∞ -ring spectrum R, a perfect R-module is a compact object in the ∞-category of left R-modules.Equivalently, the ∞-category of perfect R-modules is the smallest stable sub-∞-category of all left R-modules that contains R and is closed under retracts.The goal of this section is to show that the spectra tmf(Γ), in the cases we defined them, are perfect tmf[ 1 n ]-modules.The key technical tool is the following proposition.
Let furthermore M be a perfect R-module.Then τ ≥k M is a perfect τ ≥0 R-module for every k ∈ Z.
Lemma 4.2.With notation as in the statement of the proposition, let X be a τ ≥0 R-module with only finitely many non-trivial homotopy groups, all finitely generated over π 0 R. Then X is a perfect τ ≥0 R-module.
Proof.By induction, we can reduce to the case that π * X is concentrated in a single degree n.Then X = Hπ n X acquires the structure of a Hπ 0 R-module and it is perfect as such because π 0 R is regular noetherian and π n X is finitely generated.As Hπ 0 R is perfect over τ ≥0 R, the same is thus true for X.
Proof of proposition.Let M be a perfect R-module.As the truth of the conclusion of the proposition is clearly preserved under retracts in M and also clear for M = 0, we can assume by induction that we have a cofiber sequence where τ ≥k N is a perfect τ ≥0 R-module for all k ∈ Z. Taking τ ≥l on the first two objects gives a diagram As the fiber of τ ≥l+1 M ′ → M ′ fulfills the conditions of the previous lemma, τ ≥l+1 M is perfect as a τ ≥0 R-module.
For a general k ∈ Z, we make a case distinction: Assume first that k ≥ l + 1.Then the fiber of τ ≥k M → τ ≥l+1 M is perfect by the previous lemma, hence τ ≥k M is perfect as well.If k ≤ l + 1, consider the fiber of τ ≥l+1 M → τ ≥k M instead.
To apply Proposition 4.1 to topological modular forms, we need the following lemma.
Lemma 4.3.For every n ≥ 1, the tmf[ Killing a 1 and a 3 gives HZ[ 1 n ] and thus HZ[ 1 n ] is also a perfect tmf[ 1 n ]-module in this case.For the general case, let X i be a collection of tmf[ 1 n ]-modules.Consider If k = 2, 3 or 6, then Φ k is an equivalence by the previous results.As for every spectrum X, there is a cofiber sequence there is a cofiber sequence of maps between mapping spectra It follows that Φ 1 is an equivalence as well and that HZ[ 1 n ] is a perfect tmf[ ].If Γ is tame, the cofiber of tmf(Γ) → τ ≥0 Tmf(Γ) is by construction Hπ 1 Tmf(Γ) and π 1 Tmf(Γ) ∼ = H 1 (M(Γ); ω).If there is a tame subgroup Γ ′ ⊂ Γ of index 2, the cofiber tmf(Γ) → τ ≥0 Tmf(Γ) agrees with Σ σ HM for M the constant Mackey functor on H 1 (M(Γ ′ ); ω) by Remark 2.25.The exact sequence given in the same remark implies that the homotopy groups of Σ σ HM are concentrated in degrees 0 and 1 and are finitely generated Z[ 1 n ]modules.
We recall from [MR99] that a connective p-complete spectrum X is called an fp-spectrum if H * (X; F p ) is finitely presented as a comodule over the dual Steenrod algebra.They show in [MR99, Proposition 3.2] that equivalently there is a finite spectrum F with non-trivial F p -homology such that the total group π * (X ⊗ F ) is finite.The following proposition can be deduced from the known F p -(co)homology of tmf (see e.g.[Rez02, Section 21]) and was already noted in [MR99] for p = 2.We prefer to give a less computational proof though.
Proposition 4.5.The p-completion of tmf is an fp-spectrum for all primes p.
Proof.We implicitly p-localize.For p = 3, [Mat16, Theorem 4.10] implies the existence of a finite spectrum W with non-trivial F p -homology such that tmf ⊗W ≃ tmf 1 (3).Choose a complex V such that ) with k 0 , k 1 and k 2 positive integers.As TMF 1 (3) is Landweber exact, the sequence p, v 1 , v 2 and hence the sequence ] is an integral domain, the sequence is also regular on π * tmf 1 (3).Thus, is a finitely generated Z/p k 0 -algebra and of Krull dimension 0. Hence it is of finite length as a Z/p k 0 -module and thus finite.Essentially the same argument works for p = 3 if we choose instead a complex W ′ with tmf ⊗W ′ ≃ tmf 1 (2) as in [Mat16,Theorem 4.13].
