Calculation of local Fourier transforms for formal connections
aa r X i v : . [ m a t h . AG ] J u l Calculation of local Fourier transforms for formal connections
Jiangxue FangChern Institute of Mathematics, Nankai University, Tianjin 300071, P. R. [email protected]
Abstract
We calculate the local Fourier transforms for formal connections. In particular, weverify an analogous conjecture of Laumon and Malgrange ([6] 2.6.3).
Mathematics Subject Classification (2000) : Primary 14F40.
Let k be an algebraic closed field of characteristic zero and let k (( t )) be the field of formal Laurentseries in the variable t . A formal connection on k (( t )) is a pair ( M, t∂ t ) consisting of a finitedimensional k (( t ))-vector space M and a k -linear map t∂ t : M → M satisfying t∂ t ( f m ) = t∂ t ( f ) m + f t∂ t ( m )for any f ∈ k (( t )) and m ∈ M . In [2], S. Bloch and H. Esnault define local Fourier transforms F (0 , ∞ ) , F ( ∞ , , F ( ∞ , ∞ ) for formal connections, by analogy with the ℓ -adic local Fourier transformconsidered in [6]. In [6], 2.6.3, Laumon and Malgrange give conjectural formulas of local Fouriertransforms for a class of Q ℓ -sheaf. This results are proved by Lei Fu ([4]). In this paper, weprove an analogous conjecture of local Fourier transform for formal connections. Actually, we cancalculate local Fourier transforms for any formal connections.A key technical tool for the definitions of local Fourier transforms of formal connections is thenotion of good lattices pairs. By definition in [3], Lemma 6.21, a pair of good lattices V , W of M is a pair of lattices in M satisfying the following conditions(1) V ⊂ W ⊂ M (2) t∂ t ( V ) ⊂ W (3) For any k ∈ N , the natural inclusion of complexes( V t∂ t −−→ W ) → ( 1 t k V t∂ t −−→ t k W )1s a quasi-isomorphism.Good lattices pairs V , W exist. The number dim k W / V is independent of the choice of goodlattices pairs of M , and is called the irregularity of M .For any f ∈ k (( t )) , denote by [ f ] the formal connection on k (( t )) consisting of a one dimensional k (( t ))-vector space with a basis e and a k -linear map t∂ t : k (( t )) e → k (( t )) e satisfying t∂ t ( ge ) = ( t∂ t ( g ) + f g ) e for any g ∈ k (( t )) . Two such objects [ f ] and [ f ′ ] are isomorphic if and only if f − f ′ ∈ tk [[ t ]] + Z .Therefore the non-negative integer max(0 , − ord t ( f ))is a well-defined invariant of the isomorphic class of [ f ], and is called the slope of [ f ]. Let p bethe slope of [ f ]. One can verify k [[ t ]] e, t − p k [[ t ]] e is a good lattices pair of [ f ]. So the irregularitycoincides with the slope for any one dimensional formal connection. The definition of slopes forarbitrary formal connections is given in [5], (2.2.5). The irregularity of a formal connection coincidewith the sum of its slopes. Any formal connection has a unique slope decomposition. So the slopeof an irreducible formal connection is equal to its irregularity divided by its dimension. A formalconnection is called regular if the irregularity of this connection is equal to 0.Throughout this paper, r and s are to be positive integers. Let t ′ be the Fourier transformcoordinate of t . Write z = t and z ′ = t ′ . Let[ r ] : k (( t )) ֒ → k (( r √ t ))be the natural inclusion of fields. Let T = r √ t and let α be a formal Laurent series in k (( T )) oforder − s with respect to T . Let R be a regular formal connection on k (( T )). In this paper, wecalculate the local Fourier transform F (0 , ∞ ) (cid:16) [ r ] ∗ (cid:16) [ T ∂ T ( α )] ⊗ k (( T )) R (cid:17)(cid:17) . Similarly, let k (( z )) be the field of formal Laurent series in the variable z. Let[ r ] : k (( z )) ֒ → k (( 1 r √ t ))be the natural inclusion of fields. Let Z = r √ t and let α be a formal Laurent series in k (( Z )) oforder − s with respect to Z . Let R be a regular formal connection on k (( Z )). We also calculate2he local Fourier transforms F ( ∞ , (cid:16) [ r ] ∗ (cid:16) [ Z∂ Z ( α )] ⊗ k (( Z )) R (cid:17)(cid:17) if r > s ; F ( ∞ , ∞ ) (cid:16) [ r ] ∗ (cid:16) [ Z∂ Z ( α )] ⊗ k (( Z )) R (cid:17)(cid:17) if r < s. We refer the reader to [2] for the definitions and properties of local Fourier transforms. Themain results of this paper are the following three theorems.
Theorem 1.
Given a formal Laurent series α in k (( r √ t )) of order − s with respect to r √ t , considerthe following system of equations ( ∂ t ( α ( r √ t )) + t ′ = 0 ,α ( r √ t ) + tt ′ = β ( r + s √ t ′ ) . (1.1) Using the first equation, we find an expression of r √ t in terms of r + s √ t ′ . We then substitute thisexpression into the second equation to get β ( r + s √ t ′ ) , which is a formal Laurent series in k (( r + s √ t ′ )) of order − s with respect to r + s √ t ′ . Let T = r √ t and let Z ′ = r + s √ t ′ . For any regular formalconnection R on k (( T )) , we have F (0 , ∞ ) (cid:16) [ r ] ∗ (cid:16) [ T ∂ T ( α )] ⊗ k (( T )) R (cid:17)(cid:17) = [ r + s ] ∗ (cid:16) [ Z ′ ∂ Z ′ ( β ) + s ⊗ K (( Z ′ )) R (cid:17) , where the right R means the formal connection on k (( Z ′ )) after replacing the variable T with Z ′ . Theorem 2.
