Module Structure of the Space of Holomorphic Polydifferentials Adam - PowerPoint PPT Presentation

Module Structure of the Space of Holomorphic Polydifferentials Adam Wood Department of Mathematics University of Iowa Conference on Geometric Methods in Representation Theory November 24, 2019

Outline Setting General Problem Result Overview of Technique Consequences and Applications

Setting k algebraically closed field (usually assume char( k ) = p ) X smooth projective curve over k G finite group acting on X Ω X sheaf of relative differentials of X over k For integer m ≥ 1, Ω ⊗ m = Ω X ⊗ O X · · · ⊗ O X Ω X X � �� � m times

General Problem Definition Define the space of holomorphic m-polydifferentials of X over k to be the global sections of the sheaf Ω ⊗ m X . Remarks: ◮ Zeroth cohomology gives global sections, denote by H 0 ( X , Ω ⊗ m X ) ◮ Ω ⊗ m ⇒ H 0 ( X , Ω ⊗ m is G -equivariant = X ) is a representation of X G � g ( X ) if m = 1 ◮ dim k H 0 ( X , Ω ⊗ m X ) = (2 m − 1)( g ( X ) − 1) otherwise ∼ ◮ Ω ⊗ m = O X ( mK X ), where K X is a canonical divisor on X X ◮ If m = 1, refer to H 0 ( X , Ω X ) as the space of holomorphic differentials

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 Can vary: ◮ Value of m ◮ Ramification of the cover X → X / G ◮ Type of group G

Previous Work Tamagawa (1951), unramified cover X → X / G , G cyclic Nakajima (1976), tamely ramified cover X → X / G Bleher, Chinburg, and Kontogeorgis (preprint, 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 |

Representation Theory G = P ⋊ C , k field of characteristic p | P | = p n , | C | = c Representation theory of G over k is well known Simple kG -modules are the simple kC -modules There are c · p n isomorphism classes of indecomposable representations of G , all uniserial Determined by socle and dimension

Technique Galois cover of curves X → X / G Wild Tame X / G X Y Y = X / Q , Q = � σ � , subgroup of P generated by Sylow p -subgroups of inertia groups Define M ( j ) = Kernel of action of ( σ − 1) j on M Understand ( H 0 ( X , Ω ⊗ m X )) ( j +1) / ( H 0 ( X , Ω ⊗ m X )) ( j ) as k [ G / Q ]-modules

Wild Cover π : X → Y Get effective divisor D j on Y so that π ∗ Ω ⊗ m , ( j +1) /π ∗ Ω ⊗ m , ( j ) ∼ = O Y ( D j ) ⊗ O Y Ω ⊗ m X X Y Recall Riemann-Hurwitz formula π ∗ Ω X = π ∗ D − 1 X / Y ⊗ O Y Ω Y Compare ( H 0 ( X , Ω ⊗ m X )) ( j +1) / ( H 0 ( X , Ω ⊗ m X )) ( j ) and H 0 ( Y , π ∗ Ω ⊗ m , ( j +1) /π ∗ Ω ⊗ m , ( j ) ) X X

Quotients Get injective map X )) ( j ) ֒ → H 0 ( Y , π ∗ Ω ⊗ m , ( j +1) /π ∗ Ω ⊗ m , ( j ) ( H 0 ( X , Ω ⊗ m X )) ( j +1) / ( H 0 ( X , Ω ⊗ m ) X X Riemann-Roch Theorem = ⇒ dimensions agree X )) ( j ) ∼ = H 0 ( Y , π ∗ Ω ⊗ m , ( j +1) /π ∗ Ω ⊗ m , ( j ) ( H 0 ( X , Ω ⊗ m X )) ( j +1) / ( H 0 ( X , Ω ⊗ m ) X X Understand H 0 ( Y , O Y ( D j ) ⊗ O Y Ω ⊗ m Y ) as a k [ G / Q ]-module

Tame Cover Y → X / G tamely ramified cover with Galois group G / Q O Y ( D j ) ⊗ O Y Ω ⊗ m ∼ = O Y ( D j + mK Y ) Y ⇒ H 1 ( Y , O Y ( D j ) ⊗ O Y Ω ⊗ m Riemann-Roch Theorem = Y ) = 0 ⇒ H 0 ( Y , O Y ( D j ) ⊗ O Y Ω ⊗ m Nakajima (1986) = Y ) projective k [ G / Q ]-module, gives formula for Brauer character

Building H 0 ( X , Ω ⊗ m X ) Know k [ G / Q ]-module structure of ( H 0 ( X , Ω ⊗ m X )) ( j +1) / ( H 0 ( X , Ω ⊗ m X )) ( j ) All indecomposable kG -module are uniserial = ⇒ get kG -module decomposition of H 0 ( X , Ω ⊗ m 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 projective Verifies result of K¨ ock (2004) for weakly ramified covers

Modular Curves, p = 3, m ≡ 0 mod 3 Write ℓ + 1 = 3 n · 2 · m ′′ T t (0 ≤ t ≤ ( m ′′ − 1) / 2), γ 1 , γ 2 , η G Simple kG -modules are T 0 , � ( m ′′− 1) / 2 � m − a � � (2 m − 1) ℓ + 5 − 14 m H 0 ( X , Ω ⊗ m P ( � ) = + c m P ( T 0 ) ⊕ T t ) ⊕ � γ 1 , β � P ( γ 1 ) ⊕ � γ 2 , β � P ( γ 2 ) X 6 12 t =0 ( m ′′− 1) / 2 � � � η G , β � P ( η G ) ⊕ i m Str ( n ) ⊕ (1 − i m ) U 0 , 3 n − 1 ⊕ ⊕ U t , 2 · 3 n − 1 η t =1 where � 1 if m ≡ 0 mod 6 m ≡ a mod 6 , δ m = mod 6 , − 1 if m ≡ 3 � � − 1 if m ≡ 0 mod 6 1 if m ≡ 0 mod 6 c m = mod 6 , i m = mod 6 , 0 if m ≡ 3 0 if m ≡ 3 � (2 m − 1) ℓ − 19+ δ m 12 − 10 m if ℓ ≡ 1 mod 8 24 � γ 1 , β � = (2 m − 1) ℓ − 19 − 10 m if ℓ ≡ 5 mod 8 24 � (2 m − 1) ℓ +17 − 10 m if ℓ ≡ 1 mod 8 24 � γ 2 , β � = (2 m − 1) ℓ +17 − δ m 12 − 10 m if ℓ ≡ 5 mod 8 24 � (2 m − 1) ℓ − 1 − δ m 6 − 10 m if η ( s ) = − 1 � η G , β � = 12 (2 m − 1) ℓ − 1+ δ m 6 − 10 m if η ( s ) = 1 12

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.