Computing zeta functions of certain varieties in larger characteristic

Transcription

1 Computing zeta functions of certain varieties in larger characteristic New York University 16th March 2010 Effective methods in p-adic cohomology Mathematical Institute, University of Oxford

2 Notation q = p a X = variety over F q Z X (T ) = exp r 1 #X (F q r ) r T r Q(T ) Question: how to compute Z X (T ) efficiently when p is large? (Time = bit-complexity)

3 Previous results Schoof (1985) and descendants: smooth projective curve of degree d in time (a log p) do(1). Lauder (2004): degree d smooth hypersurface X P n in time p 2+ɛ poly(d n a). (Note: dense input size is d n a log p bits)

4 Previous results H. (2007): genus g hyperelliptic curve in time p 0.5+ɛ poly(ga). e.g. g = 3, q is feasible (30 hours on single CPU) Minzlaff (2008): superelliptic curve in time p 0.5+ɛ poly(ga).

5 Main question Can we obtain p 0.5+ɛ complexity for varieties more general than superelliptic curves?

6 Main result Theorem (almost) Let X P n F q be a smooth projective hypersurface of degree d > n. Then Z X (T ) can be computed in time p 0.5+ɛ poly((d n a) n ) (or something roughly of this form). Still some details to be checked...

7 Implementation A toy implementation in Sage has existed for about two weeks. Ran on a random degree 4 hypersurface in P 3 : where deg P = 21. Z X (T ) = ((1 T )(1 qt )(1 q 2 T )P(T )) 1 Precision parameters selected experimentally to determine P(T ) unambiguously (not proved correct). F 11 : 29 hours. Checked #X (F 11 r ) for 1 r 4 against more-or-less naive point count in Magma. F 101 : 80 hours. Checked for 1 r 2. F 1009 : 239 hours. Checked for r = 1. F : maybe in time for CRM workshop in April...

8 AKR algorithm Algorithm based on AKR = Abbott Kedlaya Roe ( Bounding Picard numbers of surfaces using p-adic cohomology, 2007). X = {f = 0} P n F q defined by f F q [x 0,..., x n ], deg f = d U = P n \X σ q = q-th power Frobenius P(T ) = det(1 q 1 σ q T H n rig (U)) Then n 1 1 Z X (T ) = P(T ) ( 1)n 1 q i T. i=0 So it suffices to compute P(T ) Z[T ].

9 AKR algorithm Plan: compute in Hrig n (U) using the Monsky Washnitzer theory. f = lift of f to Zq [x 0,..., x n ] Ũ = lift of U defined by f, i.e. with coordinate ring à = degree 0 piece of Z q [x 0,..., x n, z] where z = f 1, deg z = d. Then H n rig(u) = H n dr (Ũ/Q q).

10 AKR algorithm Explicit description of H n dr (Ũ/Q q) (Griffiths): let Ω = n ( 1) i x i dx 0 (omit dx i ) dx n. i=0 Then H n dr (Ũ/Q q) is the quotient of z m GΩ : m 1, deg G = md n 1 by (z m i G mz m+1 G i f )Ω : 0 i n, m 1, deg G = md n. (Here i = / x i.)

11 AKR algorithm A theorem of Macaulay implies that (since X smooth!) where α = (n + 1)(d 2) + 1. ( 0 f,..., n f ) (x0,..., x n ) α, This yields a reduction algorithm for computing in H n dr : if deg F α then F = i G i( i f ) for some Gi, and then z m+1 F Ω 1 m zm i ( i G i )Ω. Some subset of the monomials for deg F < α forms a basis for HdR n ; can be computed explicitly by linear algebra.

12 AKR algorithm What about Frobenius? Let A = weak completion of Ã: elements are power series j 0 G jz j, where G j Q q [x 0,..., x n ], deg G j = jd, satisfying overconvergence condition lim inf j v p (G j )/j > 0. Lift (absolute) Frobenius to A via x σ i = x p i (0 i n) and z σ = f σ = ( f p ( f p f σ )) 1 = z p (1 z p ( f p f σ )) 1 = z p j 0 z pj ( f p f σ ) j. (Converges in A since p f p f σ.)

