Computing zeta functions in families of Ca,b curves using

Weil ConjecturesMonsky-Washnitzer Cohomology

Ca,b curvesRelative Cohomology and Deformation

Computing zeta functions in families of Ca,b

curves using deformation

Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren

December 2007 - Berlin

Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation



Weil Conjectures

Monsky-Washnitzer Cohomology

Ca,b curves

Relative Cohomology and Deformation




The Zeta Function and Weil Conjectures

Let C be smooth projective curve over Fq, then zeta function ofC is

Z (t) = Z (C; t) = exp

( ∞∑r=1

Nrt r

r

)with Nr the number of points on C with coordinates in Fqr .Weil Conjectures:

I Z (t) is rational function over Z: P(t)(1−t)(1−qt)

I P(t) =∏2g

i=1(1− αi t) with g genus of C and |αi | =√

q

I P(t) =∑2g

i=0 ai t i with a0 = 1, a2g = qg and ag+i = qiag−i

I Nr = qr + 1−∑2g

i=1 αri and P(1) is the order of Jac(C/Fq)




Computational Approaches

I l-adic compute zeta mod small primes l + CRT.I Need explicit description of l-torsion of abelian varietyI Practical for genus 1 and somewhat for genus 2 curves

(Schoof, Pila, A-H, . . . )

I p-adic compute zeta mod high power of pI Canonical Lift / AGM:

I Ordinary abelian varieties admitting lift of FrobeniusI Elliptic curves over Fpn : Satoh, Mestre, Harley, . . .I Hyperelliptic curves over F2n : Mestre, Lercier-Lubicz, . . .

I p-adic Cohomology




Algebraic de Rham Cohomology

I Let A be a ring, e.g. the coordinate ring of a curveI The module of Kaher differentials D1(A) isI Generated over A by symbols da with a ∈ A with rules

d(a + b) = da + db

d(a · b) = adb + bda

I Elements of dA are called exact




Algebraic de Rham Cohomology

I X smooth affine curve over field K with coordinate ring

A = K[x , y ]/(f (x , y))

I Since f (x , y) = 0 get ( ∂f∂x dx + ∂f

∂y dy) = 0, so

D1(A) =(A dx + A dy)

(A( ∂f∂x dx + ∂f

∂y dy))

I First algebraic de Rham cohomology group is

H1DR(A) =

D1(A)

dA





I C smooth affine curve over Fq with coordinate ring

A = Fq[x , y ]/(C(x , y))

I Let Qq be unramified extension of Qp with valuation ring Zq

I Lift C(x , y) to C(x , y) ∈ Zq[x , y ]

I Dagger ring A† of A := Zq[x , y ]/(C(x , y)) is

A† := Zq〈x , y〉†/(C(x , y)) ,

I Zq〈x , y〉† consists of power series∑

ri,jx iy j ∈ Zq[[x , y ]]

∃δ, ε ∈ R, ε > 0,∀(i , j) : ordpri,j ≥ ε(i + j) + δ.





I Note: Zq〈x , y〉† is closed under integrationI Monsky-Washnitzer cohomology:

D1(A†)d(A†)

⊗Zq Qq,

I Lefschetz-fixed point theorem: exists a Zq-algebraendomorphism Fq : A† → A† such that

ZC(T ) =det(

I− qF∗−1q

∣∣∣ H1MW (A/Qq)

)1− qT

.

I Since q = pn, have F = Fσn−1

p · Fσn−2

p · · ·Fσp · Fp.




Ca,b curves

I Ca,b curve C over finite field Fq,

C(x , y) = ya + cb,0xb +∑

ai+bj<ab

ci,jxiy j (cb,0 6= 0).

I Absolutely irreducible and genus is g = (a−1)(b−1)2 .

I Unique degree 1 place Q at infinity and vQ(x) = −a,vQ(y) = −b.

