Effective Convergence Bounds for Frobenius Structures on Connections

Effective Convergence Bounds for Frobenius Structures on Connections

KIRANS. KEDLAYA(*) - JANTUITMAN(**)

ABSTRACT- Consider a meromorphic connection onP¹over ap-adic field. Inmany cases, such as those arising from Picard-Fuchs equations or Gauss-Manin con- nections, this connection admits a Frobenius structure defined over a suitable rigid analytic subspace. We give an effective convergence bound for this Fro- benius structure by studying the effect of changing the Frobenius lift. We also give an example indicating that our bound is optimal.

1. Introduction

In recent years, much work has gone into usingp-adic cohomology as an effective tool for numerical computation of zeta functions (and some re- lated quantities) of algebraic varieties over finite fields. One important technique in this field is thedeformation methodof Lauder, inwhich one computes the zeta function of a variety by fitting it into a one-parameter family of varieties, constructing the associated Picard-Fuchs equation or Gauss-Manin connection, then exploiting the existence of a Frobenius structure on this differential equation to reduce the problem to another member of the family (the initial condition, so to speak). Another important technique is thefibration method, also introduced by Lauder, in which one uses similar techniques to compute the zeta function of the total space of a one-parameter family, again starting from a single fiber. See [4, 5] for further discussion.

(*) Indirizzo dell'A.: Department of Mathematics, University of California, SanDiego, 9500 GilmanDrive #0112, La Jolla, CA 92093-0112, USA.

E-mail: kedlaya@ucsd.edu

(**) Indirizzo dell'A.: Mathematical Institute, University of Oxford, 24-29 St.

Giles', Oxford OX1 3LB. UK.

E-mail: tuitman@maths.ox.ac.uk

To execute the deformationand fibrationmethods inpractice, it is necessary to have not just the existence of a Frobenius structure, but explicit bounds on its convergence within a given residue disc. Concretely, these bounds are needed to enable the reconstruction of a rational function from a power series expansion, by bounding the degrees of its zero and pole divisors. One can often obtain crude bounds by direct calculations, but it is essential to have more accurate bounds in order to limit the required intermediate precision needed to achieve a final result to a given level of accuracy.

A technique for obtaining accurate bounds has been suggested by the first author in the preprint [3], under the assumptions (satisfied in many cases in practice) that the differential equation has at most one singularity in any residue disc, and that the exponents of the local monodromy at such a singularity are p-adically integral. The idea is to exploit the parallel transport of Frobenius structures between two choices of a Frobenius lift, to reduce the questionof convergence withina givenresidue disc to the same question with the Frobenius lift centered around the singularity in the disc, a questionwhich canbe solved rather easily.

However, the bound given in [3, Theorem 6.5.10] is not best possible. A stronger bound was claimed in the original (2008) manuscript of [3], but the second author discovered that the proof was incomplete, as it relied on some unjustified assertions about the convergence of solutions of p-adic differential equations. In this paper, we give a corrected version of the original argument, thus giving a stronger version of [3, Theorem 6.5.10].

We also provide a numerical example which indicates that the resulting bound is sharp.

2. The theorem

We first introduce some notation and terminology.

Letpdenote a prime,na positive integer, andFqthe finite field with qˆpⁿ elements. We write Qq for the unique unramified extension of degree n of the field of p-adic numbers Qp, an d Zq for the ring of in- tegers ofQ_q. LetUbe anopendense subscheme ofP¹_Qq with nonempty complementZ. Suppose thatE is a vector bundle onU equipped with a connectionr, an d lettdenote some coordinate on P¹_Qq.

We writesfor the standardp-th power Frobenius lift onP¹_Qqthat is, the (semilinear) map that lifts thep-th power Frobenius map onP¹_Fqand satisfies s(t)ˆt^p.

Let V denote the rigid analytic subspace ofP¹_Qq which is the comple- ment of the union of the open disks of radius 1 around the points ofZ, an d O^y(U) the ring of functions that converge on some strict neighbourhood of V. A Frobenius structure on E with respect to s is anisomorphism F :sE ! E of vector bundles with connection defined on some strict neighbourhood ofV.

We fix a basis [v1;. . .;vr] of sections of E on U, define matrices N2Mr(O(U)) andF2Mr(Oy(U)) such that

rv_jˆX^r

iˆ1

N_ijv_idt;

FvjˆX^r

iˆ1

Fijvi;

and call these the matrices ofrandF. Note, however, thatrandFare not O(U)- andOy(U)-linear, respectively. Instead,rsatisfies the Leibniz rule, andF iss-semilinear as a map fromEto itself.

