Zeta functions with p -adic cohomology David Roe Harvard University - PowerPoint PPT Presentation

Zeta functions with p -adic cohomology David Roe Harvard University / University of Calgary Geocrypt 2011 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 1 / 28

Outline Hyperelliptic Curves 1 Beyond dimension 1 2 Algorithm for hypersurfaces 3 Timings 4 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 2 / 28

Hyperelliptic Curves p -adic point counting Kedlaya [Ked01] gives an algorithm for computing the number of F q -rational points on a hyperelliptic curve using p-adic cohomology. Suppose that X is a hyperelliptic curve of genus g , whose affine locus is defined by the equation y 2 = f ( x ) for some f ( x ) ∈ F q [ x ] . Kedlaya’s key idea is that we can determine the size of X ( F q ) from the action of Frobenius on a Weil cohomology theory applied to X . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 3 / 28

Hyperelliptic Curves Notation We first work with a more general smooth projective X . Let U be an affine open in X (for hyperelliptic curves we will set U as the subset of the standard affine chart with y � = 0). Set ¯ A as the coordinate ring of U , and choose a smooth Z q -algebra A with A ⊗ Z q F q = ¯ A . In the curve case A = Z q [ x , y , y − 1 ] / ( y 2 − f ( x )) . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 4 / 28

Hyperelliptic Curves Monsky-Washnitzer cohomology Unfortunately, we cannot lift Frobenius to an endomorphism of A : we need to p -adically complete A somehow. The full completion is too big, so instead we use the weak completion A † . Fix x 1 , . . . , x n ∈ A whose images in ¯ A generate it over F q . Then � ∞ A † = � a n P n ( x 1 , . . . , x n ) : v p ( a n ) ≥ n , n = 0 � and ∃ c > 0 with deg ( P n ) < c ( n + 1 ) for all n The Monsky-Washnitzer cohomology of U is the cohomology of the algebraic de Rham complex over A † ⊗ Z q Q q . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 5 / 28

Hyperelliptic Curves A † for hyperelliptic curves We can be more explicit for hyperelliptic curves. For P ( x ) ∈ Z q [ x ] , let v p ( P ) be the minimum valuation of any coefficient. Then ∞ v p ( P n ( x )) v p ( P − n ( x )) A † = � P n ( x ) y n : lim inf � � > 0 , lim inf > 0 . n n n →∞ n →∞ n = −∞ David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 6 / 28

Hyperelliptic Curves Lifting Frobenius We define a lift of Frobenius σ : A † → A † by setting σ is the standard Frobenius on coefficients in Z q , σ ( x ) = x p , and defining σ ( y ) by � 1 / 2 1 + σ ( f ( x )) − f ( x ) p � σ ( y ) = y p y 2 p ∞ � ( σ ( f ( x )) − f ( x ) p ) i � 1 / 2 = y p � y pi i i = 0 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 7 / 28

Hyperelliptic Curves Lefschetz fixed-point theorem The key theorem which will allow us to use this cohomology theory to count rational points is the following. Theorem Suppose that ¯ A is smooth and integral of dimension n over F q , and that the weak completion A † of ¯ A admits a Frobenius F lifting the q-Frobenius on ¯ A. Then the number of homomorphisms ¯ A → F q is given by n ( − 1 ) i Tr ( q n F − 1 | H i ( A ; Q q ) . � i = 0 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 8 / 28

Hyperelliptic Curves Kedlaya’s Algorithm The plan: Write down a basis for H 1 ( A ; Q q ) and apply Frobenius to each 1 basis element. Subtract coboundaries in order to write these images in terms of 2 the original basis, obtaining a matrix M for the p -power Frobenius. Determine a matrix M ′ for the q -power Frobenius by taking a 3 product of conjugates of M . Recover the zeta function (or the cardinality of X ( F q ) ) from the characteristic polynomial of M ′ and the Weil conjectures. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 9 / 28

Hyperelliptic Curves A basis for H 1 ( A ; Q q ) A priori, our one-forms have the shape ∞ d n � � a i , n x i dx / y n . n = −∞ i = 0 In fact, we can determine that x i dx � 2 g − 1 x i dx � 2 g − 1 � � ∪ y 2 y i = 0 i = 0 is a basis for H 1 ( A ; Q q ) using the following reduction formulas. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 10 / 28