I Various subclasses of Ca,b curves:I Hyperelliptic curves: a = 2 and b = 2g + 1I Superelliptic curves: ci,j = 0 for j > 0




Ca,b curves - Monsky-Washnitzer

I The affine curve C has coordinate ring A := Fq[x , y ]/(C).I Take any lift C(x , y) with same Newton polygon as C(x , y)

I Let A† be the dagger ring of A := Zq[x , y ]/(C)

I Elements of A† can be represented as∑a−1

l=0∑+∞

k=0 ak ,lxky l

and the valuation of ak ,l grows linearly with k .

I Basis for H1(A/Qq)

xky l dx for k = 0, . . . , b − 2 and l = 1, . . . , a − 1

I Reduction formulae to express any differential on this basis




Ca,b curves - Reduction Formulae

I Can always exchange dy-differentials into dx-differentialsby partial integration and differentiation

I Note: x r ysdx for 0 < s < a totally ordered by pole atinfinity for r ≥ b − 1

I The differential

ωr ,s = x r−(b−1)ysdC−d

x r−(b−1)

aa + s

ya+s +∑

ai+bj<ab

jci,j

s + jx iys+j

is exact and has pole order determined by λx r ysdx andλ 6= 0




Ca,b curves - Frobenius on A†

I The necessary conditions on the Frobenius σ on A† are

xσ ≡ xp mod p and yσ ≡ yp mod p and Cσ(xσ, yσ) = 0

I Main idea: lift Frobenius on x and y simultaneously suchthat denominator in the Newton iteration is invertible in A†.

I Let Z ∈ A† such that xσ = xp + αZ and yσ = yp + βZ , then

Cσ(xσ, yσ) = Cσ(xp + αZ , yp + βZ ) = 0 and Z ≡ 0 mod p

I Since C non-singular, then can compute α, β, γ ∈ A with

1 = α

(∂C∂x

)p

+ β

(∂C∂y

)p

+ γC




Ca,b-curves - Zeta Function

I The action of Fp on a differential form xky ldx is given by

F∗p (xky l dx) ≡ (xσ)k (yσ)l dxσ.

I Substituting power series for xσ and yσ, we can writeF∗

p (xky ldx) on basis of H1(A/Qq) using the reductionformulae.

I This gives matrix Fp which is an approximation of theaction of F∗

p on H1(A/Qq).

I The polynomial χ(t) := t2gP(1/t) can then beapproximated by the characteristic polynomial ofFpFσ

p · · ·Fσn−1

p .




Relative CohomologyI C(x , y , t) ∈ Fq[t ][x , y ] family of smooth curves over Spec S

with S = Fq[t , r(t)−1].

t0, r(t0) 6= 0

t1, r(t1) = 0

Spec A

Spec At0

SpecS




Relative Cohomology

I Let A = S[x , y ]/(C(x , y , t)) and for all t0 ∈ Fq withr(t0) 6= 0 write At0

= A/(t − t0)

I Aim: describe how the action of Frobenius onH1

MW (At0/Qq) alters as t0 varies.

I Let C(x , y , t) and r(t) be lifts of C(x , y , t) and r(t)I Define S† = Zq〈t , r(t)−1〉† = Zq〈t , z〉†/(zr(t)− 1) along

with the S†-module

A† =Zq〈t , r(t)−1, x , y〉†

(C(x , y , t))




Relative Cohomology

I Construction of relative MW cohomology group

H1MW (A/S†Qq

) =D1

t (A†)dt(A†)

⊗Zq Qq

I With dt exterior derivation on A† over S†, i.e. no derivationwith respect to t (t considered constant)

I Can define Zq-algebra endomorphism Fq on A† that liftsthe Frobenius action Fq on A

I Action on t can be chosen Fq(t) = tq




Relative Cohomology

I Fq induces map F∗q on H1

MW (A/S†Qq)

I Let t0 ∈ Fq be a non-zero of r(t) and let t0 ∈ Zq beTeichmuller lift, i.e. unique root of X q − X ∈ Zq[X ] thatreduces to t0 mod p.

I H1MW (At0

/Qq) equals H1MW (A/S†Qq

)/(t − t0)

I F∗q induces map on H1

MW (At0/Qq) equal to Frobenius

action of before




Deformation

I Looked at action of Frobenius on all fibers at once, but nothow the action on different fibers is related!

