Rigid Cohomology for Algebraic Stacks David Zureick-Brown Emory University, Department of Mathematics and Computer Science 2012 Joint Meetings Special Session on Arithmetic Geometry Boston, MA January 7, 2012 Slides available at http://www.mathcs.emory.edu/~dzb/slides/
Rigid Cohomology Problem (posed by Kiran Kedlaya): for Algebraic Stacks David Zureick-Brown Basic Problem Develop a theory of Rigid Cohomology for Algebraic Applications Stacks Rigid Cohomology for Schemes The Overconvergent Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Problem (posed by Kiran Kedlaya): for Algebraic Stacks David Zureick-Brown Basic Problem Develop a theory of Rigid Cohomology for Algebraic Applications Stacks ; i.e., Rigid Cohomology for Schemes ◮ (Coefficients) – define some notion of overconvergent The Overconvergent isocrystal ; Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Problem (posed by Kiran Kedlaya): for Algebraic Stacks David Zureick-Brown Basic Problem Develop a theory of Rigid Cohomology for Algebraic Applications Stacks ; i.e., Rigid Cohomology for Schemes ◮ (Coefficients) – define some notion of overconvergent The Overconvergent isocrystal ; Site ◮ (Cohomology) – define some notion of cohomology of Overconvergent cohomology for an overconvergent isocrystal; Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Problem (posed by Kiran Kedlaya): for Algebraic Stacks David Zureick-Brown Basic Problem Develop a theory of Rigid Cohomology for Algebraic Applications Stacks ; i.e., Rigid Cohomology for Schemes ◮ (Coefficients) – define some notion of overconvergent The Overconvergent isocrystal ; Site ◮ (Cohomology) – define some notion of cohomology of Overconvergent cohomology for an overconvergent isocrystal; Algebraic Stacks Cohomology ◮ Construct variants (e.g., cohomology supported in a supported in a closed substack closed subscheme); Cohomological descent
Rigid Cohomology Problem (posed by Kiran Kedlaya): for Algebraic Stacks David Zureick-Brown Basic Problem Develop a theory of Rigid Cohomology for Algebraic Applications Stacks ; i.e., Rigid Cohomology for Schemes ◮ (Coefficients) – define some notion of overconvergent The Overconvergent isocrystal ; Site ◮ (Cohomology) – define some notion of cohomology of Overconvergent cohomology for an overconvergent isocrystal; Algebraic Stacks Cohomology ◮ Construct variants (e.g., cohomology supported in a supported in a closed substack closed subscheme); Cohomological ◮ Weil formalism (e.g., excision, Gysin, trace formulas). descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Geometric Langlands for GL n ( F p ( C )): Rigid Cohomology for Schemes The Overconvergent Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Geometric Langlands for GL n ( F p ( C )): Rigid Cohomology for Schemes ◮ Lafforgue constructs a ‘compactified moduli stack of The Overconvergent shtukas’ X (actually a compactification of a Site stratification of a moduli stack of shtukas). Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Geometric Langlands for GL n ( F p ( C )): Rigid Cohomology for Schemes ◮ Lafforgue constructs a ‘compactified moduli stack of The Overconvergent shtukas’ X (actually a compactification of a Site stratification of a moduli stack of shtukas). Overconvergent cohomology for ◮ The ℓ -adic ´ etale cohomology of ´ etale sheaves on X Algebraic Stacks realize a Langlands correspondence between certain Cohomology supported in a Galois and automorphic representations. closed substack Cohomological descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Rigid Cohomology Geometric Langlands for GL n ( F p ( C )): for Schemes ◮ ℓ = p is bad for ´ etale cohomology. The Overconvergent Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Rigid Cohomology Geometric Langlands for GL n ( F p ( C )): for Schemes ◮ ℓ = p is bad for ´ etale cohomology. The Overconvergent Site ◮ X is a singular, separated Artin stack, so crystalline