Suppose r > s . Given a formal Laurent series α in k (( r √ t )) of order − s with respectto r √ t , consider the following system of equations ( ∂ t ( α ( r √ t )) + t ′ = 0 ,α ( r √ t ) + tt ′ = β ( r − s √ t ′ ) . (1.2) Using the first equation, we find an expression of r √ t in terms of r − s √ t ′ . We then substitute thisexpression into the second equation to get β ( r − s √ t ′ ) , which is formal Laurent series in k (( r − s √ t ′ )) oforder − s with respect to r − s √ t ′ . Let Z = r √ t and let T ′ = r − s √ t ′ . For any regular formal connection R on k (( Z )) , we have F ( ∞ , (cid:16) [ r ] ∗ (cid:16) [ Z∂ Z ( α )] ⊗ k (( Z )) R (cid:17)(cid:17) = [ r − s ] ∗ (cid:16) [ T ′ ∂ T ′ ( β ) + s ⊗ k (( T ′ )) R (cid:17) , where the right R means the formal connection on k (( T ′ )) after replacing the variable Z with T ′ . heorem 3. Suppose r < s . Given a formal Laurent series α in k (( r √ t )) of order − s with respectto r √ t , consider the following system of equations ( ∂ t ( α ( r √ t )) + t ′ = 0 ,α ( r √ t ) + tt ′ = β ( s − r √ t ′ ) . (1.3) Using the first equation, we find an expression of r √ t in terms of s − r √ t ′ . We then substitute thisexpression into the second equation to get β ( s − r √ t ′ ) , which is a formal Laurent series in k (( s − r √ t ′ )) of order − s with respect to s − r √ t ′ . Let Z = r √ t and let Z ′ = s − r √ t ′ . For any regular formalconnection R on k (( Z )) , we have F ( ∞ , ∞ ) (cid:16) [ r ] ∗ (cid:16) [ Z∂ Z ( α )] ⊗ k (( Z )) R (cid:17)(cid:17) = [ s − r ] ∗ (cid:16) [ Z ′ ∂ Z ′ ( β ) + s ⊗ k (( Z ′ )) R (cid:17) , where the right R means the formal connection on k (( Z ′ )) after replacing the variable Z with Z ′ . When R is trivial, the above three theorems are conjectured by Laumon and Malgrange ([6]2.6.3) except the term s is missing in the conjecture. Any formal connection on k (( t )) is a direct sumof indecomposable connections. As in [1], section 5.9, any indecomposable connection M = N ⊗ R ,where R is regular and N = [ d ] ∗ L where L is a one dimensional connection on a finite extension[ d ] : k (( t )) → k (( t d )). So we can calculate local Fourier transform for all formal connections. Acknowledgements.
It is a great pleasure to thank my advisor Lei Fu for his guidanceand support during my graduate studies. In [8], Claude Sabbah proves these results of localFourier transforms for formal connections with a geometric method. Our method is elementaryand directly.
Given a formal Laurent series α in the variable r √ t of order − s , consider the system of equations(1.1). We express r √ t as a formal Laurent series in r + s √ t ′ of order 1 using the first equation andthen substitute this expression into the second equation to get β ∈ k (( r + s √ t ′ )). We have ∂ t ′ ( β ) = ∂ t ′ (cid:0) α ( r √ t ) + tt ′ (cid:1) = ∂ t (cid:0) α ( r √ t ) (cid:1) dtdt ′ + t ′ dtdt ′ + t (2.1)= (cid:16) ∂ t (cid:0) α ( r √ t ) (cid:1) + t ′ (cid:17) dtdt ′ + t = t. It follows that β is a formal Laurent series in r + s √ t ′ of order − s . Let T = r √ t and Z ′ = r + s √ t ′ . Set a ( T ) = − T s t∂ t ( α ) and b ( Z ′ ) = Z ′ s t ′ ∂ t ′ ( β ) . a ( T ) is a formal power series in T of order 0 and b ( Z ′ ) is a formal power series in Z ′ of order0. From the system of equations (1.1) and (2.1), we get (cid:26) a ( T ) = ( TZ ′ ) r + s b ( Z ′ ) = ( TZ ′ ) r . (2.2)To prove Theorem 1, it suffices to prove the following theorem. Theorem 1 ′ . Given a formal power series a ( T ) = P i ≥ a i T i with a i ∈ k and a = 0 , solve thesystem of equations (2.2) to get b ( Z ′ ) = P i ≥ b i Z ′ i for some b i ∈ k . Then b s = rr + s a s and F (0 , ∞ ) (cid:16) [ r ] ∗ [ − r ( a T − s + a T − s + . . . + a s )] (cid:17) = [ r + s ] ∗ [ − ( r + s )( b Z ′− s + b Z ′ − s + . . . + b s ) + s . In fact, suppose Theorem 1 ′ holds. Let c be an element in k. By remark 2.2 we shall prove later,for a ( T ) = − T s t∂ t ( α ) − cr T s , we can get a solution b ( Z ′ ) of the system of equations (2.2) such that b ( Z ′ ) ≡ Z ′ s t ′ ∂ t ′ ( β ) − cr + s Z ′ s mod. Z ′ s +1 . Then F (0 , ∞ ) (cid:16) [ r ] ∗ [ T ∂ T ( α ) + c (cid:17) = F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rT − s ( − T s t∂ t ( α ) − cr T s )] (cid:17) = [ r + s ] ∗ [ − ( r + s ) Z ′− s ( Z ′ s t ′ ∂ t ′ ( β ) − cr + s Z ′ s )]= [ r + s ] ∗ [ Z ′ ∂ Z ′ ( β ) + c ] . So Theorem 1 holds for R = [ c ]. As in [1], section 5.9, every irreducible regular formal connection N on k (( T )) is [ d ] ∗ L , where L is a one dimensional formal connection on a finite extension [ d ] : k (( T )) → k (( T d ) . So L is regular, we have L = [ c ] for some c ∈ k . Then N = [ d ] ∗ [ c ] = ⊕ ≤ i ≤ d [ c + id ] . We have d = 1 because N is irreducible. This shows that every irreducible regular formalconnection is isomorphic to the one dimensional connection [ c ] for some c ∈ k. So every regularformal connection is a successive extension of connections of the type [ c ]. Since F (0 , ∞ ) is functorieland exact, Theorem 1 holds for any regular formal connection R on k (( T )). Remark . If a s = 0, then there exists α ∈ k (( r √ t )) such that a ( T ) = − T s t∂ t ( α ) . Using the firstequation of (2.2), we find an expression of T in terms of Z ′ . We then substitute this expressioninto the second equation of (2.2) to get b ( Z ′ ) . This expression also satisfies the first equation of51.1). We then substitute this expression into the second equation of (1.1) to get β ( Z ′ ) . By (2.1),we have b ( Z ′ ) = X i ≥ b i Z ′ i = Z ′ s t ′ ∂ t ′ ( β ) . This shows b s = 0. Remark . Solving the first equation of (2.2), we get T = P i ≥ λ i Z ′ i +1 with λ = r + s √ a . The solution is not unique and different solutions differ by an r + s -th root of unity. As longas λ is chosen to be an r + s -th root of a , for each i , λ i depends only on a , . . . , a i . We have b ( Z ′ ) = ( P i ≥ λ i Z ′ i ) r , and for each i , b i depends only on λ , . . . , λ i . Therefore as long as we fixan r + s -th root of a , for each i , b i depends only on a , . . . , a i . So to prove Theorem 1 ′ , we canassume a ( T ) = P ≤ i ≤ s a i T i . Remark . Solving the first equation of (2.2), we get T = P i ≥ λ i Z ′ i +1 for some λ j ∈ k. Then λ is an r + s -th root of a . Then P i ≥ b i Z ′ i = ( P i ≥ λ i Z ′ i ) r . Choose a ′ , . . . , a ′ s ∈ k such that a ′ i = a i for all 0 ≤ i < s and a ′ s = 0. For a ( T ) = P ≤ i ≤ s a ′ i T i , consider the system of equations (2.2) ifthe variable T is changed by T . Using the first equation, we can express T as P i ≥ λ ′ i Z ′ i +1 with λ ′ = λ . Then we have P i ≥ b ′ i Z ′ i = ( P i ≥ λ ′ i Z ′ i ) r . Remark 2.1 shows b ′ s = 0. Since a i = a ′ i for0 ≤ i < s , we have λ i = λ ′ i for all 0 ≤ i < s . That is, T ≡ T mod. Z ′ s +1 and T ≡ T ≡ λ Z ′ mod. Z ′ . Comparing coefficients of Z ′ s on both sides of X i ≥ a i T i = (cid:16) X i ≥ λ i Z ′ i (cid:17) r + s and X ≤ i ≤ s a ′ i T i = (cid:16) X i ≥ λ ′ i Z ′ i (cid:17) r + s , we have a s λ s = ( a s − a ′ s ) λ s = ( r + s )( λ s − λ ′ s ) λ r + s − . Comparing coefficients of Z ′ s on both sides of X i ≥ b i Z ′ i = (cid:16) X i ≥ λ i Z ′ i (cid:17) r and X i ≥ b ′ i Z ′ i = (cid:16) X i ≥ λ ′ i Z ′ i (cid:17) r , we have b s = b s − b ′ s = r ( λ s − λ ′ s ) λ r − . This proves b s = rr + s a s . emark . Set f = a T − s + a T − s + . . . + a s . Let H = { σ ∈ Gal (cid:0) k (( T )) /k (( t )) (cid:1) | σ ( f ) = f } . We call f is irreducible with respect to the Galois extension k (( T )) /k (( t )) if H = 1. Then f isirreducible if and only if the connection [ r ] ∗ [ − rf ] is irreducible. Lemma 2.5.
