Rings of modular forms and a splitting of $${{\,\mathrm{TMF}\,}}_0(7)$$

Abstract

Among topological modular forms with level structure, TMF 0 (7) at the prime 3 is the first example that had not been understood yet. We provide a splitting of TMF 0 (7) at the prime 3 as TMF-module into two shifted copies of TMF and two shifted copies of TMF 1 (2). This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level n and introduce cubical versions of moduli stacks of elliptic curves with level structure.While our computation of mf(Γ 1 (7); Z) is not difficult to obtain, the reader should compare the simplicity of its expression with the presentation Rustom [50, Section 3.1] is forced to give by ignoring the elements of weight 1.After these explicit computations, we also prove more structural results about rings of modular forms. Theorem 1.3. For every n ≥ 2, the ring mf(Γ 1 (n); Z[ 1 n ]) is normal. Using results from [41], we give moreover a criterion for mf(Γ 1 (n); Z[ 1 n ]) being Cohen-Macaulay, which is satisfied for all 2 ≤ n ≤ 28.These commutative algebra results help us to develop cubical analogues of the stacks M 1 (n). To this purpose recall that M 1 (n) is usually defined as the normalization of the compactified moduli stack of elliptic curves M ell in M 1 (n). The stack M ell embeds into the larger stack M cub of all curves that can locally be described by a cubic Weierstraß equation. We can define M 1 (n) cub as the normalization of M cub in M 1 (n). We show:is Cohen-Macaulay. Our reason to consider these cubical stacks is that vector bundles seem to be easier to study on M cub than on M ell , a view inspired by work of Mathew [36]. This line of thought is already implicit though in the classification of vector bundles over weighted projective lines in [40, Proposition 3.4], where the idea was to extend them to vector bundles on the non-separated stack A 2 /G m . For M cub this idea takes the form that there is a nice smooth cover Spec A → M cub with A = Z[a 1 , a 2 , a 3 , a 4 , a 6 ] given by Weierstraß curves and thus quasi-coherent sheaves on M cub become equivalent to comodules over a certain explicit Hopf algebroid (A, Γ). For explicit calculations, this outweighs the disadvantage that M cub is neither Deligne-Mumford nor separated.Our wish for such explicit calculations was motivated by the results from [41].

Additive decompositions for rings of modular forms

“…There are explicit formulae for the shifts in (2) in terms of dimensions of spaces of modular forms, which follow from our results in Section 4. An example of a similar decomposition result in a non-tame situation has been obtained in [MO20]. Regarding non-liftable modular forms, we remark that Mestre was the first to construct an example of such, namely a mod-2 eigenform of weight 1 for Γ 0 (1429).…”

“…[Tsu86] or [Got20]) have proven results about the Cohen-Macaulayness of (Hilbert, Siegel and vector-valued) modular forms over the complex numbers, the author is not aware of previous work on this question over a field of different characteristic or with integral coefficients. This result has proven useful in [MO20] when studying moduli of cubic curves. Let us say a few words on how we obtain our main theorems.…”

Strictly Commutative Complex Orientations of $Tmf_1(N)$

“…Conversely if E ′ is an elliptic curve with an arithmetic Γ 1 (N )-level structure, the quotient E ′ /β arith N (µ N ) can be equipped with a naive Γ 1 (N )-level structure. For more on this we refer the reader to Chapter II of [Kat76] or the appendix of [MO20].…”

Topological Mathieu Moonshine

“…(That paper shows that Γ(J ) and suspension together generate Pic(Tmf • ) ∼ = Z 24 × Z. Other exotic elements are studied in [HM17,MO20]. )…”