Frobenius polynomials for Calabi-Yau equations
aa r X i v : . [ m a t h . AG ] S e p Frobenius polynomials for Calabi-Yauequations
Kira Samol and Duco van Straten15 September 2008
Abstract
We describe a variation of Dwork’ s unit-root method to deter-mine the degree four Frobenius polynomial for members of a 1-modulusCalabi-Yau family over P in terms of the holomorphic period near apoint of maximal unipotent monodromy. The method is illustrated ona couple of examples from the list [3]. For singular points we find thatthe Frobenius polynomial splits in a product of two linear factors anda quadratic part 1 − a p T + p T . We identify weight four modularforms which reproduce the a p as Fourier coefficients. Given a projective morphism f : X −→ P with smooth generic n − R n − f ∗ ( Q X ) restricts to a Q -local system H overthe smooth locus S ⊂ P of f and hence determines, after the choice of abase-point s ∈ S , a monodromy representation π ( S, s ) −→ Aut ( H s ). Thelocal system H carries a non-degenerate ( − n − -symmetric pairing < − , − > : H ⊗ H −→ Q S induced by the intersection form in the fibres. Hence we can identify H withits dual and the monodromy representation lands in a symplectic ( n − n − H underlies a variationof Hodge structures (VHS), polarised by < − , − > , see [20].We call a sub-VHS L ⊂ H a CY ( n ) -local system if the local monodromyaround 0 ∈ P \ S is unipotent and consists of a single Jordan block of size n .Hence, L is irreducible of rank n and the non-vanishing sub-quotients Gr W k ( k = 0 , , . . . , n −
1) of the monodromy weight filtration all have dimension1
K. Samol and D. van Straten equal to one. The Hodge filtration F · of the limiting mixed Hodge structureat 0 is opposite to the weight filtration [10],[19]. If ω is a section of thesmallest Hodge space F n − and γ a local section of L near 0, then the periodfunction f := < γ, ω > is holomorphic near 0 and satisfies a linear differential equation of order n ,called the associated Picard-Fuchs equation.We call a linear differential operator of order nP = ddx n + a n − ( x ) ddx n − + . . . + a ( x ) ∈ Q ( x )[ ddx ]a CY(n)-operator if1. P has maximal unipotent monodromy at 0 (MUM).2. P is self-dual in the sense that P = ( − n exp( − n Z a n − ) ◦ P ∗ ◦ exp( 2 n Z a n − ) , where ◦ means the composition of differential operators.3. P has a convergent power series solution f ( x ) ∈ Z [[ x ]] with f (0) = 1.The first condition implies that the operator P is irreducible and can (afterleft multiplication by x n ) be written in the form θ n + xP ( θ ) + x P ( θ ) + . . . + x d P d ( θ ) , where θ := x ddx and P i ( θ ) ∈ Q [ θ ] is a polynomial in θ . We remark thatexp( R a n − ) ∈ Q ( x ) precisely if the differential Galois group of P belongs to SL ( n ). In the second condition P ∗ is the formal adjoint of P . The conditionis equivalent to the condition that the transformed operator˜ P = exp( 1 n Z a n − ) ◦ P ◦ exp( − n Z a n − ) = ddx n + 0 ddx n − + . . . satisfies ˜ P = ( − n ˜ P ∗ which translates into ⌊ ( n − / ⌋ differential-polynomial conditions on thecoefficients a i . These express the conditions that the differential Galois groupof P is in the symplectic or orthogonal group. For n = 4 one finds thecondition of [2]: a = 12 a a − a + a ′ − a a − a ′′ robenius Polynomials Gal ( P ) = Sp (4) in general, see [7]. In [3] one finds a list with more than 350examples of such fourth order operators.Because of the MUM-condition, the solution f ( x ) from the third conditionis unique and conversely determines the operator P . As f is a G -function,the operator P is a G -operator and hence by a theorem of Katz is of fuchsiantype with rational exponents, see [4].A Picard-Fuchs operator that arises from a geometrical situation as sketchedabove will satisfy the first two conditions and the period function f will be aso called G-function, see [4]. It would therefore perhaps seem more natural torequire f to be a G -function. However, requiring integrality of the solutioncovers all interesting examples and helps fixing the coordinate x . In [2] for n = 4 further integrality properties for the mirror map and Yukawa couplingwere required.CY(2)-operators arise from families of elliptic curves, CY(3)-operators arisefrom families of K h = 1 that are studied in mirror symmetry,[8].Dwork and Bombieri have conjectured conversely that all G-operators comefrom geometry. So one may ask: is the local system of solutions Sol ( P ) of aCY(n)-operator of the form C ⊗ L , where L is a CY(n)-local system in theabove sense? When can one achieve L = H ? If L = H , can one find a family f : X −→ P with generic fibre a Calabi-Yau n − Z and consider the reductionof X −→ P modulo some prime number p . The object L ⊂ R n − f ∗ ( Q l ) ( l = p now defines an l -adic sheaf on P , lisse (that is, smooth) in some subset S . Inparticular, for each point s : Spec ( k ) −→ S one has an action of Gal ( k/k ) onthe stalk L s . Hence one obtains a Frobenius element F rob s ∈ Aut ( L s ) and P s ( T ) := det(1 − T.F rob s ) ∈ Z [ T ]determines a factor of the zeta function of the reduction X s mod p .To get a computational handle on these Frobenius polynomials it turns outto be useful to change to a de Rham-type description of the cohomology, [16].It was Dwork who realised early that there is a tight interaction between theGauss-Manin connection and the Frobenius operator. This leads in generalto a relation between periods and the zeta function and in 1958 he gave hisfamous p − adic analytic formula for the Frobenius polynomial in terms of a K. Samol and D. van Straten solution of the Picard-Fuchs differential equation for the Legendre family ofelliptic curves, which we will now review.The affine part of the Legendre family is given by X s : y = x ( x − x − s ) , where s = 0 ,