Corollary 4.6.The p-completion of tmf(Γ) for a congruence subgroup Γ of level n and p not dividing n is an fp-spectrum.
For implications involving duality we refer to [MR99] and for an implication for the Hurewicz image in H * (Ω ∞ tmf(Γ); F p ) to [Kuh18, Theorem 1.7].

All tmf(Γ) Q are formal
The goal of this section is to show that the E ∞ -rings tmf(Γ) Q are formal.While this statement is interesting in its own right, we also need it for further pursuing compactness questions in the following subsection.We begin with the following consequence of Goerss-Hopkins obstruction theory.
Proposition 4.7.Let A and B be E ∞ -HQ-algebras such that π * A is smooth as a Q-algebra.Then Proof.According to [GH04, Section 4] or [PV22, Section 6] with E = HQ, there is an obstruction theory for lifting a morphism π * A → π * B to a morphism A → B, where the obstructions lie in Ext n+1,n π * A (L E∞ π * A/Q , π * B), where L E∞ denotes the E ∞ -cotangent complex.As we are working rationally, this coincides with other forms of the cotangent complexes.In particular, we obtain from the smoothness of π Our assumption implies (M ⊗ tmf Tmf) ⊗ Tmf Tmf(Γ) = 0, hence by the faithfulness of Tmf(Γ) also M ⊗ tmf Tmf = 0. Thus, M ′ ⊗ tmf 1 (3) Tmf 1 (3) = 0.Moreover, tmf(Γ) ⊗ tmf HZ is a faithful HZ-module as its π 0 is a faithful Z-module.Thus ] and M ′ /(a 1 , a 3 ) all vanish, which implies the vanishing of M ′ .
The argument for p = 3 is similar with tmf 1 (2) in place of tmf 1 (3) and for p > 3 we can use tmf itself as π

Splittings
Our goal in this setting is to show that tmf 1 (n) often splits p-locally into small pieces.
Fixing a natural number n ≥ 2 and a prime p not dividing n, we will work throughout this section implicitly p-locally.We demand that M In general, this is a subtle condition, but it is for example always fulfilled if n ≤ 28 (see [Mei22a,Remark 3.14]).Equivalently, we can ask that H 1 (M 1 (n); ω) ∼ = π 1 Tmf 1 (n) does not have p-torsion.We note that this leaves plenty of cases where π 1 Tmf 1 (n) = 0 and hence tmf 1 (n) is not the naive connective cover of Tmf 1 (n), of which the smallest is n = 23.
By Theorem 1.3 of [Mei22b], we have a splitting of Tmf-modules, where R is Tmf 1 (3), Tmf 1 (2) or Tmf, depending on whether the prime p is 2, 3 or bigger than 3.In this splitting all n i are nonnegative.
Theorem 5.2.Under the conditions as above, we have a splitting where r = τ ≥0 R.
Proof.Consider the composition Here, the second map is just the connective cover of (5.1) (using that τ ≥0 commutes with direct sums) and the first map is the direct sum of the maps Σ 2n i r ≃ τ ≥2n i Σ 2n i R → τ ≥0 Σ 2n i R. Since all negative homotopy of R is in odd degrees, we see that f is an isomorphism on even homotopy groups.Moreover, the source has only homotopy groups in even degrees.
Recall that we defined tmf 1 (n) as a pullback where we still localize implicitly everywhere at p.This implies a fiber sequence To factor f over tmf 1 (n), it is enough to show that H 1 (Σ 2n i r; A) = 0 with any coefficients A. This is clear anyhow for n i ≥ 1, so assume n i = 0. We know that τ [0,1] r ≃ HZ and we have H 1 (HZ; A) ∼ = H 1 (S; A) = 0 (as the the cofiber of S → HZ is 1-connected).Now π * tmf 1 (n) is concentrated in even degrees and tmf 1 (n) → τ ≥0 Tmf 1 (n) induces a π * -isomorphism in even degrees.In total, we see that f induces an isomorphism on π * .