If Theorem ′ holds for irreducible f , then it holds for all f.Proof. By Remark 2.2, we can assume a ( T ) = P ≤ i ≤ s a i T i . Keep the notation in Remark 2.4.Set p = H . Then p | r . Let η be a primitive r -th root of unity in k . Then a i η rp ( i − s ) = a i for all0 ≤ i ≤ s . So a i = 0 or p | i − s . In particular, p | s since a = 0. Let τ = T p and τ ′ = Z ′ p . Then f = a τ − sp + a p τ − sp + . . . + a s and it is irreducible with respect to the Galois extension k (( τ )) /k (( t )) . For a ( τ ) = P ≤ i ≤ sp a pi τ i ,suppose b ( τ ′ ) = P i ≥ b pi τ ′ i is a solution of the following system of equation ( a ( τ ) = ( ττ ′ ) r + sp b ( τ ′ ) = ( ττ ′ ) rp . (2.3)Then b s = rr + s a s and b ( Z ′ ) = P i ≥ b pi Z ′ pi is a solution of the system of equations (2.2). For a ( τ ) = P ≤ i ≤ sp a pi τ i − jr τ sp (1 ≤ j ≤ p ), by Remark 2.2 and 2.3, we can find a solution b ( τ ′ ) ofthe system of equations (2.3) such that b ( τ ′ ) ≡ X ≤ i ≤ sp b pi τ ′ i − jr + s τ ′ sp mod. τ ′ sp +1 . Applying Theorem 1 ′ to the system of equations (2.3) for a ( τ ) = P ≤ i ≤ sp a pi τ i − jr τ sp (1 ≤ j ≤ p ),we have F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) = F (0 , ∞ ) (cid:16) [ rp ] ∗ [ p ] ∗ [ − r ( a T − s + a p T p − s + . . . + a s )] (cid:17) = M ≤ j ≤ p F (0 , ∞ ) (cid:16) [ rp ] ∗ [ − rp ( a τ − sp + a p τ p − sp + . . . + a s ) + jp ] (cid:17) = M ≤ j ≤ p [ r + sp ] ∗ [ − r + sp ( b τ ′− sp + b p τ ′ p − sp + . . . + b s ) + jp + s p ]= [ r + sp ] ∗ [ p ] ∗ [ − ( r + s )( b Z ′− s + b p Z ′ p − s + . . . + b s ) + s r + s ] ∗ [ − ( r + s )( b Z ′− s + b p Z ′ p − s + . . . + b s ) + s . f is irreducible.Let’s describe the connection F (0 , ∞ ) (cid:0) [ r ] ∗ [ − rf ] (cid:1) on k (( z ′ )).The formal connection [ − rf ] on k (( T )) consist of a one dimensional k (( T ))-vector space witha basis e and a k -linear map T ∂ T : k (( T )) e → k (( T )) e satisfying T ∂ T ( ge ) = ( T ∂ T ( g ) − rf g ) e for any g ∈ k (( T )) . Since the formal connection [ − rf ] on k (( T )) has slope s , we get k [[ T ]] e, T − s k [[ T ]] e is a good lattices pair for it. Identify [ r ] ∗ [ − rf ] with k (( T )) e as k (( t ))-vector spaces. Then the for-mal connection [ r ] ∗ [ − rf ] has pure slope sr and k [[ T ]] e, T − s k [[ T ]] e is a good lattices pair for thisconnection. The action of the differential operator t∂ t on k (( T )) e is given by t∂ t ( ge ) = ( t∂ t ( g ) − f g ) e for any g ∈ k (( T )) . So we have( ∂ t ◦ t )( T − i e ) = r − ir T − i e − ( a T − ( s + i ) e + . . . + a s T − i e ) (1 ≤ i ≤ r ) ,t · T − i e = T − ( i − r ) e ( r + 1 ≤ i ≤ r + s ) . By [2], Proposition 3.7, the map ι : k (( T )) e → F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) is an isomorphism of k -vector spaces. By [2], Lemma 2.4, ( ιT − e, . . . , ιT − ( r + s ) e ) is a basis of F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) over k (( z ′ )). Then by the relation ι ◦ t = − z ′ ∂ z ′ ◦ ι and ι ◦ ∂ t = − z ′ ◦ ι in[2], Proposition 3.7, the matrix of the connection F (0 , ∞ ) (cid:0) [ r ] ∗ [ − rf ] (cid:1) with respect to the differentialoperator z ′ ∂ z ′ and the basis ( ιT − e, . . . , ιT − ( r + s ) e ) is − r z }| { s z }| { a s a s − . . .... . . . a s z ′ . . . z ′ a a s − . . . ... a + diag { r − r , . . . , r , , . . . , } . r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) = k (( Z ′ )) ⊗ k (( z ′ )) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) with respect to the differential operator Z ′ ∂ Z ′ and the basis ( Z ′ ⊗ ιT − e, . . . , Z ′ r + s ⊗ ιT − ( r + s ) e ) is − r + sZ ′ s a s Z ′ s a s − Z ′ s − . . . . . .... . . . a s Z ′ s a . . . a s − Z ′ s − . . . ... a +( r + s )diag { r − r , . . . , r , , . . . , } + diag { , . . . , r + s } . We can write this matrix as ( r + s ) B − ( r + s ) P ≤ i ≤ s Z ′ i − s A i for some matrices A i and B withentries in k , where A = (cid:18) I s a I r (cid:19) ,B = diag { r − r , . . . , r , , . . . , } + 1 r + s diag { , . . . , r + s } . Let V be the k -vector subspace of [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) generated by Z ′ i ⊗ ιT − i e (1 ≤ i ≤ r + s ). With respect to this basis, V can be identified with the k -vector space of column vectors in k of length r + s . The action of the differential operator Z ′ ∂ Z ′ on elements of V can be written as Z ′ ∂ Z ′ ( v ) = ( r + s ) B ( v ) − ( r + s ) X ≤ i ≤ s Z ′ i − s A i ( v ) . Lemma 2.6.
