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Representation Theory of the Space of Holomorphic Polydifferentials - - PowerPoint PPT Presentation
Representation Theory of the Space of Holomorphic Polydifferentials - - PowerPoint PPT Presentation
Representation Theory of the Space of Holomorphic Polydifferentials Adam Wood Department of Mathematics University of Iowa Joint Mathematics Meetings AMS Contributed Paper Session on Algebra and Algebraic Geometry, I January 17, 2020
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Group Actions on Curves
Let X be a curve over k and let G be a finite group. An action of G on X is either
◮ A morphism σ : G ×k X → X ◮ An action of G on the function field k(X)
If σ(g, x) = g · x, then define an action on k(X) by (g · f )(x) = f (g−1 · x) for all f ∈ k(X), g ∈ G, and x ∈ X
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Space of Holomorphic Polydifferentials
The sheaf of relative differentials of X over k, denoted by ΩX, is a quasicoherent sheaf on X, also referred to as the cotangent sheaf For an integer m ≥ 1, define Ω⊗m
X
= ΩX ⊗OX · · · ⊗OX ΩX
- m times
Define the space of holomorphic m-polydifferentials of X over k to be the global sections of Ω⊗m
X , denote by H0(X, Ω⊗m X )
H0(X, Ω⊗m
X ) is a k-vector space with
dimkH0(X, Ω⊗m
X ) =
- g(X)
if m = 1 (2m − 1)(g(X) − 1)
- therwise
If G acts on X, then G acts on H0(X, Ω⊗m
X ) =
⇒ H0(X, Ω⊗m
X ) is a
representation of G
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General Problem
Question (Hecke, 1928): How does H0(X, Ω⊗m
X ) decompose into a
direct sum of indecomposable representations of G? Solved if char(k) = 0 (Chevalley and Weil, 1934) Assume that char(k) = p
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Previous Work
Nakajima (1976), tamely ramified cover X → X/G Bleher, Chinburg, and Kontogeorgis (2017), m = 1, G has cyclic Sylow p-subgroups Karanikolopoulos (2012), m > 1, G cyclic p-group
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Result
Theorem
Let k be a perfect field of prime characteristic p and let G be finite group acting on a curve X over k. Assume that G has cyclic Sylow p-subgroups. For m > 1, the module structure of H0(X, Ω⊗m
X ) is
determined by the inertia groups of closed points x ∈ X and their fundamental characters. Assume k is algebraically closed Conlon induction theorem = ⇒ assume that G = P ⋊ C, P cyclic p-group, C cyclic group with p ∤ |C|
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Technique
Assume G = P ⋊ C, k algebraically closed field of characteristic p |P| = pn, |C| = c Representation theory of G over k is “nice” Galois cover of curves X → X/G X Y X/G
Wild Tame
Y = X/Q Q = σ, subgroup of P generated by Sylow p-subgroups of inertia groups of closed points of X
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Modular Curves
ℓ = p prime, X(ℓ) modular curve of level ℓ, k algebraically closed, char(k) = p Get smooth projective model X of X(ℓ) over k G = PSL(2, Fℓ) acts on X H0(X, Ω⊗m
X ) gives space of weight 2m holomorphic cusp forms
For p = 3, proof of theorem gives method for determining the decomposition of H0(X, Ω⊗m
X ) as a direct sum of indecomposable
kG-modules Uses Green correspondence, known structure of G, and known ramification of X → X/G
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Modular Curves, p = 3
The decomposition of H0(X, Ω⊗m
X ) depends on m mod 6
If m ≡ 2 mod 3, then H0(X, Ω⊗m
X ) is a projective kG-module
Example of result of K¨
- ck (2004) for weakly ramified covers
Has implications for congruences between modular forms
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References
J.L. Alperin. Local Representation Theory, Cambridge University Press, 1986. Frauke M. Bleher, Ted Chinburg, and Artistides Kontogeorgis. “Galois structure of the holomorphic differentials of curves”. 2019. arXiv:1707.07133. Sotiris Karanikolopoulos. “On holomorphic polydifferentials in positive characteristic”. Mathematische Nachrichten, 285(7):852-877, 2012. Bernhard K¨
- ck. “Galois structure of Zariski cohomology for weakly ramified