1. Over C , the relative de Rham cohomology H dR of the family isfree of rank 2, and the Hodge filtration Fil H dR is generated by the differential ω := dxy . Let ∇ be the Gauss-Manin connection on H dR . Then, ω satisfies the differ-ential equation s ( s − ω ′′ + (1 − s ) ω ′ − ω = 0 , where ω ′ = ∇ ( ω ). Let f be the unique solution in C [[ s ]] to the above differ-ential equation satisfying f ( s ) = 1. f is then given by the hypergeometricseries f ( s ) = F (cid:18) , , , s (cid:19) = ∞ X j =0 (cid:0) (cid:1) j j ! ! s j . Now let s ∈ F p a such that f ( p − / ( s ) = 0, where f ( p − / is the truncationof f up to degree ( p − /
2. Let ˆ s be the Teichm¨uller lifting of s to W ( F p a ).The formal power series h ( s ) := f ( s ) f ( s p )converges at ˆ s and can be evaluated there. If ǫ = ( − ( p − / , the element π := ǫ a f (ˆ s ) f (ˆ s p ) ...f (ˆ s p a − )is a reciprocal zero of the Frobenius polynomial, and the zeta function of X s is given by ζ ( X s , T ) = (1 − πT )(1 − p a /πT )(1 − T )(1 − p a T ) . Thus, Dwork found a way to derive a formula for the Frobenius polynomial,which does only depend (up to ǫ ) on the solution of the Picard-Fuchs differen-tial equation. The geometrical origin of ǫ lies in the geometry of the singularfibre X , which has a node with tangent cone x + y = 0, which splits over F p precisely when ǫ = 1.In this paper we will consider the following Question:
Given a CY(n)-operator P of f : X −→ P defined over Z , is robenius Polynomials P s ( T )?We describe a method to solve this problem for n = 4 (modulo “ ǫ ”) andillustrate the procedure on some non-trivial examples. We give a short introduction to the theory of Hodge F -crystals, which pro-vides a framework to formalise the interaction between the Gauss-Manin con-nection and the Frobenius operator. (see [22], [15], [9],[24]).Let k be a perfect field of characteristic p >
0, and let W ( k ) be the ring ofWitt vectors of k . Let A be the ring W ( k )[ z ][ g ( z ) − ], where g is a polynomialin z (which will be specified later according to the actual situation), and let A n be the ring A/p n +1 A . By A ∞ := lim ← A/p n +1 A , we denote the p − adiccompletion of A .Let σ be the absolute Frobenius on k , given by σ ( x ) = x p . Following [24], wedefine Definition 2.1
1. An F − crystal over W ( k ) is a free W ( k ) − module H offinite rank with a σ − linear endomorphism F : H → H such that F ⊗ Q p : H ⊗ Q p → H ⊗ Q p is an isomorphism. If F iself isan isomorphism, we call H a unit-root F − crystal.2. A Hodge F − crystal over W ( k ) is an F − crystal H equipped with a fil-tration by free W ( k ) − submodules H = Fil H ⊃ Fil H ⊃ ... ⊃ Fil N − H ⊃ Fil N H = 0 (called the Hodge filtration on H ) which satisfies F (Fil i H ) ⊂ p i H forall i . The Frobenius automorphism σ on k lifts canonically to an automorphism σ on W ( k ).There are different lifts of the Frobenius σ on A ∞ , which restrict to σ on W ( k ) and reduce to the p − th power map modulo p . Let φ : A ∞ → A ∞ besuch a lift of Frobenius. Definition 2.2 An F − crystal over A ∞ is a finitely generated free A ∞ − module H with an integrable and p − adically nilpotent connection ∇ : H → Ω A ∞ /W ( k ) ⊗ A H. K. Samol and D. van Straten such that for every lift φ : A ∞ → A ∞ of Frobenius, there exists a homomor-phism of A ∞ − modules F ( φ ) : φ ∗ H → H such that the square H ∇ / / F ( φ ) φ ∗ (cid:15) (cid:15) Ω A ∞ /W ( k ) ⊗ H φ ⊗ F ( φ ) φ ∗ (cid:15) (cid:15) H ∇ / / Ω A ∞ /W ( k ) ⊗ H is commutative, and such that F ( φ ) ⊗ Q p : φ ∗ H ⊗ Q p → H ⊗ Q p is anisomorphism. If F ( φ ) itself is an isomorphism, we call H a unit-root crystal. From now on, to simplify the notation, we set F := F ( φ ) φ ∗ . Definition 2.3
A divisible Hodge F − crystal H is an F − crystal H equippedwith a filtration by free A ∞ − submodules H = Fil H ⊃ Fil H ⊃ ... ⊃ Fil N − H ⊃ Fil N H (called the Hodge filtration on H ) which satisfies1. ∇ Fil i H ⊂ Ω A ∞ /W ( k ) ⊗ A ∞ Fil i − H F (Fil i H ) ⊂ p i H . Proposition 2.1