Suppose f is irreducible in the sense of Remark 2.4. Given α , . . . , α s ∈ k , thefollowing three conditions are equivalent: (1) F (0 , ∞ ) (cid:0) [ r ] ∗ [ − rf ] (cid:1) = [ r + s ] ∗ [ − ( r + s ) P ≤ i ≤ s α i Z ′ i − s ] . (2) [ − ( r + s ) P ≤ i ≤ s α i Z ′ i − s ] is a subconnection of [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) . (3) There exist an integer N and v , . . . , v s ∈ V such that v = 0 and (cid:26) P ≤ i ≤ k ( A i − α i ) v k − i = 0 (0 ≤ k ≤ s − P ≤ i ≤ s − ( A i − α i ) v s − i + ( A s − B − α s − Nr + s ) v = 0 . (2.4) Proof.
Since f is irreducible, the connection [ r ] ∗ [ − rf ] on k (( t )) is irreducible with pure slope sr .By [2], Proposition 3.14, the connection F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) on k (( z ′ )) is irreducible with pure slope9 r + s . As in the proof of [2], Lemma 3.3, we have F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) = [ r + s ] ∗ [ − ( r + s ) X ≤ i ≤ s ̺ i Z ′ i − s ]for some ̺ , . . . , ̺ s ∈ k with ̺ = 0 . Let µ be a primitive ( r + s )-th root of unity in k . Then[ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) = M ≤ j ≤ r + s [ − ( r + s )( µ − js ̺ Z ′− s + µ j (1 − s ) ̺ Z ′ − s + . . . + ̺ s )] . So there are r + s one dimensional subconnections of [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) which are notisomorphic to each other.(1) ⇒ (2) is trivial. For (2) ⇒ (1), assume that [ − ( r + s ) P ≤ i ≤ s α i Z ′ i − s ] is a subconnectionof [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) . Then[ − ( r + s ) X ≤ i ≤ s α i Z ′ i − s ] = [ − ( r + s ) X ≤ i ≤ s µ j ( i − s ) ̺ i Z ′ i − s ]for some 1 ≤ j ≤ r + s . Then F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17) = [ r + s ] ∗ [ − ( r + s ) X ≤ i ≤ s µ j ( i − s ) ̺ i Z ′ i − s ]= [ r + s ] ∗ [ − ( r + s ) X ≤ i ≤ s α i Z ′ i − s ] . For (2) ⇒ (3), assume that [ − ( r + s ) P ≤ i ≤ s α i Z ′ i − s ] is a subconnection of [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) . This means that there is a nonzero map of connections φ : [ − ( r + s ) X ≤ i ≤ s α i Z ′ i − s ] → [ r + s ] ∗ (cid:16) F (0 , ∞ ) (cid:16) [ r ] ∗ [ − rf ] (cid:17)(cid:17) . The connection [ − ( r + s ) P ≤ i ≤ s α i Z ′ i − s ] consist of a one dimensional k (( Z ′ ))-vector space witha basis ε and a k -linear map Z ′ ∂ Z ′ : k (( Z ′ )) ε → k (( Z ′ )) ε satisfying Z ′ ∂ Z ′ ( gε ) = (cid:16) Z ′ ∂ Z ′ ( g ) − ( r + s ) g X ≤ i ≤ s α i Z ′ i − s (cid:17) ε for any g ∈ k (( Z ′ )) . Suppose φ ( ε ) = P ≤ i Z ′ i + N v i for some integer N and some v i ∈ V with10 = 0. Then − ( r + s ) X ≤ i ≤ s α i Z ′ i − s X ≤ i Z ′ i + N v i = φ (cid:0) Z ′ ∂ Z ′ ( ε ) (cid:1) = Z ′ ∂ Z ′ (cid:0) φ ( ε ) (cid:1) = Z ′ ∂ Z ′ (cid:0) X ≤ i Z ′ i + N v i (cid:1) = X ≤ i Z ′ i + N (cid:0) ( r + s ) B + i + N (cid:1) v i − ( r + s ) X ≤ i Z ′ i + N X ≤ j ≤ s Z ′ j − s A j ( v i ) . Comparing coefficients of Z ′ i , for N − s ≤ i ≤ N on each side, we get the system of equations(2.4). This proves (2) ⇒ (3). So for α = µ − sj ̺ , α = µ (1 − s ) j ̺ , . . . , α s = ̺ s , the system ofequations (2.4) holds for some N ∈ Z and some v , . . . , v s ∈ V with v = 0. These ( s + 1)-tuples( µ − sj ̺ , µ (1 − s ) j ̺ , . . . , ̺ s ) (1 ≤ j ≤ r + s ) are pairwise distinct, since f is irreducible. Lemma 2.7shows that there are at most r + s ( s + 1)-tuples ( α , . . . , α s ) such that the system of equations(2.4) holds for N = 0 and some v , . . . , v s ∈ V with v = 0. This proves (3) ⇒ (2). Hensel’s lemma.
Let E be a finite dimensional k -vector space. Suppose D is a k [[ t ]] -linearendomorphism of E ⊗ k k [[ t ]] . Write the action of D on elements of E : D ( v ) = X i ≥ t i D i ( v ) , for unique elements D i ∈ End k ( E ) . Suppose the characteristic polynomial of D has a simple root α in k . Then (1) The equation ( D − α )( u ) = 0 has a solution α ∈ k [[ t ]] with constant term α and = u ∈ E ⊗ k k [[ t ]] . In this case, α is uniquelydetermined by α . (2) Let k be a positive integer. The following systems of equations X ≤ i ≤ j ( D i − α i ) u j − i = 0 (0 ≤ j ≤ k ) has a solution α , . . . , α k ∈ k ; u , . . . , u k ∈ E with u = 0 . In this case, α , . . . , α k are uniquelydetermined by α .Proof. The proof is similar to that of [9], Proposition 7, p. 34.
Lemma 2.7.
