Richard Hain

Suppose that r and n are non-negative integers satisfying r + n > 0. Denote the moduli stack over SpecZ of smooth elliptic curves with n marked points and r non-zero tangent vectors by M1,n+r⃗. Here the n marked points and the anchor points of the r tangent vectors are distinct. Taking each tangent vector to its anchor point defines a morphism M1,n+r⃗ → M1,n+r that is a principal Gm-bundle. The Deligne-Mumford compactification [8] of M1,n will be denoted by M1,n. It is also defined over SpecZ. In this paper, we are primarily concerned with the cases (n, r) = (1, 0), (0, 1) and (2, 0). For a Z-algebra A, we denote by M1,n+r⃗/A the stack M1,n+r⃗ ×SpecZ SpecA. Similarly, the pullback of M1,n to SpecA will be denoted by M1,n/A. The universal elliptic curve over M1,1 will be denoted by E . Note that M1,2 is E with its identity section removed. The extension E → M1,1 of the universal elliptic curve to M1,1 is obtained by1 glueing in the Tate curve ETate → SpecZ[[q]] (cf. [41, Ch. V]). It is simply M1,2. The unique cusp of M1,1 is the moduli point of the nodal cubic. Denote it by eo. The standard line bundle L over M1,1 is the conormal bundle of the zero section of E → M1,1. Sections of L⊗n over M1,1 are modular forms of weight n, and those that vanish at the cusp eo are cusp forms of weight n. The restriction of the relative tangent bundle of E to its identity section is the dual Ľ of L. The moduli space M1,⃗1 is Ľ′, the restriction of Ľ to M1,1 with its zero-section removed. This is isomorphic to the complement L′ of the zero-section of L. We will also identify M1,1 with the identity section of E . With this convention, eo also denotes the identity of the nodal cubic in E and also the corresponding point of the partial compactification Ľ of M1,⃗1. An explicit description of L′ over M1,1 can be deduced from the discussion [28, Chapt. 2] and the formulas in [40, Appendix A]. Then M1,1 is the quotient stack Gm\\L. When 2 and 3 are invertible in A, LA is the scheme LA = AA − {0} = SpecA[u, v]− {0}. The point (u, v) corresponds to the plane cubic y = 4x − ux− v and the abelian differential dx/y. The Gm-action is λ : (u, v) 7→ (λ−4u, λ−6v). In this case, M1,1 and M1,1 are the quotient stacks M1,1/A = Gm\\LA and M1,1/A = Gm\\(AA −D−1(0)), where D = u − 27v is (up to a factor of 4) the discriminant of the cubic. Similarly, when 2 and 3 are invertible in A, M1,2/A is the quotient of the scheme {(u, v, x, y) ∈ AA × AA : y = 4x − ux− v, (u, v) ̸= 0}