Vector Bundles on the Moduli Stack of Elliptic Curves
Vector Bundles on the Moduli Stack of Elliptic Curves
As is well known, there is classification of line bundles on the moduli stack of elliptic curves over a nearly arbitrary base scheme in the paper The Picard group of
I am now interested in a similar classification scheme for higher dimensional vector bundles (i.e. etale-locally free quasi-coherent sheaves of finite rank). I am especially interested in the prime 3, so you may assume, 2 is inverted, or even that we work over
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I think, I can prove that every vector bundle
on the moduli stack over localized at for is an extension of the form where is a line bundle and a vector bundle of one dimension smaller than (this may hold also for , but I haven't checked). In a paper of Tilman Bauer (Computation of the homotopy of the spectrum tmf) Ext groups of the so called Weierstraß Hopf algebroid are computed, which should amount to a computation of the Ext groups of the line bundles on the moduli stack of elliptic curves if one inverts . It follows than that every vector bundle on the moduli stack of elliptic curves over is isomorphic to a sum of line bundles for if I have not made a mistake. But for , there are many non-trivial Ext groups and I did not manage to see which of the occuring vector bundles are isomorphic. -
One can try to find explicit examples of a non-trivial higher dimensional vector bundles. A candidate was suggested to me by M. Rapoport: for every elliptic curve
over a base scheme we have an universal extension of by a vector bundle. Take the Lie algebra of this extension and we get a canonical vector bundle over . As explained in the book Universal Extensions and One Dimensional Crystalline Cohomology by Mazur and Messing, this is isomorphic to the deRham cohomology of . This vector bundle is an extension of and and lies in a non-trivial Ext group. But I don't know how to show that this bundle is non-trivial.
I should add that I am more a topologist than an algebraic geometer and stand not really on firm ground in this topic. I would be thankful for any comment on the two strategies or anything else concerning a possible classification scheme.
K2-Think
The classification of vector bundles on the moduli stack of elliptic curves ( \mathcal{M}{1,1} ) over ( \mathbb{Z}{(3)} ) (with 2 inverted) involves nuanced differences at the prime 3 compared to primes ( p > 3 ). Here's a structured summary:
1. Line Bundles and General Strategy
The Picard group of ( \mathcal{M}_{1,1} ) is generated by the line bundle ( \omega ) (relative differen...
GPT-5
Two very different behaviors show up depending on whether you invert 6 or not. Over Z[1/6] (and hence over Z(p) for p>3) the situation is completely rigid and all vector bundles split as sums of powers of the Hodge line bundle ω. At p=3 (and similarly at p=2) there are many nontrivial extensions and indeed indecomposables of all ranks.
Here are precise statements and proofs/sketches that address ...
o3
Below I shall denote by M := M_{1,1} the Deligne‐Mumford stack which classifies elliptic curves, by
ω the Hodge bundle and by Vect(M_R ) the category of vector bundles (= locally free, finite rank, 𝒪‐modules) on the base–change M_R := M×Spec R of M to a commutative ring R.
Write μ_n for the finite flat group scheme of n-th roots of unity.
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- Invertin...
Claude 3.7 Sonnet
Vector Bundles on the Moduli Stack of Elliptic Curves
Your question touches on a challenging area in algebraic geometry and algebraic topology. Let me address both your strategies and suggest some additional perspectives.
On your first strategy:
Your approach of using extensions is promising. If every vector bundle on the moduli stack can be expressed as an extension of a line bundle and a...
o4 mini
Here is what is known (and what one can prove) about vector bundles on the stack
M₁,₁ — the moduli of elliptic curves — under the two regimes “2,3 invertible” vs. “3‐torsion present.”
- The tame case: 2,3 ∈ Oₓ× (equivalently working over ℤ[1/6] or over any ℤ[1/6]–algebra)
• In this world every stabilizer (μ₂ at j=1728 and μ₃ at j=0) is a linearly‐reductive group‐scheme. One has the f...
DeepSeek R1
First, I need to understand the context. The moduli stack of elliptic curves is tricky because it's a stack, not a scheme, so vector bundles ...
Gemini 2.5 Pro
This is an excellent and deep question that touches upon the arithmetic of moduli spaces, with strong connections to chromatic homotopy theory. Your two proposed strategies are precisely the right ways to think about this problem. Let's break down each of them.
Summary
The short answer is:
- Over a field of characteristic
, or over , every vector bundle on $\mathcal{...
Claude Opus 4
This is a very interesting question about vector bundles on the moduli stack of elliptic curves. Let me address both of your strategies and provide some additional context.
Strategy 1: Extension approach
Your observation that every vector bundle admits a filtration by line bundles is a good starting point. The key insight is that for
o3 Pro
Let 𝔐 = 𝔐_{1,1} denote the Deligne–Mumford stack that classifies elliptic curves and write ω for the Hodge (= sheaf of invariant 1–forms) line bundle. We discuss algebraic, locally free, finite rank quasi–coherent sheaves (“vector bundles’’) on 𝔐 over various bases.
- Two general facts.
0.1 If X is an affine scheme and G is an affine group scheme, every vector bundle on [X/G] is...
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