Let H be a divisible Hodge F − crystal where H/ Fil H isfree of rank one. Then ∧ H is a divisible Hodge F − crystal, with homomor-phism of A ∞ − modules p ∧ F : ∧ H → ∧ H and with Hodge filtration given by Fil i − ( ∧ H ) = i X k =0 Fil k H ∧ Fil i − k H for i ≥ .Proof: Since H/ Fil H is of rank one, Fil ∧ Fil = Fil ∧ Fil .Let a ∈ Fil k H and b ∈ Fil i − k H .Then, a ∧ b ∈ Fil i − ( ∧ H ) and1 p ∧ F ( a ∧ b ) = 1 p F a ∧ F b ∈ p p k H ∧ p i − k H = p i − ∧ H. robenius Polynomials i ≥ ∇ ( a ∧ b ) = ∇ ( a ) ∧ b + a ∧ ∇ ( b ) ∈ Ω A ∞ /W ( k ) ⊗ A ∞ Fil i − ( ∧ H ) . Let k ′ be a perfect field extension of k and let e : A → k ′ be a k − morphism.Let e ( z ) = α , let α be the Teichm¨uller lifting of α in W ( k ′ ) and let e : A ∞ → W ( k ′ ) be the W ( k ) − morphism with e ( z ) = α . By H α , we denote theTeichm¨uller representative H α := H ⊗ ( A ∞ ,e ) W ( k ′ ) of the crystal H at thepoint e , which is an F − crystal with corresponding map F α := e ∗ F . If H isa Hodge F − crystal, then so is H α .On W ( k ′ )[[ z − α ]], we put the natural connection ∇ and choose the lift ofFrobenius given by φ ( z ) = z p . Theorem 2.1 ([24], Theorem 2.1 or [15], Theorem 4.1)Let ¯ k be the algebraic closure of k , and let H be a divisible Hodge F − crystalover A ∞ .If H/ Fil H is of rank one and if for every k − morphism e : A → ¯ k with e ( z ) = α and α ∈ W (¯ k ) a Teichm¨uller lifting of α , H α contains a directfactor of rank one, transversal to Fil H α , which is fixed by the map inducedby F on H α , then there exists a unique unit-root F-subcrystal U of H suchthat H = U ⊕ Fil H as A ∞ modules.Suppose that over A ∞ , U is locally generated by u . Write F ( u ) = r ( z ) u for r ( z ) ∈ A ∗∞ . Then we have1. Let e : A → k ′ be a k − morphism to a perfect field extension k ′ of k with e ( z ) = α where u is defined. Let α be the Teichm¨uller lifting of α . Then there exists an f ∈ W ( k ′ )[[ z − α ]] such that v := f · u ∈ W ( k ′ )[[ z − α ]] ⊗ A ∞ H is horizontal with regard to ∇ and the quotient f /f φ is in fact the expansion of an element in A ∞ .2. There exists c ∈ W (¯ k ) such that c · v ∈ W (¯ k ) ⊗ W ( k ) H is fixed by F and r ( z ) = ( cf ) / ( cf ) φ . The fact that f /f φ ∈ A ∞ although f ∈ W ( k )[[ z − α ]] means that f /f φ is alocal expression of a “global” function. Although f itself does only convergein a neighbourhood of α , the global function expressed by the ratio f /f φ converges at any Teichm¨uller point in Spec( A ∞ ). Now let P be a CY(4)-operator. We assume that P is the Picard-Fuchsoperator on a rank 4 submodule H ⊂ H dR ( X/S ∞ ) for some family f : X → S ∞ of smooth CY-threefolds. K. Samol and D. van Straten
Let k be the finite field with p r elements. From now on, we have S ∞ =Spec( A ∞ ), where A = W ( k )[ z ][( zs ( z ) g ( z )) − ] for some polynomials g ( z ) and s ( z ). We assume that over the roots of s ( z ), the family becomes singular. Wewill specify the polynomial g ( z ) later (see section 2.4); it will be chosen in away such that over each Teichm¨uller point α ∈ S ∞ , the Frobenius polynomialon H α is of the form P := 1 + aT + bpT + ap T + p T . with four different reciprocal roots r , pr , p /r , p /r , where r and r are p − adic units. Hence, giving a formula for the polynomial P is equivalent to giving formulas for the p − adic units r and r .In general, if f : V → V is a homomorphism of vector spaces, then theeigenvalues of ∧ f : ∧ V → ∧ V are given by products ab , where a and b areeigenvalues of f corresponding to linearly independent eigenvectors.Let α ∈ S , and let α ∈ S ∞ be the Teichm¨uller lifting of α . By Proposition2.1, the Frobenius automorphism on each fiber ∧ H α of the crystal ∧ H is given by p ∧ F α , where F α is the Frobenius on H α ⊂ H dR ( X α ). Theeigenvalues of the relative Frobenius ( ∧ F α ) r on the fibres ∧ H α are of theform a α b α /p , where a α and b α are eigenvalues of the relative Frobenius F rα onthe corresponding fibre H α . Thus, if r is the unit root on a fibre H α , andˆ r is the unit root on the corresponding fibre ∧ H α , then the roots of theFrobenius polynomial det(1 − T F rα ) on H α are given by r , p ˆ r /r , p r / ˆ r , p /r . (1)We will give p − adic analytic formulas for the unit roots r and ˆ r . Let P be a CY(4)-operator. The differential equation P y = 0 can be writtenin the form y (4) + a y (3) + a y (2) + a y (1) + a y = 0 , where the coefficients a i satisfy the following relation: a = 12 a a − a + a ′ − a a ′ − a ′′ . (2) robenius Polynomials Proposition 2.2 (see [24]) Let P be a CY(4) differential operator and let ( H, ∇ ) be a Q ( z ) / Q differential module. Let ω ∈ H such that ∇ ω + a ∇ ω + a ∇ ω + a ∇ ω + a ω = 0 and let f ∈ Q [[ z ]] be a formal solution to the differential equation P y = 0 . If Y := exp (cid:0) / R a (cid:1) ∈ Q [[ z ]] , then the following element u ∈ H ⊗ Q [ z ] Q [[ z ]] is horizontal with regard to ∇ : u = Y [ f ∇ ( ω ) − f ′ ∇ ( ω ) + f ′′ ∇ ( ω ) − f ′′′ ω ] + ( Y a − Y ′ )[ f ∇ ( ω ) − f ′′ ω ]+ ( Y a − ( Y a ) ′ + Y ′′ )[ f ∇ ( ω ) − f ′ ω ] . (3) Proof:
See [24]. The proof is by direct computation, using (2).Now let Q be a CY(5)-operator. The differential equation Q y = 0 can bewritten in the form y (5) + b y (4) + b y (3) + b y (2) + b y (1) + b y = 0 . Proposition 2.3
The operator Q satisfies the second condition for CY(5) ofthe introduction, if and only if the coefficients b i ( z ) satisfy the relations b = 35 b b − b + 32 b ′ − b b ′ − b ′′ (4) and b = 12 b ′ − b b + 15 b b − b b ′′ + 25 b ′′′ b + 45 b ′′ b ′ + 16125 b ′ b (5)+ 1225 ( b ′ ) b − b ′′ b + 825 b b ′′ − b ′ b ′ − b b ′ − b ′′′ + 163125 b + 15 b ′′′′ − b b ′ b . Proof:
By direct calculation, for details we refer to [7].
