Newton polygons of L -functions of polynomials x d +a x d−1 with p≡−1modd
aa r X i v : . [ m a t h . N T ] O c t NEWTON POLYGONS OF L -FUNCTIONS OF POLYNOMIALS x d + ax d − WITH p ≡ − d YI OUYANG, SHENXING ZHANG
Abstract.
For prime p ≡ − d and q a power of p , we obtain the slopesof the q -adic Newton polygons of L -functions of x d + ax d − ∈ F q [ x ] with respectto finite characters χ when p is larger than an explicit bound depending onlyon d and log p q . The main tools are Dwork’s trace formula and Zhu’s rigidtransform theorem. Main results
Let q = p h be a power of the rational prime number p . Let v be the normalizedvaluation on Q p with v ( p ) = 1. For a polynomial f ( x ) ∈ F q [ x ], let ˆ f ∈ Z q [ x ] beits Teichm¨uller lifting. For a finite character χ : Z p → C × p of order p m χ , define the L -function L ∗ ( f, χ, t ) = exp ∞ X m =1 S ∗ m ( f, χ ) t m m ! , (1)where S ∗ m ( f, χ ) is the exponential sum S ∗ m ( f, χ ) = X x ∈ µ qm − χ (Tr Q qm / Q p ˆ f ( x )) (2)and µ n is the group of n -th roots of unity. Then L ∗ ( f, χ, t ) is a polynomial of degree p m χ − d by Adolphson-Sperber [AS] and Liu-Wei [LWe]. We denote NP q ( f, χ, t ) the q -adic Newton polygon of L ∗ ( f, χ, t ).We fix a character Ψ : Z p → C × p of order p , and denote L ∗ ( f, t ) = L ∗ ( f, Ψ , t )and NP q ( f, t ) = NP q ( f, Ψ , t ). When p ≡ d , it is well-known that NP q ( f, t )coincides the Hodge polygon with slopes { i/d : 0 ≤ i ≤ d − } .Let a be a nonzero element in F q . For f ( x ) = x d + ax s ( s < d ), Liu-Niu andZhu obtained the slopes of NP q ( f, t ) for p large enough under certain conditionsin [LN2, Theorem 1.10] and [Z2], but these conditions are not so easy to check.For f ( x ) = x d + ax , Zhu, Liu-Niu and Ouyang-J. Yang obtained the slopes in [Z2,Theorem 1.1], [LN1, Theorem 1.10] and [OY, Theorem 1.1], see also R. Yang [Y, § q ( f, χ, t ) when the order of χ is large enough. In this way for p sufficientlylarge, they can obtain the slopes of NP q ( f, χ, t ) based on the slopes of NP q ( f, χ , t )with χ a character of order p . In [N], Niu gave a lower bound of the New-ton polygon NP q ( f, χ, t ). In [OY, Theorem 4.3], Ouyang–Yang showed that if theNewton polygon of L ∗ ( f, t ) is sufficiently close to its Hodge polygon, the slopes ofNP q ( f, χ, t ) for χ in general follow from the slopes of NP q ( f, t ). As a consequence Mathematics Subject Classification.
Primary 11; Secondary 14.Corresponding author: S. Zhang. Email: [email protected]. they obtained the slopes of NP q ( x d + ax, χ, t ) when p is bigger than an explicitbound depending only on d and h .Our main results are the following two theorems. Theorem 1.
Let f ( x ) = x d + ax d − be a polynomial in F q [ x ] with a = 0 . Let N ( d ) = d +34 for q = p and d for general q . If p ≡ − d and p > N ( d ) , the q -adic Newton polygon of L ∗ ( f, t ) has slopes { w , w , . . . , w d − } , where w i = ( p +1) id ( p − , if i < d ; ( p +1) i − dd ( p − = , if i = d ; ( p +1) i − dd ( p − , if i > d . Remark. (1) For general p , write pi = dk i + r i with 1 ≤ i, r i ≤ d −
1. If r i > s for any 1 ≤ i ≤ s , then one can decide that the first s + 1 slopes of NP q ( f, t ) are { , k +1 p − , . . . , k s +1 p − } by our method for sufficiently large p . For the rest of slopes,one needs to calculate the determinants of submatrices of “Vandermonde style”matrices.(2) The slopes in our case coincide Zhu’s result in [Z2]. Theorem 2.