SinceFis a morphism of vector bundles with connection, it is horizontal with respect to r. This implies that the matrices N and F satisfy the differential equation

NF‡dF

dt ˆds(t)

dt Fs(N)ˆpt^{p 1}Fs(N):

…1†

Now letzbe a geometric point ofZ, and suppose that the entries of Nhave at most a simple pole atz. Whenris a Gauss-Manin connection, by the regularity theorem we canalways choose the basis [v1;. . .;vr] so that this is the case (where the choice will in general depend onz). The exponents of r at z with respect to [v₁;. . .;v_r] are defined as the eigenvalues of the matrix (t z)Nj_tˆz. When r is a Gauss-Manin con- nection or admits a Frobenius structure, these are known to be rational numbers.

Letj:jdenote the norm onOy(U) induced by the supremum norm onV, andv_p(:) the corresponding discrete valuation, so thatj:j ˆp ^vp(:). Extend both of these to M_r(Oy(U)) inthe usual way, i.e., as the maximum and minimum over the entries, respectively.

THEOREM2.1. Let z be an unramified geometric point of Z, and as- sume that Z does not contain anyother points with the same reduction modulo p. Suppose that[v₁;. . .;v_r]is a basis ofEwith respect to which the matrix N ofrhas at most a simple pole at z, and the exponentsfl₁;. . .;l_rg

of r at z are contained inQ\Zp. Assume thatE admits a Frobenius structureFwith respect tos, and letFbe the matrix ofFwith respect to the basis[v1;. . .;vr]. For i2N, put

f(i)ˆmaxf(vp(F)‡vp(F ¹))dlog_p(i)e;

(r 1)v_p(N)‡(v_p(F)‡v_p(F ¹))blog_p(i)cg;

and define

cˆ 0 if v_p(N)0

minf0;i‡f(i):i2Ng if vp(N)50:

(

For m2N, put

g(m)ˆmaxfi2Nji‡v_p(F)‡c‡f(i)5mg;

and define

a1ˆ b pmin

i fl_ig ‡max

i fl_igc;

a2ˆ

0 if N does not have a pole at z;

0 if z2 f0;1g;

g(m) otherwise:

8>

<

>:

Then F is congruent modulo p^m to a matrix of rational functions of order greater than or equal to (a₁‡pa₂) at z (that is, the entries of the difference between the two matrices all have p-adic valuation at least m).

The proof proceeds inseveral steps. We start with the following lemma.

LEMMA2.2. Let Nˆ P¹

iˆ 1N_itⁱbe an rr matrix such that tN converges on the open unit disk and N ₁is a nilpotent matrix. LetFˆ P¹

iˆ 1F_itⁱbe an rr matrix that converges on some open annulus of outer radius1.

Suppose that N;Fsatisfyequation(1).ThenF_iˆ0for all i50, so thatF converges on the whole open unit disk.

PROOF. See [2, Proposition17:5:1]. p

When the exponents of N at 0 are not necessarily zero, this can be generalized as follows.

LEMMA2.3. Let Nˆ P¹

iˆ 1N_itⁱbe an rr matrix such that tN converges on the open unit disk and the eigenvalues l₁;. . .;l_r of N ₁ are rational numbers with denominators coprime to p. LetFˆ P¹

iˆ 1Fitⁱ be an rr matrix that converges on some open annulus of outer radius1. Suppose that N;Fsatisfyequation(1).ThenF_iˆ0whenever

i5pmin

j fl_jg max

j fl_jg:

PROOF. First we may adjoint^1=k forkcoprime top(if necessary), to reduce to the case wherel₁;. . .;l_r2Z. Inthat case, by applying so-called shearing transformations, one can find an invertiblerrmatrixW over Qq(t) such that the matrix

N⁰ˆW ¹NW‡W ¹dW dt

still has (at most) a simple pole attˆ0, but now with all exponents equal to 0. Moreover, one can ensure that t^bW andt ^aW ¹ do not have a pole at tˆ0, foraˆmin_jfl_jgandbˆmax_jfl_jg. More details onthis canbe found in [3, Lemma 5.1.6]. If we change basis to the basis given by the columns of W, then

N!N⁰;

F!F⁰ˆW ¹Fs(W):

Now Lemma 2.2 canbe applied to the pairN⁰;F⁰, so thatF⁰_iˆ0 for alli50.

SinceFˆWF⁰s(W ¹), this implies thatF_iˆ0 for alli5pa b. p Recall that we have chosenthe standardp-th power Frobenius lifts.

However, we could just as well have chosena different lift. The following lemma allows one to change from one Frobenius lift to another.

LEMMA 2.4. Let D denote the differential operator on E defined by rvˆ Dvdt. Suppose thatEadmits a Frobenius structureF₁:s₁E ! E with respect to a Frobenius lifts1, and lets2be some other Frobenius lift.