I Need to take into account derivation with respect to tI Construct MW-complex on surface

0 → A† d→ D1(A†) d→ D2(A†) → 0

I Gauss-Manin connection

∇ : H1MW (A/S†Qq

) → H1MW (A/S†Qq

) : ω 7→ ∇(ω)




Gauss-Manin Connection

I Gauss-Manin connection

∇ : H1MW (A/S†Qq

) → H1MW (A/S†Qq

) : ω 7→ ∇(ω)

I Let ω be represented by ω ∈ D1(A†), then can rewrite

d(ω) = ϕ ∧ dt

I ϕ modulo dt and dt(A†) then is ∇(ω)




Gauss-Manin Connection

I Theorem: ∇ ◦ F∗q = qtq−1 ◦ F∗

q ◦ ∇I Proof: commutativity of diagram

D1(A†)d- D2(A†)

D1(A†)

F∗q

?d- D2(A†).

F∗q

?




Differential Equation

I Let s1, . . . , sd be an S†Qq-basis of H1

MW (A/S†Qq)

I Let F , N be (d × d)-matrices with entries in S†Qqsuch that

F∗q (sj) =

d∑i=1

Fi,jsi , ∇(sj) =d∑

i=1

Ni,jsi

for j = 1, . . . , d .I Theorem of Gauss-Manin connection then gives rise to

first-order differential equation

N · F +ddt

F = qtq−1 · F · N(tq).




Differential Equation

I Similar differential equation for matrix Fp

N · Fp +ddt

Fp = ptp−1 · Fp · Nσ(tp)

I Solve this starting from matrix Fp (t0) of easy fiber

I Note: Fp matrix over S†Qqso overconvergent

I Matrix Fp of different fiber then simply is Fp (ti)




Algorithm in Practice

I Two applications:I compute zeta function of given Ca,b curveI generate Ca,b curve with nearly prime Jacobian

I Given curve C1(x , y) embed this in family

C(x , y , t) = tC1(x , y) + (1− t)C0(x , y)

with C0 easy fiber:I C0(x , y) = ya + xb + 1 for p - a, bI C0(x , y) = ya + xb + y for p | bI C0(x , y) = ya + xb + x for p | a

I Computing Fp(0) easy since can use superellipticalgorithm




Example: Elliptic Curves

I Family: C(x , y , t) = y3 − x3 − x − t over F5n

I Basis of H1MW (A/S†Qq

) is {ydx , xydx}I Matrix Fp(0) modulo 520(

60982419361512 00 83213137279115

)I Connection matrix N over Q is(

5t6(t2+4/27)

−79(t2+4/27)

527(t2+4/27)

7t6(t2+4/27)

)




Example: Elliptic Curves

I Solve diff equation N · Fp + ddt Fp = ptp−1 · Fp · Nσ(tp)

I Giving matrix over S†Qq= 〈t , 1/(t2 + 4/27)〉

I Frobenius matrix of any fiber t i simply evaluate Fp at tiI Note: t i can be in extension field of Fp

I Example: evaluate at 1 given Fp(1) of y2 − x3 − x − 1(9076222447517 76647025390324

26756651910595 15618474818105

)




Results of quick hack in Magma

equation t ∈ Fpn genus g precomp time/curveY 3 + X 4 + (t + 1)XY + 1 259 3 553s 14.5sY 3 + X 5 + X 2 + t + 1 243 4 135s 6.5sY 3 + X 4 + (t + 1)XY + 1 337 3 1064s 13sY 3 + X 5 + XY + tY + 1 329 4 4128s 22sY 3 − X 4 + tX 2 + t − 1 523 3 30.5s 2sY 3 − X 5 − X 2 + tX − 1 519 4 837s 20sY 3 + X 4 + tX − 1 5200 3 515s 538s

I Ref: C-implementation of Kedlaya version for first 2 linesgives 5000 and 7000 seconds per curve




Conclusion

I Deformation very useful for curves tooI Easier to use than extending Kedlaya to more general

classesI Also works for non-degenerate curvesI Faster than Kedlaya both for random curves and certainly

to generate Ca,b curves with nearly prime order


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