Proposition 2.4
Let Q be a CY(5) differential operator and let ( H, ∇ ) be a Q ( z ) / Q differential module. Let η ∈ H such that ∇ η + b ∇ η + b ∇ η + b ∇ η + b ∇ η + b η = 0 and let F ∈ Q [[ z ]] be a formal solution to the differential equation Q y = 0 .If Y := exp (cid:0) / R b (cid:1) ∈ Q [[ z ]] , then the following element u ∈ H ⊗ Q [ z ] Q [[ z ]]0 K. Samol and D. van Straten is horizontal with regard to ∇ : u = Y [ F ∇ ( η ) − F ′ ∇ ( η ) + F ′′ ∇ ( η ) − F ′′′ ∇ ( η ) + F ′′′′ η ]+ ( Y b − Y ′ )[ F ∇ ( η ) − F ′ ∇ ( η ) − F ′′ ∇ ( η ) + F ′′′ η ]+ ( Y b − ( Y b ) ′ + Y ′′ )[ F ∇ ( η ) + F ′′ η ] + ( 43 (( Y b ) ′ − Y ′′ ) − αb ) F ′ ∇ ( η )+ ( 12 (( Y b ) ′ −
43 ((
Y b ) ′′ − Y ′′′ )))[ F ′ η + F ∇ ( η )]+ ( Y b −
12 ((
Y b ) ′ −
43 ((
Y b ) ′ − Y ′′′′ ))) F η, (6) Proof:
Applying the identities (4) and (6), one directly verifies that u satis-fies ∇ ( u ) = 0. Let p be a prime number. We say that a sequence ( c n ) n ∈ N satisfies the Dwork-congruences for p , if the associated sequence C ( n ) := c ( n ) /c ( ⌊ np ⌋ ) ∈ Z p satis-fies C ( n ) ≡ C ( n + mp s ) mod p s for all n, s ∈ N and m = { , , . . . , p − } and if c (0) = 1. We say that theDwork-congruences hold for a CY(n) differential operator P if the Dwork-congruences hold for the sequence ( c n ) n ∈ N of coefficients of the holomorphicsolution f ( z ) = ∞ X n =0 c n z n to the differential equation P y = 0 around z = 0. Dwork shows (see [12],Corollary 1. and 2.) that hypergeometric type numbers satisfy these Dworkcongruences for all p . Theorem 2.2 (see [12], Lemma 3.4.) Let y ( z ) = P n c n z n such that ( c n ) satisfies the Dwork congruences. Let D := { x ∈ Z p , | y ( p − ( x ) | = 1 } . Then,for all x ∈ D , y ( z ) y ( z p ) | z = x ≡ y ( p s − ( x ) y ( p s − − ( x p ) mod p s . This leads to an efficient evaluation of the left hand side at Teichm¨uller points.(Here y ( p s − ( z ) is the polynomial obtained from y ( z ) by truncation at z p s .)This crucial fact was used in all of our computations. robenius Polynomials Let P := P ( θ, z ) be a CY(4)-operator, where θ denotes the logarithmic deriva-tive z∂/∂z .As before, we assume that P is the Picard- Fuchs operator on a rank foursubmodule H ⊂ H dR ( X/S ∞ ) for a family f : X → S ∞ of smooth CY- three-folds.The rank 6= (cid:0) (cid:1) A ∞ − module ∧ H is a direct sum of an A ∞ − module G ofrank 5 and a rank 1 module. The rank 1 module is generated by a sectionthat corresponds to the pairing < − , − > and is horizontal with respect to ∇ .We construct a 5th order differential operator Q on the submodule G bychoosing Q to be the differential operator of minimal order such that forany two linearly independent solutions y ( z ) , y ( z ) of the differential equa-tion P y = 0, w := z (cid:12)(cid:12)(cid:12)(cid:12) y y y ′ y ′ (cid:12)(cid:12)(cid:12)(cid:12) is a solution of Q w = 0 . Proposition 2.5
The operator Q satisfies the first and the second conditionof CY(5).Proof: The statement that Q satisfies the first condition of CY(5) is thecontent of [2], Proposition 4. A direct computation shows that since P is a CY (4) − operator, the coefficients of Q satisfy the equations (6) and (4), sothe second condition of CY(5) holds.In all examples it was found that the operator Q also has an integral power se-ries solution, and thus satisfies the third condition of CY(5). For the moment,however, we are unable to prove this is general so we Conjecture 2.1
The differential operator Q , constructed from a CY(4)-operator P as above, satisfies the third condition of CY(5). So if Conjecture 2.1 holds true, the differential operator Q is a CY(5)-operator. Q can be expressed in terms of ∧ P ( θ, z ) as Q ( θ, z ) = ∧ P ( θ − , z ) . For the differential operators P and Q , we use the same notation with coef-ficients a i and b i as in section 2.2.2 K. Samol and D. van Straten
Proposition 2.6
Let Q be the CY(5)-operator constructed above, and let ω ∈ H such that ∇ ω + a ∇ ω + a ∇ ω + a ∇ ω + a ω = 0 . Then, the element η := zω ∧ ∇ ω ∈ G satisfies ∇ η + b ∇ η + b ∇ η + b ∇ η + b ∇ η + b η = 0 . Proof:
The proposition follows by a straightforward calculation, applying therelations between the coefficients a i of the CY(4)-operator P and the coeffi-cients b i of the CY(5)-operator Q listed in [1].It still remains to point out how to choose the polynomial g ( z ) in the defini-tion of the ring W ( k )[ z ][( zs ( z ) g ( z )) − ] to obtain divisible Hodge F − crystals H ⊂ H dR ( X/S ∞ ) and G ⊂ ∧ H which satisfy the conditions of Theorem 2.1.The following conjecture was crucial for the choice of the polynomial g ( z ): Conjecture 2.2
1. Let f be the solution of the differential equation P y = 0 around z = 0 with f (0) = 1 . If the coefficients c n in the expansion f ( z ) = ∞ X n =0 c n z n satisfy the Dwork congruences, then H satisfies the conditions of The-orem (2.1) if the polynomial g ( z ) in the definition of A ∞ is chosen as g ( z ) := f ( p − ( z ) .2. Let F ( z ) be the solution of the differential equation Q y = 0 around z = 0 with F (0) = 1 . If the coefficients d n in the expansion F ( z ) = ∞ X n =0 d n z n satisfy the Dwork congruences, then the sub- F - crystal G ⊂ ∧ H satis-fies the conditions of Theorem (2.1) if the polynomial g ( z ) in the defi-nition of A ∞ is chosen as g ( z ) := F ( p − ( z ) . According to the conjecture, it seems to be the right thing to choose g ( z ) = f ( p − ( z ) F ( p − ( z ). So from now on, we fix the ring A ∞ by A := W ( k )[ z ][( zs ( z ) f ( p − ( z ) F ( p − ( z )) − ] . robenius Polynomials z = α with f ( p − ( α ) = 0 mod p and F ( p − ( α ) = 0 mod p , in theexamples we considered, we were able to compute the Frobenius polynomialexplicitly.For each pair of CY(4) and CY(5) operators we treat in this paper, thefunctions Y = exp (cid:18) / Z a (cid:19) and Y = exp (cid:18) / Z b (cid:19) satisfy Y ∈ Q ( z ) and Y ∈ Q ( z ). Thus, in each of the examples we considered,the following proposition holds: Proposition 2.7