Given α , . . . , α s ∈ k , there exist v , . . . , v s ∈ V such that v = 0 and (cid:26) P ≤ i ≤ k ( A i − α i ) v k − i = 0 (0 ≤ k ≤ s − , P ≤ i ≤ s − ( A i − α i ) v s − i + ( A s − B − α s ) v = 0 (2.5)11 f and only if there exist v ′ , . . . , v ′ s ∈ V such that v ′ = 0 and (cid:26) P ≤ i ≤ k ( A i − α i ) v ′ k − i = 0 (0 ≤ k ≤ s − , P ≤ i ≤ s − ( A i − α i ) v ′ s − i + ( A s − r + s r +2 s − α s ) v ′ = 0 . (2.6) Moreover, there are at most r + s ( s + 1) -tuples ( α , . . . , α s ) in k such that the system of equations(2.5) (resp. (2.6)) holds for some v , . . . , v s ∈ V with v = 0 (resp. v ′ , . . . , v ′ s ∈ V with v ′ = 0 ).Proof. Let µ be a primitive ( r + s )-th root of unity in k . We fix an ( r + s )-th root a r + s of a . Forany 1 ≤ j ≤ r + s, set e j to be the column vector ( µ j a r + s , . . . , µ j ( r + s − a r + s − r + s , a ) and ε j the rowvector ( µ − j a − r + s , . . . , µ − j ( r + s − a − r + s − r + s , a − ). Then A · e j = µ rj a rr + s · e j , ε j · A = µ rj a rr + s · ε j , ε i · e j = ( r + s ) δ ij . Set d = ( r, s ). We get ker( A − µ rj a rr + s ) is generated by those e k with r + s | ( k − j ) d , andim( A − µ rj a rr + s ) is generated by the other e k ’s. Thenim( A − µ rj a rr + s ) = { v ∈ V | ε k · v = 0 for all k satisfying r + s | ( k − j ) d } . For the only if part, suppose the system of equations (2.5) holds for some v , . . . , v s ∈ V with v = 0.In particular, ( A − α ) v = 0. Then α = µ rj a rr + s for some integer j and then v = P r + s | ( i − j ) d γ i e i for some γ i ∈ k . For any 1 ≤ k, l ≤ r + s , we have ε k · ( B − r + s r + 2 s ) e l = X ≤ i ≤ r r − ir µ i ( l − k ) + X ≤ i ≤ r + s ir + s µ i ( l − k ) − r + s r + 2 s X ≤ i ≤ r + s µ i ( l − k ) . If k = l , ε k · ( B − r + s r + 2 s ) e l = X ≤ i ≤ r r − ir + X ≤ i ≤ r + s ir + s − r + s r + 2 s X ≤ i ≤ r + s . Suppose k = l and r + s | ( l − k ) d . Let ξ = µ l − k . Then ξ d = 1 and ξ = 1. For any d | n , we have P ≤ i ≤ n ξ i = 0 and hence P ≤ i ≤ n iξ i = nd P ≤ i ≤ d iξ i . So we have ε k · ( B − r + s r + 2 s ) e l = − r X ≤ i ≤ r iξ i + 1 r + s X ≤ i ≤ r + s iξ i = − r rd X ≤ i ≤ d iξ i + 1 r + s r + sd X ≤ i ≤ d iξ i = 0 . So ε k · ( B − r + s r +2 s ) v = 0 if r + s | ( k − j ) d. Therefore ( B − r + s r +2 s ) v = ( A − α ) v for some v ∈ V .Then v ′ = v , . . . , v ′ s − = v s − , v ′ s = v s − v satisfy the system of equations (2.6). Reversing the12bove argument, we get the if part. So for the last assertion, it suffices to show that the sameassertion holds for the following system of equations X ≤ i ≤ k ( A i − α i ) v k − i = 0 for any 0 ≤ k ≤ s. (2.7)Suppose the system of equations (2.7) holds for some α , . . . , α s ∈ k and some v , . . . , v s ∈ V with v = 0. There exists an integer 1 ≤ j ≤ r + s such that α = µ rj a rr + s and v = P r + s | ( i − j ) d γ i e i for some γ i ∈ k with γ j = 0. The system of equations (2.7) is equivalent to the following equation (cid:16) X ≤ i ≤ s A i Z ′ i − X ≤ i ≤ s α i Z ′ i (cid:17)(cid:16) X ≤ i ≤ s v i Z ′ i (cid:17) ≡ Z ′ s +1 . There exist ρ = µ j a r + s , ρ , . . . , ρ s ∈ k such that X ≤ i ≤ s α i Z ′ i ≡ (cid:16) X ≤ i ≤ s ρ i Z ′ i (cid:17) r mod. Z ′ s +1 . Let Γ = a s Z ′ s . . .... 1 a and Γ = a . Then P ≤ i ≤ s A i Z ′ i = Γ r , A = Γ r and hence (cid:16) Γ − X ≤ i ≤ s ρ i Z ′ i (cid:17)(cid:16) X ≤ k ≤ r − (cid:16) X ≤ i ≤ s ρ i Z ′ i (cid:17) k Γ r − − k (cid:17)(cid:16) X ≤ i ≤ s v i Z ′ i (cid:17) ≡ Z ′ s +1 . Write (cid:16) X ≤ k ≤ r − (cid:16) X ≤ i ≤ s ρ i Z ′ i (cid:17) k Γ r − − k (cid:17)(cid:16) X ≤ i ≤ s v i Z ′ i (cid:17) = X ≤ i u i Z ′ i for some u i ∈ V. Then u = X ≤ k ≤ r − ρ k Γ r − − k X r + s | ( i − j ) d γ i e i = X r + s | ( i − j ) d γ i · X ≤ k ≤ r − µ jk a kr + s µ i ( r − − k ) a r − − kr + s e i = rµ j ( r − γ j a r − r + s e j = 0 . (cid:16) Γ − X ≤ i ≤ s ρ i Z ′ i (cid:17)(cid:16) X ≤ i ≤ s u i Z ′ i (cid:17) ≡ Z ′ s +1 . (2.8)Since ρ is a simple root of the characteristic polynomial of Γ , by Hensel’s lemma, ρ , . . . , ρ s areuniquely determined by ρ . So α , . . . , α s are uniquely determined by ρ = µ j a r + s (1 ≤ j ≤ r + s ).This proves the last assertion.Now we are ready to prove Theorem 1 ′ . By Remark 2.2, we assume that a ( T ) = P ≤ i ≤ s a i T i . Then the first equation of (2.2) means that TZ ′ is a root in k [[ Z ′ ]] of the polynomial λ r + s − X ≤ i ≤ s a i Z ′ i λ i ∈ k [[ Z ′ ]][ λ ] . This polynomial is exactly the characteristic polynomial of Γ. The characteristic polynomial ofΓ is the polynomial λ r + s − a which has no multiple roots, then by Hensel’s lemma, Γ has aneigenvector P i ≥ Z ′ i v i corresponding this eigenvalue TZ ′ with v = 0. Since P ≤ i ≤ s Z ′ i A i = Γ r ,we have (cid:16) X ≤ i ≤ s Z ′ i A i (cid:17)(cid:16) X ≤ i Z ′ i v i (cid:17) = (cid:16) TZ ′ (cid:17) r (cid:16) X ≤ i Z ′ i v i (cid:17) = (cid:16) X ≤ i b i Z ′ i (cid:17)(cid:16) X ≤ i Z ′ i v i (cid:17) . So X ≤ i ≤ k ( A i − b i ) v k − i = 0 for any 0 ≤ k ≤ s. Recall that P ≤ i ≤ s a i T i − s is assumed to be irreducible. Then by Lemma 2.6 and 2.7, we have F (0 , ∞ ) (cid:16) [ r ] ∗ [ − r ( a T − s + a T − s + . . . + a s )] (cid:17) = [ r + s ] ∗ [ − ( r + s )( b Z ′− s + b Z ′ − s + . . . + b s − r + s r + 2 s )]= [ r + s ] ∗ [ − ( r + s )( b Z ′− s + b Z ′ − s + . . . + b s ) + s . Suppose r > s . Given a formal Laurent series α in the variable r √ t of order − s , consider thesystem of equations (1.2). We express r √ t as a formal power series in r − s √ t ′ of order 1 using thefirst equation, and then substitute this expression into the second equation to get β ∈ k (( r − s √ t ′ )).Similar to equation (2.1), we have ∂ t ′ ( β ) = t. It follows that β is a formal Laurent series in r − s √ t ′ of order − s . Let Z = r √ t and let T ′ = r − s √ t ′ . Set a ( Z ) = Z s t∂ t ( α ) and b ( T ′ ) = − T ′ s t ′ ∂ t ′ β. a ( Z ) is a formal power series in Z of order 0 and b ( T ′ ) is a formal power series in T ′ of order0. From the system of equations (1.2), we get ( a ( Z ) = − ( T ′ Z ) r − s b ( T ′ ) = − ( T ′ Z ) r . (2.9)Similar to Theorem 1 and 1 ′ , to prove Theorem 2, it suffices to show the following theorem. Theorem 2 ′ . Suppose r > s . Given a formal power series a ( Z ) = P i ≥ a i Z i with a i ∈ k and a = 0, suppose b ( T ′ ) = P i ≥ b i T ′ i with b i ∈ k is a solution of the system of equations (2.9). Wehave b s = rr − s a s and F ( ∞ , (cid:16) [ r ] ∗ [ − r ( a Z − s + a Z − s + . . . + a s )] (cid:17) = [ r − s ] ∗ [ − ( r − s )( b T ′− s + b T ′ − s + . . . + b s ) + s . Proof.