Remark 5.3.The condition that π 1 Tmf 1 (n) ∼ = H 1 (M 1 (n); ω) does not have p-torsion is actually necessary in the preceding theorem.One can indeed show that Tmf 1 (n) can be recovered as tmf 1 (n) ⊗ tmf Tmf.Thus a p-local tmf-linear splitting of tmf 1 (n) into shifted copies of r implies a p-local splitting of Tmf 1 (n) into copies of R. As the latter has torsionfree homotopy groups, such a splitting can indeed only occur if the homotopy groups of Tmf 1 (n) are p-torsionfree as well.
We want to show that we can change Φ so that this is true.Using Φ, the C 2 -spectrum Tmf 1 (n) gets the structure of a Tmf 1 (3)-module.Thus, Tmf 1 (3)-module maps N i=0 Σ n i ρ Tmf 1 (3) → Tmf 1 (n) correspond to a sequence of classes x i ∈ π C 2 n i ρ Tmf 1 (n) by considering the images of 1 ∈ π C 2 n i ρ Σ n i ρ Tmf 1 (3).Denote the sequence corresponding to Φ by e 0 , . . ., e N .By possibly reordering, we can assume n 0 = 0. We construct a new map Φ ′ : N i=0 Σ n i ρ Tmf 1 (3) → Tmf 1 (n) corresponding to x 0 , x 1 , . . ., x N with x i = e i for i > 0 and x 0 corresponding to the image of u ∈ Z in (5.7),where u maps to res C 2 e (e 0 ) along the isomorphism Z ∼ = π e 0 tmf 1 (n) → π e 0 Tmf 1 (n).As Φ ′ and Φ induce the same map on underlying homotopy groups, the map Φ ′ is an equivalence.By construction, r(x 0 ) = 0. Thus the map i factors indeed over tmf 1 (n).As before, the map Σ n i ρ tmf 1 (3) → tmf 1 (n) induces an isomorphism on underlying homotopy groups.Both source and target are strongly even and thus the map is a C 2 -equivariant equivalence by [HM17, Lemma 3.4].
diagonal arrows are the Goerss-Hopkins-Miller and Hill-Lawson sheaves of ring spectra.The horizontal arrow N is a normalization construction (see e.g.[HL16, Proposition 2.27]).The canonical map O top (N (U )) → O top (U ) for U → M ell étale comes from the fact that U ⊂ N (U ) is an open substack and the Hill-Lawson sheaf restricts to the Goerss-Hopkins-Miller sheaf.Applying the left diagonal arrow to (M(Γ), t) gives a C 2 -action on TMF(Γ).Doing the same with the composite of the right diagonal arrow and the horizontal arrow produces the C 2 -action on Tmf(Γ).Moreover, we obtain a C 2 -map Tmf(Γ) → TMF(Γ).
and slice-ktruncated G-spectra.Then the inclusion CAlg(Sp [0,k] G ) → CAlg(Sp G ) admits for every k ≥ 0 a left adjoint, which agrees on the level of underlying G-spectra with the slice truncation τ ≤k .Proof.Connective G-spectra form a presentable ∞-category with compact generators the Σ ∞ G/H + .We obtain Sp [0,k] G by localizing Sp ≥0 G at the collection S of maps C → 0 for C a slice cell of dimension greater than k.By [Lur09, Proposition 5.5.4.15], operads are adjoint.Since commutative algebras in such an ∞-operad C ⊗ are defined as sections of C ⊗ → NFin * as maps of operads, we see that the resulting maps between CAlg(Sp [0,k] G ) and CAlg(Sp ≥0 G ) are indeed adjoint.Here, we use the characterization of an adjunction given by [RV16], namely the existence of a unit and counit, satisfying the triangle identities up to homotopy.Proposition 2.21.Let R be a connective E ∞ -ring C 2 -spectrum with π C 2 0 = Z S being a localization of Z and η = 0 ∈ π C 2 2.2], i.e. the kernel O C (−(P )) of O C → P * O S is a line bundle.Given any linear combination of sections P i , we denote by O C ( i n i (P i )) the corresponding tensor product of line bundles.Lemma 3.4.Let n ≥ 2 and S be a Z[ 1 n ]-scheme.Furthermore let C/S be a generalized elliptic curve with 0-section e : S → C and a chosen point P : S → C of exact order n in the smooth locus.(a) The pullback of e * O C ((P ) − (e)) to S is canonically isomorphic to ω C/S = e * Ω 1 C/S .(b) Let λ be an invariant differential on C. Then there exists a unique meromorphic function h on C with an n-fold zero at e and an n-fold pole at P as only pole whose restriction along e coincides with λ n under the identification of the previous part.(c) There exists a degree-n étale cover q : C ′ → C by a generalized elliptic curve and a meromorphic function f on C ′ with f n = q * h.Proof.For the proof of (a), note that O C (−(e)) is the ideal sheaf associated to the closed immersion e and the pullback e * O C ((P ) − (e)) coincides with O C (−(e))/O C (−(e)) 2 viewed as an O S -module.Indeed: We can cover S by opens of the form U ∩ S, where U ∼ = Spec R is an affine open in C not intersecting the image of P .The section e corresponds to an element s ∈ R and U ∩ S ∼ = Spec S/s.Then e * O C ((P ) − (e))(U ∩ S) is the S/s-module sS ⊗ S S/s, which is canonically isomorphic to the S/s-module sS/s 2 S.