Let r ( z ) = f ( z ) f ( z p ) and ˆ r ( z ) = F ( z ) F ( z p ) . Assuming that Conjecture 2.2 holds, if | αs ( α ) f ( p − ( α ) F ( p − ( α ) | = 1 , thereexist constants ǫ and ǫ ∈ W (¯ k ) such that the p − adic units r ( α ) and ˆ r ( α ) determining the Frobenius polynomial on H α ⊂ H dR ( X α ) are given by r ( α ) = ( ǫ (1 − σ )4 ) ... + σ r − r ( α ) r ( α p ) ...r ( α p r − ) and ˆ r ( α ) = ( ǫ (1 − σ )5 ) σ + ... + σ r − ˆ r ( α )ˆ r ( α p ) .... ˆ r ( α p r − ) . If we assume furthermore that ( ǫ − σ ) σ + ... + σ r − = ( ǫ − σ ) σ + ... + σ r − = 1 , the p − adic units are given by r ( α ) = r ( α ) r ( α p ) ...r ( α p r − ) and ˆ r ( α ) = ˆ r ( α )ˆ r ( α p ) .... ˆ r ( α p r − ) . Proof:
There exists an ω ∈ H such that the horizontal section w.r.t. ∇ isgiven by formula (3), while on G , it is given by formula (6), where η = zω ∧∇ ω by Proposition 2.6. These sections u and u play the role of the section v = f · u in Theorem 2.1. Hence, the section u in the theorem is given by( f Y ) − u and ( F Y ) − u respectively, where Y = exp (cid:0) / R a (cid:1) ∈ Q ( z )and Y = exp (cid:0) / R b (cid:1) ∈ Q ( z ). Since (cid:18) Y ( z ) Y ( z p ) | z = α (cid:19) σ + ... + σ r − = 1 and (cid:18) Y ( z ) Y ( z p ) | z = α (cid:19) σ + ... + σ r − = 1 , K. Samol and D. van Straten by Theorem 2.1 there exist constants ǫ and ǫ ∈ W (¯ k ) (where ¯ k denotes thealgebraic closure of F p ) such that r ( α ) = ( ǫ (1 − σ )4 ) ... + σ r − r ( α ) r ( α p ) ...r ( α p r − )and ˆ r ( α ) = ( ǫ (1 − σ )5 ) σ + ... + σ r − ˆ r ( α )ˆ r ( α p ) .... ˆ r ( α p r − ) . Now we assume that the constants satisfy( ǫ − σ ) σ + ... + σ r − = ( ǫ − σ ) σ + ... + σ r − = 1 . (7)Then, the proposition follows. We will apply the method explained in the previous section to compute Frobe-nius polynomials for some special fourth order operators. These operatorsbelong to the list [3]. A typical example is operator 45 from that list: θ − x (2 θ + 1) (cid:0) θ + 7 θ + 2 (cid:1) − x (2 θ + 1) (2 θ + 3) This operator is a so-called
Hadamard product of two second order operators.
The
Hadamard product of two power series f ( x ) := P n a n x n and g ( x ) = P n b n x n is the power-series defined by the coefficient-wise product: f ∗ g ( x ) := X n a n b n x n It is a classical theorem, due to Hurwitz, that if f and g satisfy linear dif-ferential equations P and Q resp., then f ∗ g satisfies a linear differentialequation P ∗ Q . Only in very special cases, the Hadamard product of twoCY-operators will again be CY, but it is a general fact that if f and g satisfydifferential equations of geometrical origin , then so does f ∗ g . For a proof,we refer to [4]. Here we sketch the idea. The multiplication map m : C ∗ × C ∗ −→ C ∗ , ( s, t ) s.t can be compactified to a map µ : ^P × P −→ P robenius Polynomials , ∞ ) and ( ∞ ,
0) of P × P . Given two families X −→ P and Y −→ P over P , we define a new family X ∗ Y −→ P , asfollows. The cartesian product X × Y maps to P × P and can be pulled backto X ∗ Y over ^P × P . Via the map µ we obtain a family over P . If n resp. m is the fibre dimension of X −→ P resp. Y −→ P , then X ∗ Y −→ P hasfibre dimension n + m + 1. The local system H n + m +1 of X ∗ Y −→ P containsthe convolution of the local systems of X −→ P and Y −→ P . Note thatthe critical points of X ∗ Y −→ P are, apart from 0 and ∞ , the productsof the critical values of the factors. In down-to-earth terms, if X −→ P and Y −→ P are defined by say Laurent polynomials F ( x ) and G ( y ) resp., thenthe fibre of X ∗ Y −→ P over u is defined by the equations F ( x ) = s, G ( y ) = t, s.t = u If the period functions for X −→ P and Y −→ P are represented as f ( s ) = Z γ Res ( ωF ( x ) − s ) = X n a n s n g ( t ) = Z δ Res ( ηG ( y ) − t ) = X m b m t m then Z γ × δ × S ω ∧ η ∧ ds ∧ dt ( F ( x ) − s )( G ( y ) − t )( st − u ) = Z S X a n s n b m t m duu = X a n b n u n = f ( u ) ∗ g ( u )is a period of X ∗ Y −→ P .For example, if we apply this construction to the rational elliptic surfaces X = Y with singular fibres of Kodaira type I over 0 and I over ∞ and twofurther fibres of type I , we obtain a family X ∗ Y −→ P , with generic fibrea Calabi-Yau 3-fold with h = 1 and χ = 164. C Y (2) -operators
We will use Hadamard-products of some very special CY(2)-operators ap-pearing in [2] from which we also take the names. These operators all areassociated to extremal rational elliptic surfaces X −→ P with non-constantj-function. Such a surface has three or four singular fibres, [18]. The six caseswith three singular fibres fall into four isogeny-classes and each of these givesrise to a Picard-Fuchs operator of hypergeometric type (named A,B,C,D) andone obtained by performing a M¨obius transformation interchanging ∞ withthe singular point = 0 (named e,h,i,j).6 K. Samol and D. van Straten