The proof of b s = rr − s a s is similar to that of Theorem 1 ′ . Using the first equation of (2.9),we can express Z as a formal power series in the variable T ′ of order 1. We then substitute thisexpression into the second equation to get b ( T ′ ) is a formal power series in T ′ with nonzero constantterm. That is, b = 0. Let ζ be an r -th root of − k and let Z = ζ · Z . Let [ − ] : k (( z )) → k (( z ))be the automorphism of k -algebra defined by z
7→ − z. From the system of equations (2.9), we get ( P i ≥ b i T ′ i = ( T ′ Z ) r P i ≥ ζ i − s a i Z i = ( T ′ Z ) r − s . Since b = 0, by Theorem 1 ′ , we have F (0 , ∞ ) (cid:16) [ r − s ] ∗ [ − ( r − s )( b T ′− s + b T ′ − s + . . . + b s ) + s (cid:17) = [ r ] ∗ [ − r ( ζ − s a Z − s + ζ − s a Z − s + . . . + a s ) + s s − ] ∗ [ r ] ∗ [ − r ( a Z − s + a Z − s + . . . + a s )]= F (0 , ∞ ) (cid:16) F ( ∞ , (cid:16) [ r ] ∗ [ − r ( a Z − s + a Z − s + . . . + a s )] (cid:17)(cid:17) . The theorem holds by [2], Proposition 3.10.
Suppose r < s . Given a formal Laurent series α in the variable r √ t of order − s , consider thesystem of equations (1.3). We express r √ t as a formal Laurent series in s − r √ t ′ of order 1 using the15rst equation and then substitute this expression into the second equation to get β ∈ k (( s − r √ t ′ )).Similar to equation (2.1), we have ∂ t ′ ( β ) = t. It follows that β is a formal Laurent series in s − r √ t ′ of order − s . Let Z = r √ t and Z ′ = s − r √ t ′ . Set a ( Z ) = Z s t∂ t ( α ) and b ( Z ′ ) = Z ′ s t ′ ∂ t ′ ( β ) . Then a ( Z ) is a formal power series in Z of order 0 and b ( Z ′ ) is a formal power series in Z ′ of order0. From the system of equations (1.3), we get (cid:26) a ( Z ) = − ( ZZ ′ ) s − r b ( Z ′ ) = ( Z ′ Z ) r . (3.1)Similar to Theorem 1 and 1 ′ , to prove Theorem 3, it suffices to show the following theorem. Theorem 3 ′ . Suppose s > r . Given a formal power series a ( Z ) = P i ≥ a i Z i with a i ∈ k and a = 0, solve the system of equations (3.1) to get b ( Z ′ ) = P i ≥ b i Z ′ i for some b i ∈ k . Then b s = rs − r a s and F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − r ( a Z − s + a Z − s + . . . + a s )] (cid:17) = [ s − r ] ∗ [ − ( s − r )( b Z ′− s + b Z ′ − s + . . . + b s ) + s . Lemma 3.1.
Set h = a Z − s + a Z − s + . . . + a s . We can reduce Theorem 3 ′ to the case where s ≥ r and where h is irreducible with respect to the Galois extension k (( Z )) /k (( z )) .Proof. The proof of b s = rs − r a s is similar to that of Theorem 1 ′ and the proof of the last assertionis similar to that of Lemma 2.5. If s < r , then s > s − r ). Let ζ be an r -th root of − k andlet Z = ζ · Z . From the system of equations (3.1), we get ( P i ≥ b i Z ′ i = − ( Z ′ Z ) r P i ≥ ζ i − s a i Z i = ( Z Z ′ ) s − r . We prove b = 0 similarly as in Theorem 2 ′ . Applying this theorem to [ s − r ] ∗ [ − ( s − r )( b Z ′− s + . . . + b s ) + s ], we have F ( ∞ , ∞ ) (cid:16) [ s − r ] ∗ [ − ( s − r )( b Z ′− s + b Z ′ − s + . . . + b s ) + s (cid:17) = [ r ] ∗ [ − r ( ζ − s a Z − s + ζ − s a Z − s + . . . + a s ) + s s − ] ∗ [ r ] ∗ [ − r ( a Z − s + a Z − s + . . . + a s )]= F ( ∞ , ∞ ) (cid:16) F ( ∞ , ∞ ) (cid:0) [ r ] ∗ [ − rh ] (cid:1)(cid:17) . The lemma holds by [2], Proposition 3.12 (iv). 16rom now on, we assume h is irreducible.Let’s describe the formal connection F ( ∞ , ∞ ) (cid:0) [ r ] ∗ [ − rh ] (cid:1) on k (( z ′ )) . The formal connection [ − rh ] on k (( Z )) consist of a one dimensional k (( Z ))-vector space witha basis e ′ and a k -linear map Z∂ Z : k (( Z )) e ′ → k (( Z )) e ′ satisfying Z∂ Z ( ge ′ ) = ( Z∂ Z ( g ) − rhg ) e ′ for any g ∈ k (( Z )) . Since the formal connection [ − rh ] on k (( Z )) has slope s , we get k [[ Z ]] e ′ , Z − s k [[ Z ]] e ′ is a good lattices pair for it. Identify [ r ] ∗ [ − rh ] with k (( Z )) e ′ as k (( z ))-vector spaces. So the con-nection [ r ] ∗ [ − rh ] on k (( z )) has pure slope sr and k [[ Z ]] e ′ , Z − s k [[ Z ]] e ′ is a good lattices pair forthis connection. The action of the differential operator z∂ z on k (( Z )) e ′ is given by z∂ z ( ge ′ ) = ( z∂ z ( g ) − hg ) e ′ for any g ∈ k (( Z )) . Then for any i ∈ Z , we have z ∂ z ( Z − ( r + i ) e ′ ) = − r + ir Z − i e ′ − ( a Z − ( i + s ) e ′ + . . . + a s Z − i e ′ ) . By [2], Proposition 3.12 (ii), the map ι : k (( Z )) e ′ → F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17) is an isomorphism of k -vector spaces. As in [2], Proposition 3.14, ( ιZ − e ′ , . . . , ιZ − ( s − r ) e ′ ) is abasis of F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17) over k (( Z ′ )). By the relation ι ◦ z ∂ z = z ′ ◦ ι and − ι ◦ z = z ′ ∂ z ′ ◦ ι in [2], Proposition 3.12 (iii), we have z ′ ∂ z ′ ( ιZ − ( i + s − r ) e ′ ) = − ιZ − ( i + s ) e ′ = a