Overconvergent cohomology won’t work. cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Original Motivation (Langlands) for Algebraic Stacks David Zureick-Brown Basic Problem Applications Rigid Cohomology Geometric Langlands for GL n ( F p ( C )): for Schemes ◮ ℓ = p is bad for ´ etale cohomology. The Overconvergent Site ◮ X is a singular, separated Artin stack, so crystalline Overconvergent cohomology won’t work. cohomology for Algebraic Stacks ◮ Generalizing rigid cohomology to Artin stacks would Cohomology extend Lafforgue’s work to the ℓ = p case. supported in a closed substack Cohomological descent
Rigid Cohomology Other Applications for Algebraic Stacks David Zureick-Brown Basic Problem Applications Rigid Cohomology Applications : for Schemes ◮ Geometric Langlands for GL n ( F p ( C )); The Overconvergent Site ◮ Logarithmic rigid cohomology and crystalline Overconvergent fundamental groups; cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
Rigid Cohomology Other Applications for Algebraic Stacks David Zureick-Brown Basic Problem Applications Rigid Cohomology Applications : for Schemes ◮ Geometric Langlands for GL n ( F p ( C )); The Overconvergent Site ◮ Logarithmic rigid cohomology and crystalline Overconvergent fundamental groups; cohomology for Algebraic Stacks ◮ Arithmetic Statistics – Cohen-Lenstra heuristics for Cohomology p -divisible groups. supported in a closed substack Cohomological descent
� � Rigid Cohomology Rigid Cohomology Setup (for Schemes) for Algebraic Stacks David Zureick-Brown P n , an i ] X [ ⊂ Q p Basic Problem Applications sp sp Rigid Cohomology for Schemes j � P n X � � F p The Overconvergent Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
� � Rigid Cohomology Rigid Cohomology Setup (for Schemes) for Algebraic Stacks David Zureick-Brown P n , an i ] X [ ⊂ Q p Basic Problem Applications sp sp Rigid Cohomology for Schemes j � P n X � � F p The Overconvergent Site � � H i rig ( X ) := H i ] X [ , i − 1 Ω • P n , an Overconvergent Q p cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
� � Rigid Cohomology Rigid Cohomology Setup (for Schemes) for Algebraic Stacks David Zureick-Brown P n , an i ] X [ ⊂ Q p Basic Problem Applications sp sp Rigid Cohomology for Schemes j � P n X � � F p The Overconvergent Site � � H i rig ( X ) := H i ] X [ , i − 1 Ω • P n , an Overconvergent Q p cohomology for Algebraic Stacks Cohomology Isoc X := { ( M , ∇ ) ∈ MIC W } / ∼ supported in a closed substack Cohomological descent
� � Rigid Cohomology Rigid Cohomology Setup (for Schemes) for Algebraic Stacks David Zureick-Brown P n , an i ] X [ ⊂ Q p Basic Problem Applications sp sp Rigid Cohomology for Schemes j � P n X � � F p The Overconvergent Site � � H i rig ( X ) := H i ] X [ , i − 1 Ω • P n , an Overconvergent Q p cohomology for Algebraic Stacks Cohomology Isoc X := { ( M , ∇ ) ∈ MIC W } / ∼ supported in a closed substack Cohomological Isoc † X := { ( M , ∇ ) ∈ Isoc X s.t. ∇ is ‘overconvergent’ } descent
� � Rigid Cohomology Rigid Cohomology Setup (for Schemes) for Algebraic Stacks David Zureick-Brown P n , an i ] X [ ⊂ Q p Basic Problem Applications sp sp Rigid Cohomology for Schemes j � P n X � � F p The Overconvergent Site � � H i rig ( X ) := H i ] X [ , i − 1 Ω • P n , an Overconvergent Q p cohomology for Algebraic Stacks Cohomology Isoc X := { ( M , ∇ ) ∈ MIC W } / ∼ supported in a closed substack Cohomological Isoc † X := { ( M , ∇ ) ∈ Isoc X s.t. ∇ is ‘overconvergent’ } descent (OK to replace P n with a formal scheme which is smooth and proper over Spf Z p .)
� � Example: X = A 1 Rigid Cohomology for Algebraic F p Stacks David i ] A 1 ( P 1 Q p ) an Zureick-Brown F p [ ⊂ Basic Problem sp sp Applications j Rigid Cohomology A 1 � P 1 � � for Schemes F p F p The Overconvergent Site Overconvergent cohomology for Algebraic Stacks Cohomology supported in a closed substack Cohomological descent
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