Name Operator a n A θ − x (2 θ + 1) n )! n ! B θ − x (3 θ + 1)(3 θ + 2) (3 n )! n ! C θ − x (4 θ + 1)(4 θ + 3) (4 n )!(2 n )! n ! D θ − x (6 θ + 1)(6 θ + 5) (6 n )!(3 n )!(2 n )! n ! Name Operator a n e θ − x (32 θ + 32 θ + 12) + 256 x ( θ + 1) n P k ( − k (cid:0) − / k (cid:1)(cid:0) − / n − k (cid:1) h θ − x (54 θ + 54 θ + 21) + 729 x ( θ + 1) n P k ( − k (cid:0) − / k (cid:1)(cid:0) − / n − k (cid:1) i θ − x (128 θ + 128 θ + 52) + 4096 x ( θ + 1) n P k ( − k (cid:0) − / k (cid:1)(cid:0) − / n − k (cid:1) j θ − x (864 θ + 864 θ + 372) + 18664 x ( θ + 1) n P k ( − k (cid:0) − / k (cid:1)(cid:0) − / n − k (cid:1) The six cases with four singular fibres are the Beauville surfaces ([6]) andalso form four isogeny classes and lead to the six Zagier-operators, called(a,b,c,d,f,g).These are also of the form θ − x ( aθ + aθ + b ) − cx ( θ + 1) but now the discriminant 1 − ax − cx is not a square, so the operator hasfour singular points.Name Operator a n a θ − x (7 θ + 7 θ + 2) − x ( θ + 1) P k (cid:0) nk (cid:1) c θ − x (10 θ + 10 θ + 3) + 9 x ( θ + 1) P k (cid:0) nk (cid:1) (cid:0) kk (cid:1) g θ − x (17 θ + 17 θ + 6) + 72 x ( θ + 1) P i,j n − i ( − i (cid:0) ni (cid:1)(cid:0) ij (cid:1) d θ − x (12 θ + 12 θ + 4) + 32 x ( θ + 1) P k (cid:0) nk (cid:1)(cid:0) kk (cid:1)(cid:0) n − kn − k (cid:1) f θ − x (9 θ + 9 θ + 3) + 27 x ( θ + 1) P k ( − k n − k (cid:0) n k (cid:1) (3 k )! k ! b θ − x (11 θ + 11 θ + 3) − x ( θ + 1) P k (cid:0) nk (cid:1) (cid:0) n + kn (cid:1) The ten products A ∗ A , etc. form 10 of the 14 hypergeometric families from[3]. The 16 products A ∗ e etc. are not hypergeometric, but also have threesingular fibres. The 24 operators A ∗ a etc. have, apart from 0 and ∞ twofurther singular fibres. The operators a ∗ a etc. have four singular fibres apartfrom 0 and ∞ . Observations:
1) The Dwork-congruences hold for the operators a, b, . . . , j . For the Apery-sequence (case b) this was also conjectured in [24]. (It follows from [12] robenius Polynomials
A, B, C, D satisfy the Dwork-congruences). It follows that the Dwork-congruences hold for all fourth order Hadamard products within this group.2) For the hypergeometric cases A ∗ A , etc, and the cases A ∗ a , etc. the Dwork-congruences also hold for the associated fifth order operator, although evenfor the simplest examples like the quintic threefold, this is not at all obvious.In the case of the quintic, the holomorphic solution around z = 0 to the fifthorder differential equation is given by the formula F ( z ) = P ∞ n =0 A n z n , where A n := n X k =0 (5 k )! k ! n − k )!( n − k )! (1 + k ( − H k + 5 H n − k + 5 H k − H n − k ) ))and H k is the harmonic number H k = P kj =1 1 j . Thus, by the formula it is noteven obvious that the coefficients A n are integers.3) In fact, the Dwork-congruences hold for almost all fourth order operatorsfrom the list [3]. It is an interesting problem to try to prove these experimentalfacts. On the other hand, it is clear that they cannot hold in general fordifferential operators of geometrical origin: if we multiply f with a rationalfunction of x we obtain a (much more complicated) CY-operator for whichthe congruences in general will not hold. In the hypergeometric cases we reproduced results obtained in [23]. In theappendix of [21], the results of our calculations on the 24 operators which areHadamard products like A ∗ a etc. are collected. We computed coefficients( a, b ) of the Frobenius polynomial P ( T ) = 1 + aT + bpT + ap T + p T for all primes p between 3 and 17 and for all possible values of z ∈ F ∗ p .In our computations, we assumed that Conjecture 2.2 holds true and tookthe constants (7) appearing in the formula for the unit root to be one. Togenerate the tables of coefficients in [21], we used the programming languageMAGMA. We computed with an overall p − adic accuracy of 500 digits. Thiswas necessary, since in the computation of the power series solutions to thedifferential equations P y = 0 and Q y = 0, denominators divisible by largepowers of p occured during the calculations (although the solutions themselveshave integral coefficients). The occurance of large denominators reduces the p − adic accuracy in MAGMA, and thus we had to compute with such a highoverall accuracy to obtain correct results in the end. For the unit rootsthemselves, we computed the ratio f ( z ) ( p − f ( z p ) ( p − | z = α mod p K. Samol and D. van Straten with p − adic accuracy modulo p . We checked our results for the tuples ( a, b )determined the absolute values of the complex roots of the Frobenius polyno-mial, which by the Weil conjectures should have absolute value p − / . Need-less to say, this was always fulfilled. In this section, we describe the computational steps we performed in MAGMAfor one specific example. We consider the operator A ∗ a , which is nr. 45 fromthe list [3].We compute the Frobenius polynomial for p = 7 and α = 2 ∈ F with 4digits of 7 − adic precision, i.e. modulo 7 . Since 2 = − and 2 = in F , α is not a singular point of the differential equation.First of all, we computed the truncated power series solution f ( p s +1 − ( z ) tothe differential equation P y = 0 , and obtained f (7 − ( z ) = 1 + 8 z + 360 z + 22400 z + 1695400 z + 143011008 z + ... Thus, f (7 − ( α ) = 1 ∈ F is nonzero. Let α (4) be the Teichm¨uller lifting of α with 7 − adic accuracy of 4 digits. Evaluating f in this point, we obtain f (7 − ( α (4) ) ≡ and f (7 − (( α (4) ) ) ≡ . Thus, the unit root of the Frobenius polynomial is r := f (7 − ( α (4) ) f (7 − (( α (4) ) ) ≡