Hyperelliptic Curves Reduction in cohomology Suppose B ( x ) ∈ Z q [ x ] . Then we can write B ( x ) = R ( x ) f ( x ) + S ( x ) f ′ ( x ) and this gives 2 S ′ ( x ) dx B ( x ) dx ≡ R ( x ) dx + y s y s − 2 ( s − 2 ) y s − 2 allowing us to collect terms in the n = 1 and n = 2 components. Moreover, the relation [ S ( x ) f ′ ( x ) + 2 S ′ ( x ) f ( x )] dx / y ≡ 0 with S ( x ) = x m − 2 g then allows us to reduce the degree of the coefficient of dx / y and dx / y 2 . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 11 / 28

Beyond dimension 1 Zeta functions X ⊂ P n F q smooth, given by f ∈ F q [ x 0 , . . . , x n ] , deg ( f ) = d . � ∞ � # X ( F q n ) T n � Z X ( T ) = exp n n = 1 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 12 / 28

Beyond dimension 1 Zeta functions X ⊂ P n F q smooth, given by f ∈ F q [ x 0 , . . . , x n ] , deg ( f ) = d . � ∞ � # X ( F q n ) T n � Z X ( T ) = exp n n = 1 2 n − 2 P i ( T ) ( − 1 ) i + 1 , � Z X ( T ) = i = 0 where P i ( T ) = det ( 1 − TF i | H i ( X )) . This works when H ∗ is a Weil cohomology theory, where each H i ( X ) comes equipped with a Frobenius. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 12 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces equipped with endomorphisms F i with P i ( T ) = det ( 1 − TF i | H i ( X )) . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces equipped with endomorphisms F i with P i ( T ) = det ( 1 − TF i | H i ( X )) . Lefschetz: for any m , # X ( F q m ) = � 2 dim ( X ) ( − 1 ) i Tr ( F m i | H i ( X )) . i = 0 David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces equipped with endomorphisms F i with P i ( T ) = det ( 1 − TF i | H i ( X )) . Lefschetz: for any m , # X ( F q m ) = � 2 dim ( X ) ( − 1 ) i Tr ( F m i | H i ( X )) . i = 0 Write H i ( X )( k ) for H i ( X ) with Frobenius q − k F i . If n = dim ( X ) , one has functorial, F -equivariant Tr X : H 2 n ( X )( n ) → K , isomorphisms if X is geometrically irreducible. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces equipped with endomorphisms F i with P i ( T ) = det ( 1 − TF i | H i ( X )) . Lefschetz: for any m , # X ( F q m ) = � 2 dim ( X ) ( − 1 ) i Tr ( F m i | H i ( X )) . i = 0 Write H i ( X )( k ) for H i ( X ) with Frobenius q − k F i . If n = dim ( X ) , one has functorial, F -equivariant Tr X : H 2 n ( X )( n ) → K , isomorphisms if X is geometrically irreducible. Associative, functorial, F -equivariant cup products so that Tr X H i ( X ) × H 2 n − i ( X )( n ) ∪ → H 2 n ( X )( n ) − − − → K is perfect. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Weil cohomology Contravariant functors H i from smooth proper varieties over F q to finite dimensional K -vector spaces equipped with endomorphisms F i with P i ( T ) = det ( 1 − TF i | H i ( X )) . Lefschetz: for any m , # X ( F q m ) = � 2 dim ( X ) ( − 1 ) i Tr ( F m i | H i ( X )) . i = 0 Write H i ( X )( k ) for H i ( X ) with Frobenius q − k F i . If n = dim ( X ) , one has functorial, F -equivariant Tr X : H 2 n ( X )( n ) → K , isomorphisms if X is geometrically irreducible. Associative, functorial, F -equivariant cup products so that Tr X H i ( X ) × H 2 n − i ( X )( n ) ∪ → H 2 n ( X )( n ) − − − → K is perfect. Rigid cohomology is an example of a Weil cohomology. David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 13 / 28

Beyond dimension 1 Notation Let U = P n F q \ X , f ∈ Z q [ x 0 , . . . , x n ] a lift of f , X the zero locus of f , U = P n Z q \ X X = X Q q , ˜ ˜ U = U Q q . David Roe ( Harvard University / University of Calgary ) Zeta functions with p -adic cohomology 14 / 28