s a ιZ − i e ′ + . . . + a a ιZ − ( i + s − e ′ + 1 a z ′ ιZ − ( r + i ) e ′ + r + ira ιZ − i e ′ . Let A = − a s a − a s − r +1 a . . . − a s − r a − a z ′ . . . − a s − r − a . . . ...1 − a a . i ∈ Z , let B i be the s × s -matrix whose entries are all zero except the (1 , s )-th entry whichis valued by − r + ira . We have( ιZ − ( i +1) e ′ , . . . , ιZ − ( i + s ) e ′ ) = ( ιZ − i e ′ , . . . , ιZ − ( i + s − e ′ )( A + B i ) . So z ′ ∂ z ′ ( ιZ − e ′ , . . . , ιZ − s e ′ ) = − ( ιZ − ( r +1) e ′ , . . . , ιZ − ( r + s ) e ′ )= − ( ιZ − e ′ , . . . , ιZ − s e ′ ) Y ≤ i ≤ r ( A + B i ) . Consider the connection[ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17) = k (( Z ′ )) ⊗ k (( z ′ )) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17) . Set ∧ = diag { Z ′ , . . . , Z ′ s } and ε ′ = ( Z ′ ⊗ ιZ − e ′ , . . . , Z ′ s ⊗ ιZ − s e ′ ). We have Z ′ ∂ Z ′ ( ε ′ ) = ε ′ · (cid:16) diag { , . . . , s } − s − rz ′ ∧ − (cid:16) Y ≤ i ≤ r ( A + B i ) (cid:17) ∧ (cid:17) = ε ′ · (cid:16) diag { , . . . , s } − s − rZ ′ s Y ≤ i ≤ r (cid:16) Z ′ ∧ − ( A + B i ) ∧ (cid:17)(cid:17) . We have Z ′ ∧ − A ∧ = − a s a Z ′ s − a s − r +1 a Z ′ s − r +1 . . . − a s − r a Z ′ s − r − a . . . − a s − r − a Z ′ s − r − . . . ...1 − a a Z ′ and Z ′ ∧ − B i ∧ is the s × s -matrix whose entries are all zero except the (1 , s )-th entry which isvalued by − r + ira Z ′ s . So we can writediag { , , . . . , s } − s − rZ ′ s Y ≤ i ≤ r (cid:16) Z ′ ∧ − ( A + B i ) ∧ (cid:17) = − ( s − r ) X i ≥ Z ′ i − s C i and (cid:0) Z ′ ∧ − A ∧ (cid:1) r = X i ≥ Z ′ i C ′ i for some matrices C i and C ′ i with entries in k . Then C i = C ′ i for all 0 ≤ i ≤ s − C ′ s − C s = diag { s − r , . . . , ss − r } − P P is the s × s -matrix whose entries are all zero except the ( i, i + s − r )-th entry which isvalued by − r + ira (1 ≤ i ≤ r ) . Let W be the k -vector space of column vectors in k of length s. Wehave
Lemma 3.2.
Suppose s ≥ r and h is irreducible with respect to the Galois extension k (( Z )) /k (( z )) .Given α , . . . , α s ∈ k with α = 0 , the following three conditions are equivalent: (1) F ( ∞ , ∞ ) (cid:0) [ r ] ∗ [ − rh ] (cid:1) = [ s − r ] ∗ [ − ( s − r ) P ≤ i ≤ s α i Z ′ i − s ] . (2) [ − ( s − r ) P ≤ i ≤ s α i Z ′ i − s ] is a subconnection of [ s − r ] ∗ F ( ∞ , ∞ ) (cid:0) [ r ] ∗ [ − rh ] (cid:1) . (3) There exist N ∈ Z and w , . . . , w s ∈ W such that w = 0 and (cid:26) P ≤ i ≤ k ( C i − α i ) w k − i = 0 (0 ≤ k ≤ s − , P ≤ i ≤ s − ( C i − α i ) w s − i + ( C s − α s − Ns − r ) w = 0 . (3.2) Proof.
Set U = W ⊗ k k (( Z ′ )) and W = W ⊗ k k [[ Z ′ ]]. Let u = ( u , . . . , u s ) be the canonical basisof W. There exists a unique connection (
U, Z ′ ∂ Z ′ ) such that the action of Z ′ ∂ Z ′ on elements of W can be written as Z ′ ∂ Z ′ ( w ) = − ( s − r ) X i ≥ Z ′ i − s C i ( w ) . The map of k (( Z ′ ))-vector spaces U → [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17) which maps each u i to Z ′ i ⊗ ιZ − i e ′ is a surjective morphism of connections. We have Z ′ s +1 ∂ Z ′ ( W ) ⊂W . Let ψ : W → W /Z ′ W ∼ = W be the canonical map. The k -linear action on W ∼ = W /Z ′ W induced by Z ′ s +1 ∂ Z ′ is − ( s − r ) C . Write Z ′ ∧ − A ∧ = X i ≥ Z ′ i D i for some matrices D i with entries in k . The characteristic polynomial of D is λ s + a λ r . So W is the direct sum of two subspaces W and W , invariant under D , and such that D | W isnilpotent, D | W is invertible. Then dim W = r and dim W = s − r . Since C = D r , we have W and W are C -invariant, and then C | W = 0, C | W is invertible. By the splitting lemma in [7],2, W is the direct sum of two free submodules W and W , invariant under Z ′ s +1 ∂ Z ′ , and suchthat W = ψ ( W ) , W = ψ ( W ) . Let U , U be the subconnections of U generated by W , W ,respectively. Then U = U ⊕ U . The induced action of Z ′ s +1 ∂ Z ′ on W is 0, so the slopes of the19onnection U are all < s . But [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17) is an s − r dimensional connectionon k (( Z ′ )) with pure slope s , we haveHom conn . (cid:16) U , [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17)(cid:17) = (0)and then U ∼ = [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17) . For any one dimensional formal connection L on k (( Z ′ )) with slope s , we haveHom conn . ( L, U ) = (0)and then Hom conn . ( L, U ) = Hom conn . ( L, U ) M Hom conn . ( L, U )= Hom conn . (cid:16) L, [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17)(cid:17) . So to find a one dimensional subconnection in [ s − r ] ∗ (cid:16) F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17)(cid:17) is equivalent to findinga one dimensional subconnection in U of slope s . By Lemma 3.3, the remainder proof is similar tothat of Lemma 2.6. Lemma 3.3.