582 mod 7 . To compute the second root of the Frobenius polynomial, we compute thetruncated power series solution F (7 − ( z ) of the fifth order differential equa-tion Q y = 0 , robenius Polynomials Q is the second exterior power of the differential operator P , given by Q = θ − z (44 + 260 θ + 628 θ + 792 θ + 560 θ + 224 θ )+ z ( − θ + 44160 θ + 71040 θ + 42240 θ + 8448 θ )+ z (4177920 + 13180928 θ + 16588800 θ + 10567680 θ + 3440640 θ + 458752 θ )+ z (100663296 + 285212672 θ + 310378496 θ + 163577856 θ + 41943040 θ + 4194304 θ ) . The solution is given by F (7 − = 1 + 44 z + 3652 z + 337712 z + 33909700 z + 3567877424 z + ...,F (7 − ( α ) = 2 ∈ F is nonzero and we compute F (7 − ( α (4) ) ≡
51 mod 7 and F (7 − (( α (4) ) ) ≡ . Thus, ˆ r := F (7 − ( α (4) ) F (7 − (( α (4) ) ) ≡ . Since the Frobenius polynomial (with 7 − adic accuracy 4) is given by P ( T ) = (1 − r T )(1 − r /r T )(1 − r / ˆ r T )(1 − /r T ) , we finally obtain P ( T ) = 7 T − · T + 7 · T − T + 1 . As expected, the complex roots of P do have complex absolute value 7 − / .Exemplarily, we now list all values (a,b) we computed for the differential op-erator A ∗ a . If there occurs a “-” in the table instead of the tuple ( a, b ),then the correponding z ∈ F p is either a zero of f ( p − or F ( p − or of both,where f was the power series solution of the fourth order differential equa-tion and F was the solution of the fifth order equation. The appearanceof ( a, b ) ′ means that the polynomial is reducible . The appearance of ( a, b ) ∗ means that the corresponding z is a singular point of the differential equation.0 K. Samol and D. van Straten p = 3 p = 5: z − − z , − ′ (28 , ∗ − (32 , ∗ p = 7: z p = 11: z z p = 13: z z p = 17: z z In some cases, the so chosen accuracy was too low, and we had to computemod p . This happened in case the parameter α ∈ F p was a critical point ofthe differential equation. But it is somewhat of a miracle that our calculationmade sense at the critical points at all. In order to understand what issupposed to happen at a singular point, recall that if the fibre X s of a family X −→ P over s ∈ P ( Q ) aquires an ordinary double point, then the Frobeniuspolynomial should factor as P ( T ) = (1 − χ ( p ) T )(1 − pχ ( p ) T )(1 − a p T + p T )for some character χ . The factor (1 − a p T + p T ) is the Frobenius polynomialon the two dimensional pure part of H . This part can be identified with the H of a small resolution ˜ X s , which then is a rigid Calabi-Yau 3-fold. Accord-ing to the modularity conjecture for such Calabi-Yau 3-folds, the coefficients a p are Fourier coefficients of a weight four modular form for some congruencesubgroup Γ ( N ), [17].This is exactly the phenomenon that occurs at the singular points of ourdifferential equations. For the hypergeometric cases we refind the results of[23]. For 16 of the 24 operators A ∗ a etc, we have two rational critical values.In 31 of the cases we are able to identify the modular form. robenius Polynomials a/b means:the b -th Hecke eigenform of level a . ’Twist of’ means: the modular formsdiffer by character. We remark that the critical points of the operators arereciprocal integers and the level of the corresponding modular form dividesthat integer. For the cases involving the operator c one usually has equalityand so the modular form for D ∗ c presumably has level 3888, which wasoutside the range of our table. Remark that all levels appearing only involveprimes 2 and 3.Case Point Form Twist of Point Form Twist of A ∗ a − /
16 8 / − /
128 64 / / B ∗ a − /
27 27 / / /
126 54 / − C ∗ a − /
64 32 / / /
512 256 / − D ∗ a − /
432 216 / / / /
16 216 / A ∗ c /
144 48 / / /
16 16 / / B ∗ c /
243 243 / − /
27 27 / − C ∗ c /
576 576 / / /
64 64 / / D ∗ c / / /
432 432 / / A ∗ d /
128 64 / / /
64 32 / − B ∗ d /
216 9 / − /
108 108 / / C ∗ d /
512 256 / − /
256 128 / / D ∗ d / / / / / / A ∗ g /
144 24 / − /
128 64 / / B ∗ g /
243 243 / / /
216 54 / / C ∗ g /
576 288 /
10 96 / /
512 256 / / D ∗ g / / / / / − The simplest modular forms appearing are the well-knonw η -products 8 / η ( q ) η ( q ) , 9 / η ( q ) . Acknowledgement.
We thank G. Almkvist and W. Zudilin for interest inthe project. We thank M. Bogner for help with calculations, S.Cynk and J.D. Yu for useful discussions. Special thanks to V. Golyshev for his idea to use“polynomial functors” to obtain information on the root of valuation one.The work of K.S. was funded by the SFB Transregio 45.2
K. Samol and D. van Straten
In this appendix we collect the results of our calculations on the 24 opera-tors Hadamard products A ∗ a , etc. We computed coefficients ( a, b ) of theFrobenius Polynomial P ( T ) = 1 + aT + bpT + ap T + p T for all primes p between 3 and 17 and for all possible values of z ∈ F ∗ p . If thereoccurs a “-” in the table instead of the tuple ( a, b ), then the correponding z ∈ F p is either a zero of f ( p − or F ( p − or of both, where f was thepower series solution of the fourth order differential equation and F wasthe solution of the fifth order equation. The appearance of ( a, b ) ′ meansthat the polynomial is reducible . The appearance of ( a, b ) ∗ means that thecorresponding z is a singular point of the differential equation. A ∗ a This is operator nr. 45 from the list [3]: θ − x (2 θ + 1) (cid:0) θ + 7 θ + 2 (cid:1) − x (2 θ + 1) (2 θ + 3) p = 3 p = 5: z − − z , − ′ (28 , ∗ − (32 , ∗ p = 7: z p = 11: z z p = 13: z z p = 17: z z robenius Polynomials B ∗ a This is operator nr. 15 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 7 θ + 2 (cid:1) − x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z ,
4) (8 , z − , − − (3 , −
22) (6 , p = 7: z p = 11: z z p = 13: z z p = 17: z z C ∗ a This is operator nr. 68 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 7 θ + 2 (cid:1) − x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7) p = 3: p = 5: z , − − z − (6 ,