ThenEalso admits a Frobenius structureF2:s₂E ! Ewith respect tos2, defined by

F2(v)ˆX¹

iˆ0

(s2(t) s1(t))ⁱF1 Dⁱ i! (v)

:

PROOF. See [2, Proposition17:3:1]. p

Finally, we need a bound on the matrices of the differential opera- tors Dⁱ

i! that appear inLemma 2.4.

LEMMA2.5. LetD⁽ⁱ⁾ be the matrix of the differential operatorDⁱ i! with respect to the basis[v1;. . .;vr], that is,

Dⁱ

i! v_kˆX^r

jˆ1

D⁽ⁱ⁾_jkv_j:

Then we have

v_p(D⁽ⁱ⁾)f(i);

where f(i)is defined as in Theorem2.1.

PROOF. Lethdenote a generic point of the disk of radius 1 aroundz.

One can verify that the Taylor series T( t‡h;v_j)ˆX¹

iˆ0

( t‡h)ⁱDⁱ(v_j) i!

defines a horizontal section of r that meetsv_j at the point h. Form the matrixMwhosej-th columnconsists of the expressionofT( t‡h;vj) in terms of the basis [v1;. . .;vr], thenexpandMˆP¹

iˆ0M_i(t h)ⁱ. Sin cerdoes not have any singularities in the open disk of radius 1 aroundh, it follows from [2, Theorem 18.3.3] that

minfv_p(M₀);. . .;v_p(M_i)g (v_p(F)‡v_p(F ¹))dlog_p(i)e;

and from [2, Remark 18.3.4] (withqˆp) that

minfvp(M0);. . .;vp(Mi)g (r 1)vp(N)‡(vp(F)‡vp(F ¹))blog_p(i)c:

SincejD⁽ⁱ⁾jattains its maximum ath, an d

Miˆ( 1)ⁱD⁽ⁱ⁾(h)‡(terms coming fromD^(j) withj5i);

we deduce the bound by induction oni. p

Now we finally get to the proof of Theorem 2.1.

PROOF OFTHEOREM2.1. We first note that in casezˆ0 orzˆ 1, the claim is clear from Lemma 2.3.

Suppose next that z is a point at which N has no pole, so that a1ˆa2ˆ0. If we use the Frobenius lifts⁰withs⁰(t z)ˆ(t z)^p, thenby Lemma 2.3 again (applied after translatingzto the origin), the Frobenius matrix F⁰ with respect to s⁰ is holomorphic at z. By Lemma 2.4 (with s₁ˆs⁰;s₂ˆs),Fis also holomorphic atz, proving the claim in this case.

Finally, suppose thatNdoes have a pole atz. Inthis case, Lemma 2.3 implies thatF⁰has order at least a1 atz. We may againuse Lemma 2.4 (with s1 ˆs⁰;s2ˆs) to convert back to the original Frobenius lift; this gives us the identity

FˆX¹

iˆ0

pⁱuⁱF⁰s⁰(D⁽ⁱ⁾);

wherepuˆ(t z)^p‡s(z) t^p(withv_p(u)0), andD⁽ⁱ⁾ againdenotes the matrix of the differential operatorDⁱ

i! with respect to the basis [v₁;. . .;v_r].

In this identity, the summand at indexihas order at least a₁ piatz, an d p-adic valuationat leasti‡v_p(F⁰)‡f(i) by Lemma 2.5. This will give the desired bound once we check that v_p(F⁰)v_p(F)‡c. To see this, apply Lemma 2.4 withs1ands2interchanged to obtain

F⁰ˆX¹

iˆ0

pⁱ

i!( u)ⁱFs(i!D⁽ⁱ⁾):

Ifvp(N)50, we get the claim by invoking Lemma 2.5 again; ifvp(N)0, we instead note thatv_p(pⁱ=i!) an dv_p(i!D⁽ⁱ⁾) are both nonnegative. p The following corollary is oftenuseful whenthe matrix N of r with respect to some basis does not have a simple pole atz.

COROLLARY 2.6. Suppose that [v1;. . .; vr] is a basis for E as in Theorem 2.1, and let [w1;. . .; wr] be another basis for E, such that vj ˆP^r

iˆ1Wijwj withW2Mr(Qq(t)). Then the matrixF⁰ ofF with respect to[w1;. . .; wr]is congruent modulop^{m‡vp(W)‡vp(W 1)}to a matrix of rational functions of order greater than or equal to

(a₁‡pa₂(m))‡ord_z(W)‡pord_z(W ¹) at z.

PROOF. The matrixF⁰satisfies

F⁰ˆWFs(W) ¹: p

REMARK2.7. Insome special cases Theorem 2.1 canstill be improved a little.