computation of this group is classical and can be found e.g. in [Mei22a, Proposition 2.13].The case of Γ 1 1 n ]-module.Theorem 4.4.Let Γ be a congruence subgroup of level n, which is tame or has a subgroup Γ ′ ⊂ Γ of index 2 with Γ ′ tame.Then tmf(Γ) is a perfect tmf[ 1 n ]-module.The same conclusion holds without the tameness hypothesis for any tmf[ 1 n ]-module R with a map R → τ ≥0 Tmf(Γ) whose fiber has finitely generated homotopy groups over Z[ 1 as in the statement of the theorem, R is thus perfect as well by Lemma 4.2.To see that tmf(Γ) satisfies the hypotheses on R, note first that every H s (M(Γ); ω ⊗t ) is a finitely generated Z[ 1 n ]-module for every s and t since M(Γ) is proper over Z[ 1 n [Mat16,], which again by smoothness is a projective π We observe using Proposition 4.7 that π 0 π 0 M • agrees with the set of ring morphisms π * O → π * O ′ , in which we can pick an isomorphism f 0 .According to [Bou89, Sections 5.2, 2.4], the vanishing ofπ i+1 π i M • ∼ = H i+1 (X , π i O) for i ≥ 1 suffices to lift f 0 to a multiplicative map O → O ′ ,which is automatically an equivalence.Remark 4.10.In the original account of the construction of O top on M ell in[DFHH14], O top Let M ∈ Mod tmf S with M ⊗ tmf S tmf(Γ) S = 0.It suffices to show that M (p) = 0 for all p not in S. Consider the case p = 2 and localize everything implicitly at 2. As tmf 1 (3) is faithful over tmf (see[Mat16, Theorem 4.10]), it suffices to show that M * A-module.Thus the Ext-groups vanish and there is no obstruction to lifting a morphism π * A → π * B to a morphism A → B. The same sources provide a spectral sequence computing π * Map CAlg (A, B), which collapses by a similar Ext-calculation and gives the result.Proposition 4.8.Let X be a smooth Deligne-Mumford stacks over Q and O an evenperiodic sheaf of E ∞ -ring spectra on X such that π 0 O ∼ = O X and the π i O X are quasicoherent.Assume further that H i+1 (X ; π i O) = 0 for all even i ≥ 1.Then O is formal, i.e. equivalent to the (sheafification of the pre)sheaf Hπ * O of graded Eilenberg-MacLane spectra.Proof.Note first that (X , O) actually defines a non-connective spectral Deligne-Mumford stack and in particular O is hypercomplete (cf.e.g.[Mei22b, Lemma B.2]).Set O ′ = Hπ * O. Choosing an étale hypercover U • → X by affines, we can compute Map CAlg X (O, O ′ ) as the totalization of the cosimplicial diagram M • = Map CAlg (O(U • ), O ′ (U • )).Q is actually formal by construction.Our argument shows that this choice was necessary, not only for M ell , but also for M(Γ).(The former was shown in a different manner already in [HL15, Proposition 4.47].)Proposition 4.11.Let Γ be a congruence group.Then the E ∞ -rings tmf(Γ) Q are formal.Proof.