6) ( − , − ∗ (8 , p = 7: z p = 11: z z p = 13: z K. Samol and D. van Straten z p = 17: z z D ∗ a This is operator nr. 62 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 7 θ + 2 (cid:1) − x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z ,
4) (8 , z , ∗ (29 , ∗ − − p = 7: z p = 11: z z p = 13: z z p = 17: z z A ∗ b This is operator nr. 25 from the list [3]: θ − x (2 θ + 1) (cid:0) θ + 11 θ + 3 (cid:1) − x (2 θ + 1) (2 θ + 3) p = 3: p = 5: z − (5 , z , − (2 , ′ ( − , p = 7: z robenius Polynomials p = 11: z z p = 13: z z p = 17: z z B ∗ b This is operator nr. 24 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 11 θ + 3 (cid:1) − x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z ,
7) (5 , z − (8 , − − (7 , − p = 7: z p = 11: z z p = 13: z z p = 17: z z C ∗ b This is operator nr. 51 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 11 θ + 3 (cid:1) − x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7)6 K. Samol and D. van Straten p = 3: p = 5: z ,
14) (5 , z , − ′ (12 , − − p = 7: z p = 11: z z p = 13: z z p = 17: z z D ∗ b This is operator nr. 63 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 11 θ + 3 (cid:1) − x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z ,
7) (5 , z − (24 ,
76) (4 ,
1) ( − , − p = 7: z p = 11: z z p = 13: z z p = 17: z robenius Polynomials z A ∗ c This is operator nr.58 from the list [3]: θ − x (2 θ + 1) (cid:0) θ + 10 θ + 3 (cid:1) + 144 x (2 θ + 1) (2 θ + 3) p = 3: p = 5: z , ∗ − z − , ∗ − ( − , −
14) (16 , − ∗ p = 7: z p = 11: z z p = 13: z z p = 17: z z B ∗ c This is operator nr.70 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 10 θ + 3 (cid:1) +81 x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z , ′ ( − , − ′ z − , −
4) ( − , ∗ (11 , − ′ ( − , p = 7: z p = 11: z z K. Samol and D. van Straten p = 13: z z p = 17: z z C ∗ c This is operator nr. 69 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 10 θ + 3 (cid:1) +144 x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7) p = 3: p = 5: z − , − ∗ ( − , z − , ∗ ( − , ′ ( − , − p = 7: z p = 11: z z p = 13: z z p = 17: z z D ∗ c This is operator nr. 64 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 10 θ + 3 (cid:1) +1296 x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z , ′ ( − , − ′ z − (19 , − ∗ ( − , ∗ − robenius Polynomials p = 7: z p = 11: z z p = 13: z z p = 17: z z A ∗ d This is operator nr. 36 from the list [3]: θ − x (2 θ + 1) (cid:0) θ + 3 θ + 1 (cid:1) + 512 x (2 θ + 1) (2 θ + 3) p = 3: p = 5: z , − − z ,
46) ( − , − − − p = 7: z p = 11: z z p = 13: z z p = 17: z z K. Samol and D. van Straten B ∗ d This is operator nr. 48 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 3 θ + 1 (cid:1) +288 x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z − , −
8) ( − , z − (21 , − ∗ − − p = 7: z p = 11: z z p = 13: z z p = 17: z z C ∗ d This is operator nr. 38 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 3 θ + 1 (cid:1) +512 x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7) p = 3: p = 5: z − ,
10) (2 , − z , − − − p = 7: z p = 11: z z p = 13: z robenius Polynomials z p = 17: z z D ∗ d This is operator nr. 65 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 3 θ + 1 (cid:1) +4608 x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z − , −
8) ( − , z − , ∗ ( − , − ∗ ( − , −
2) (14 , p = 7: z p = 11: z z p = 13: z z p = 17: z z A ∗ f This is operator nr. 133 from the list [3]: θ − x (2 θ + 1) (cid:0) θ + 3 θ + 1 (cid:1) + 432 x (2 θ + 1) (2 θ + 3) p = 3: p = 5: z , ′ ( − , − z ,
44) ( − , − ′ ( − ,
28) ( − , ′ p = 7: z K. Samol and D. van Straten p = 11: z z p = 13: z z p = 17: z z B ∗ f This is operator nr. 134 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 3 θ + 1 (cid:1) +243 x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z − ,
13) (5 , z − ,
71) (3 , − ( − , − p = 7: z p = 11: z z p = 13: z z p = 17: z z C ∗ f This is operator nr. 135 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 3 θ + 1 (cid:1) +432 x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7) robenius Polynomials p = 3: p = 5: z − , − ′ (5 , ′ z − ,
22) ( − ,
34) (6 , − p = 7: z p = 11: z z p = 13: z z p = 17: z z D ∗ f This is operator nr. 136 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 3 θ + 1 (cid:1) +3888 x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z − ,
13) (5 , z − − ( − , −
7) ( − , p = 7: z p = 11: z z p = 13: z z p = 17: z K. Samol and D. van Straten z A ∗ g This is operator nr. 137 from the list [3]: θ − x (cid:0) θ + 17 θ + 6 (cid:1) (2 θ + 1) + 1152 x (2 θ + 1) (2 θ + 3) p = 3: p = 5: z − (8 , ∗ z − ( − , ∗ ( − , ′ (16 , − ∗ p = 7: z p = 11: z z p = 13: z z p = 17: z z B ∗ g This is operator nr. 138 from the list [3]: θ − x (3 θ + 1) (3 θ + 2) (cid:0) θ + 17 θ + 6 (cid:1) +648 x (3 θ + 1) (3 θ + 2) (3 θ + 4) (3 θ + 5) p = 3: p = 5: z − , − ′ (5 , ′ z , − ∗ ( − , ∗ ( − , − p = 7: z p = 11: z z robenius Polynomials p = 13: z z p = 17: z z C ∗ g This is operator nr. 139 from the list [3]: θ − x (4 θ + 1) (4 θ + 3) (cid:0) θ + 17 θ + 6 (cid:1) +1152 x (4 θ + 1) (4 θ + 3) (4 θ + 5) (4 θ + 7) p = 3: p = 5: z − ,
10) ( − , − ∗ z − , ∗ − (18 , − ∗ ( − , p = 7: z p = 11: z z p = 13: z z p = 17: z z D ∗ g This is operator nr. 140 from the list [3]: θ − x (6 θ + 1) (6 θ + 5) (cid:0) θ + 17 θ + 6 (cid:1) +10368 x (6 θ + 1) (6 θ + 5) (6 θ + 7) (6 θ + 11) p = 3: p = 5: z − , − ′ (5 , ′ z − , ∗ (19 , − ∗ − − K. Samol and D. van Straten p = 7: z p = 11: z z p = 13: z z p = 17: z z References [1] G. Almkvist,
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