13 AKR algorithm Furthermore (z t ) σ = z ( ) t + j 1 pt z pj ( f p t 1 f σ ) j j 0 and Ω σ = p n (x 0 x n ) p 1 Ω. Therefore we obtain series expansion for σ applied to a cohomology basis element: (z t x k 0 0 x kn n Ω) σ.

14 AKR algorithm Summary of AKR algorithm: 1. Compute a basis for HdR n 2. Compute series approximations for (z t x k 0 0 x n kn Ω) σ for each basis element, to some p-adic and z-adic precision 3. Apply reduction algorithm to reduce each series back to basis elements 4. This yields matrix of absolute Frobenius; take product of conjugates to obtain matrix of q-th Frobenius 5. Characteristic polynomial is P(T ) to some p-adic precision

15 AKR algorithm AKR did not analyse complexity. Running time behaves at least like p n, because algorithm works with dense polynomials like in n variables. f p f σ

16 New algorithm First modification: use a sparse series expansion. (z t ) σ = z ( ) t + j 1 pt z pj ( f p t 1 f σ ) j j 0 N 1 ( ) t + j 1 z pt z pj ( f p t 1 f σ ) j (mod p N ) j=0 N 1 = z pt = N 1 k=0 j=0 ( t + j 1 t 1 C t,k z p(k+t) f kσ ) (1 z p f σ ) j Number of terms does not depend on p! [ C t,k = ( 1) k N 1 j=k ( t+j 1 )( j t 1 k) ]

17 New algorithm Thus (z t x k 0 0 x n kn Ω) σ is approximated by a sum of terms of the form z pm x pr xn prn 1 Ω, i.e. on a lattice with distance p between terms. Number of terms is about O((Nd) n ). To recover zeta function, need N = O(d n a) (?).

18 New algorithm Second modification: use controlled reduction. Consider a differential z m+1 x u F Ω, where deg F = α. (Multi-index notation: x u = x u 0 0 x un n.) Write F = G i ( i f ) for polynomials Gi, then z m+1 x u F Ω zm m where deg H = α d + n + 1. i (x u G i ) Ω = zm m i x u x 0 x n HΩ,

19 New algorithm Provided that d > n, we can rewrite this as (for example) z m m where deg F = α. x u x0 d n (x0 d n 1 H)Ω = zm x 1 x n m x u F Ω, (Result is slightly different if some u i 0, need to reduce in a different direction, but basically same idea.) Now iterate this process with F and u ; number of terms remains bounded as m decreases. Already this technique reduces complexity from p n to p 1! If d n, unclear how to control growth of number of terms.

20 New algorithm Third modification: use accelerated reduction. Consider the reduction described above, i.e. z m+1 x u F Ω = zm m x u F Ω. Let F and F denote the vectors of coefficients. Then F = S(m)F where S(m) is a matrix of linear polynomials in m over Z q. To iterate the reduction, want to compute the matrix product S(m p + 1) S(m 1)S(m).

21 New algorithm Fortunately there is a fast algorithm for computing such products! Let S(m) be an r r matrix of linear polynomials over a ring R. Computing S(k) S(1)S(0) naively requires O(kr ω ) ring operations (ω 3 is exponent of matrix multiplication). Better algorithm of Chudnovsky Chudnovsky (1988), using fast polynomial arithmetic, obtains ring operations. O(k 0.5+ɛ r ω ) (See also improvements by Bostan Gaudry Schost (2004).)

22 New algorithm Summary of new algorithm: 1. Compute a basis for HdR n 2. Compute sparse series approximations for (z t x k 0 0 x n kn Ω) σ for each basis element 3. Compute reduction matrices moving between adjacent lattice points 4. Apply reduction matrices to terms from step 2 to compute absolute Frobenius matrix; get P(T ) as before. For large p, all the work is in step 3.

23 Open questions 1. Can the algorithm be made more practical? countless opportunities for optimisation implementation grunt work 2. Can (d n a) n be reduced to d n a? maybe combine with deformation techniques? (can we deform in p 0.5 time?) 3. Can we drop the condition d > n? would have expected d n to be easier! 4. What about smooth affine varieties? 5. Can we drop smoothness condition? 6. Can we do better than p 0.5? does anyone know how to compute (p 1)! mod p 2 faster than O(p 0.5+ɛ )?

24 Thank you!