Suppose s ≥ r . Given α , . . . , α s ∈ k with α = 0 , there exist w , . . . , w s ∈ W suchthat w = 0 and X ≤ i ≤ k ( C i − α i ) w k − i = 0 (0 ≤ k ≤ s ) (3.3) if and only if there exist w ′ , . . . , w ′ s ∈ W such that w ′ = 0 and (cid:26) P ≤ i ≤ k ( C ′ i − α i ) w ′ k − i = 0 (0 ≤ k ≤ s − , P ≤ i ≤ s − ( C ′ i − α i ) w ′ s − i + ( C ′ s − α s − s − r s − r ) w ′ = 0 . (3.4) Moreover, there are at most s − r ( s + 1) -tuples ( α , . . . , α s ) in k such that α = 0 and the systemof equations (3.3) (resp. (3.4)) holds for some w , . . . , w s ∈ W with w = 0 (resp. w ′ , . . . , w ′ s ∈ W with w ′ = 0 ).Proof. Let η be a primitive ( s − r )-th root of unity in k . We fix an ( s − r )-th root ( − a ) s − r of − a .For any 1 ≤ j ≤ s − r , set e ′ j to be the column vector (0 , . . . , , η ( r +1) j ( − a ) r +1 s − r , . . . , η sj ( − a ) ss − r )and ε ′ j the row vector ( η − j ( − a ) − s − r , . . . , η − sj ( − a ) − ss − r ). We have C · e ′ j = η − rj ( − a ) − rs − r · e ′ j , ε ′ j · C = η − rj ( − a ) − rs − r · ε ′ j , ε ′ i · e ′ j = ( s − r ) δ ij . d = ( r, s ). Let W be as in Lemma 3.2. We have C | W = 0 . Then ker( C − η − rj ( − a ) − rs − r ) isgenerated by those e ′ k with s − r | ( k − j ) d , and im( C − η − rj ( − a ) − rs − r ) is generated by W andthe other e ′ k ’s. Soim( C − η − rj ( − a ) − rs − r ) = { w ∈ W | ε ′ k · w = 0 for all k satisfying s − r | ( k − j ) d } . For the only if part, suppose the system of equations (3.3) holds for some w , . . . , w s ∈ W with w = 0. So α = η − rj ( − a ) − rs − r for some integer j and then w = P s − r | ( i − j ) d σ i e ′ i for some σ i ∈ k . Since s ≥ r , for any 1 ≤ k, l ≤ s − r , we have ε ′ k · (diag { s − r , . . . , ss − r } − P − s − r s − r ) e ′ l = X r +1 ≤ i ≤ s is − r η ( l − k ) i − X ≤ i ≤ r r + ir η ( l − k ) i − s − r δ kl . If k = l , then ε ′ k · (diag { s − r , . . . , ss − r } − P − s − r s − r ) e ′ l = X r +1 ≤ i ≤ s is − r − X ≤ i ≤ r r + ir − s − r . If k = l and s − r | ( l − k ) d , then ( η l − k ) d = 1 and η l − k = 1. We have ε ′ k · (diag { s − r , . . . , ss − r } − P − s − r s − r ) e ′ l = s − rd X ≤ i ≤ d is − r η ( l − k ) i − rd X ≤ i ≤ d ir η ( l − k ) i = 0 . So ε ′ k · (diag { s − r , . . . , ss − r } − P − s − r s − r ) w = 0 if s − r | ( k − j ) d . Therefore(diag { s − r , . . . , ss − r } − P − s − r s − r ) w = ( C − α ) w for some w ∈ W . Then w ′ = w , . . . , w ′ s − = w s − , w ′ s = w s − w satisfy the system of equations(3.4). Reversing the above argument, we get the if part. The characteristic polynomial of D is λ s + a λ r . Each nonzero root of this polynomial is simple. Since P i ≥ Z ′ i C ′ i = ( Z ′ ∧ − A ∧ ) r and C = D r , the proof of the last assertion is similar to that of Lemma 2.7.Now we are ready to prove Theorem 3 ′ . Similar to Remark 2.2, we assume a ( Z ) = P ≤ i ≤ s a i Z i .Then the first equation of (3.1) means that Z ′ Z is a root in k [[ Z ′ ]] with nonzero constant term ofthe polynomial λ s + a a Z ′ λ s − + . . . + a s a Z ′ s + 1 a λ r ∈ k [[ Z ′ ]][ λ ] . Z ′ ∧ − A ∧ . The characteristic poly-nomial of D is λ s + a λ r which has no nonzero multiple roots, by Hensel’s lemma, Z ′ ∧ − A ∧ has an eigenvector P i ≥ Z ′ i w i corresponding this eigenvalue Z ′ Z with w = 0. Since P i ≥ Z ′ i C ′ i =( Z ′ ∧ − A ∧ ) r , we have (cid:16) X i ≥ Z ′ i C ′ i (cid:17)(cid:16) X i ≥ Z ′ i w i (cid:17) = (cid:16) Z ′ Z (cid:17) r (cid:16) X i ≥ Z ′ i w i (cid:17) = (cid:16) X i ≥ b i Z ′ i (cid:17)(cid:16) X i ≥ Z ′ i w i (cid:17) . That is, X ≤ i ≤ k ( C ′ i − b i ) w k − i = 0 for any k ≥ . Recall that s ≥ r and P ≤ i ≤ s a i Z i − s is assumed to be irreducible. By Lemma 3.2 and 3.3, wehave F ( ∞ , ∞ ) (cid:16) [ r ] ∗ [ − rh ] (cid:17) = [ s − r ] ∗ [ − ( s − r )( b Z ′− s + b Z ′ − s + . . . + b s − s − r s − r )]= [ s − r ] ∗ [ − ( s − r )( b Z ′− s + b Z ′ − s + . . . + b s ) + s . eferences [1] Beilinson, A.; Bloch, S.; Esnault, H.: ǫ -factors for Gauß-Manin determinants, Moscow Math-ematical Journal, vol. 2, (2002), 477-532.[2] Bloch, S. and Esnault, H.: Local Fourier Transforms and rigidity for D -Modules, Asian J.Math. 8 (2004), no. 4, 587–605.[3] Deligne, P.: ´Equations Diff´erentielles `a Points Singuliers R´eguliers, Lectures Notes ,Springer Verlag.[4] Fu, Lei.: Calculation of ℓ -adic local Fourier transformations, arXiv:math/0702436.[5] Katz, N.: On the calculation of some differential Galois groups, Inv. Math. (1986), 13-61.[6] Laumon, G.: Transformation de Fourier, constantes d’´equations fonctionnelles et conjecturede Weil, Publ. Math. IHES65