1. Ifs(z)ˆz^p(such azis called aTeichmuÈller lift), thens(t) s⁰(t) is divisible byt zinthe proof of Theorem 2.1. So whenwe apply Lemma 2.4, some cancellation occurs, and modulop^m the matrixFhas order greater thanor equal to (a₁‡(p 1)a₂(m)) atz.

2. Suppose thatz6ˆ0;1. If we denote the residue matrix (t z)Nj_tˆz ofNatzbyR_z, and the identity matrix byI, thenthe leading term inthe Laurent series expansion of the matrixD⁽ⁱ⁾ ofDⁱ

i! at (t z) is givenby

(Rz (i 1)I). . .(Rz I)Rz (t z) ⁱ

i! :

Inmany cases there exists S2GLr(Qq) such that S ¹RzS is diagonal.

Writingl₁;. . .;l_rfor the entries on the diagonal, the leading term can then be writtenas

S

l1(l1 1). . .(l1 (i 1))

i! 0

.. .

0 lr(lr 1). . .(lr (i 1))

i!

0 BB BB

@

1 CC CC

AS ¹(t z) ⁱ:

However, sincel₁;. . .;l_r2Q\Z_pby assumption, the matrix in the middle is easily seento have entries inQ\Z_pas well. This means that the val- uation of the leading term in the the Laurent series expansion ofD⁽ⁱ⁾atzis bounded byvp(S)‡vp(S ¹). Now whenwe apply Lemma 2.4, we see that if g(m)‡v_p(F)‡c‡v_p(S)‡v_p(S ¹)m, thenmodulop^mthe matrixFhas order greater thanor equal to (a₁‡p(a₂(m) 1)) atz. This is related to the improvement upon [2, Theorem 18.2.1] given by the theorem of Dwork and Robba on which it is based [1].

3. An example: a family of elliptic curves

We consider the family given by the affine equation y²ˆx³‡1‡(t‡1)(x²‡x);

The closure of the zero locus of this equationinP²_QP¹_Qdefines a family X=Uof elliptic curves overUˆP¹_Q f 2;2g.

The relative algebraic de Rham cohomology H¹_dR(X=U) is a vector bundle onU, and carries a natural Gauss-Manin connectionr. Moreover, H¹_dR(X=U) is of rank 2, and a basis is given by

dx y ;xdx

y

:

Let p be anodd prime number. The space H¹_dR(X=U) equipped with r coincides with the relative rigid cohomologyH¹_rig(Xp=Up) of the reduction Xp=UpofX=Umodulop, and therefore it admits a Frobenius structureF with respect to the standard liftsof thep-th power Frobenius. LetNandF be the matrices ofrandFwith respect to the above basis, respectively. We compute

Nˆ 1 t² 4

t 2

1 2

t 2‡3

2 1

2 t 2‡1

2 0

BB

@

1 CC A:

It is known thatvp(F)ˆ0, andvp(F ¹)ˆ 1 inthis case, and clearly vp(N)ˆ0, so inTheorem 2.1 we have

g(m)ˆmaxfi2Nji blog_p(i)c5mg:

3.1 ± zˆ2

At zˆ2 the exponents are f 1=4;1=4g. The residue matrix R2 is di- agonalizable by integral matrices forp6ˆ5. So by remark 2.7, forp6ˆ5 the bound from Theorem 2.1 for the order ofFmodulop^mcanbe improved to

p‡1 4

‡p(g(m) 1)

, while forpˆ5 it remains (1‡5g(m)).

Experimentally, we find that for pˆ3 the order is boun ded by 1 3(m 1), for pˆ5 it is boun ded by 1 5(m 1), and forpˆ7 it is bounded by 2 7(m 1), all for m up to 250 an d with equality for many m.

3.2 ± zˆ 2

Atzˆ 2 the exponents aref0;0g, so the bound from Theorem 2.1 for the order ofFmodulop^mis givenby pg(m).

Experimentally, we find that for pˆ3 the bound is sharp for mˆ1;2;3;6;8;17;25;52;78;159;239, for pˆ5 for mˆ1;2;3;4;5;10, 15;20;24;49;74;99;123;248, and for pˆ7 for mˆ1;2;3;4;5;6;7;14, 21;28;35;42;48;97;146;195;244, all formup to 250.

Acknowledgments. Kedlaya was supported by NSF (CAREER grant DMS-0545904, grant DMS-1101343), DARPA (grant HR0011-09-1-0048), MIT (NEC Fund, Cecil and Ida Green professorship), and UCSD (Stefan E. Warschawski professorship). Tuitmanwas supported by the European Research Council (grant 204083).

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Manoscritto pervenuto in redazione il 15 novembre 2011.