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
Setting k algebraically closed field (usually assume char( k ) = p ) X smooth projective curve over k G finite group A representation of G over k is a kG -module
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
Space of Holomorphic Polydifferentials The sheaf of relative differentials of X over k , denoted by Ω X , is a quasicoherent sheaf on X , also referred to as the cotangent sheaf For an integer m ≥ 1, define Ω ⊗ m = Ω X ⊗ O X · · · ⊗ O X Ω X X � �� � m times Define the space of holomorphic m-polydifferentials of X over k to be the global sections of Ω ⊗ m X , denote by H 0 ( X , Ω ⊗ m X ) H 0 ( X , Ω ⊗ m X ) is a k -vector space with � g ( X ) if m = 1 dim k H 0 ( X , Ω ⊗ m X ) = (2 m − 1)( g ( X ) − 1) otherwise If G acts on X , then G acts on H 0 ( X , Ω ⊗ m ⇒ H 0 ( X , Ω ⊗ m X ) = X ) is a representation of G
General Problem Question (Hecke, 1928): How does H 0 ( 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
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
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 H 0 ( 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 |
Technique Assume G = P ⋊ C , k algebraically closed field of characteristic p | P | = p n , | C | = c Representation theory of G over k is “nice” Galois cover of curves X → X / G Wild Tame X Y X / G Y = X / Q Q = � σ � , subgroup of P generated by Sylow p -subgroups of inertia groups of closed points of X
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 , F ℓ ) acts on X H 0 ( X , Ω ⊗ m X ) gives space of weight 2 m holomorphic cusp forms For p = 3, proof of theorem gives method for determining the decomposition of H 0 ( 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
Modular Curves, p = 3 The decomposition of H 0 ( X , Ω ⊗ m X ) depends on m mod 6 If m ≡ 2 mod 3, then H 0 ( X , Ω ⊗ m X ) is a projective kG -module Example of result of K¨ ock (2004) for weakly ramified covers Has implications for congruences between modular forms
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¨ ock. “Galois structure of Zariski cohomology for weakly ramified covers of curves”. American Journal of Mathematics , 126:1085-1107, 2004. Carlos J. Moreno. Algebraic Curves over Finite Fields , Cambridge University Press, 1991. Shoichi Nakajima. “Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties”. Journal of Number Theory , 22:115-123, 1986.
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