Assume f ( x ) and N ( d ) as above. For any non-trivial finite character χ , if p ≡ − d and p > max { N ( d ) , h ( d − d + 1 } , the q -adic Newton polygon of L ∗ ( f, χ, t ) has slopes { p − m χ ( i + w j ) : 0 ≤ i ≤ p m χ − − , ≤ j ≤ d − } . Preliminaries
Dwork’s trace formula.
We will recall Dwork’s work for f ( x ) = x d + ax d − .For general f , one can see [OY, § γ ∈ Q p ( µ p ) be a root of the Artin-Hasse exponential series E ( t ) = exp( ∞ X m =0 p − m t p m )such that v ( γ ) = p − . Fix a γ /d ∈ ¯ Q p . Let θ ( t ) = E ( γt ) = ∞ X m =0 γ m t m be Dwork’s splitting function. Then v ( γ m ) ≥ m/ ( p − γ m = γ m /m ! for0 ≤ m ≤ p −
1. Let F ( x ) = θ ( x d ) θ ( ax d − ) = ∞ X i =0 F i x i , then F i = X dm +( d − n = i γ m γ n a n . One can see m + n ≥ i/d and v ( F i ) ≥ id ( p − . EWTON POLYGONS 3
Set A = ( F pi − j γ ( j − i ) /d ) i,j ≥ . This is a nuclear matrix over Q q ( γ /d ) with v ( F pi − j γ ( j − i ) /d ) ≥ pi − jd ( p −
1) + j − id ( p −
1) = id .
We extend the Frobenius ϕ to Q q ( γ /d ) with ϕ ( γ /d ) = γ /d . Theorem 3 (Dwork) . Let A h = A ϕ ( A ) · · · ϕ h − ( A ) . Then L ∗ ( f, t ) = det ϕ − ( I − tA h )det ϕ − ( I − tqA h ) . Zhu’s rigid transformation theorem.
Let U = ( u ij ) i,j ≥ be a nuclearmatrix over Q q ( γ /d ). Then the Fredholm determinant det( I − tU ) is well definedand p -adic entire (see [S]). Writedet( I − tU ) = c + c t + c t + · · · . For 0 ≤ t < t < · · · < t s , denote by U ( t , . . . , t s ) the principal sub-matrixconsisting of ( t i , t j )-entries of U for 1 ≤ i, j ≤ s . In particular, denote U [ s ] = U (0 , , . . . , s − c = 1 and for s ≥ c s = ( − s X ≤ t 1. Then x [ n ] isa polynomial of x of degree n and { ( x + j )[ t ] : 0 ≤ t ≤ j − } is a basis of the spaceof polynomials of degree ≤ j − 1. Thus we can write(( k − x − j − 1] = c ( j ) + j − X t =1 c t ( j ) · ( x + j )[ t ] . YI OUYANG, SHENXING ZHANG Let x = − j , we get c ( j ) = (( k − − j ) − j − 1] = ((1 − k ) j − j − . For any 1 ≤ u ≤ j − ≤ ( k − j + u < kj ≤ k ( d − ≤ p. Hence p ∤ (1 − k ) j − u and v ( c ( j )) = 0.Let D = diag { a, a , . . . , a s } and M ′ = ( a ′ ij ) ≤ i,j ≤ s with a ′ ij = a ij a − i − j , then M ( s ) = DM ′ D. (3)Let a ′′ ij := ( ki − i − i + s )! a ′ ij . Then a ′′ ij = ( ki − i − j − · ( i + s )[ s − j ]= j − X t =0 c t ( j ) · ( i + j )[ t ] · ( i + s )[ s − j ] , = j − X t =0 c t ( j ) · ( i + s )[ s − j + t ]= j X t =1 ( i + s )[ s − t ] · c j − t ( j ) . Define c j − t ( j ) := 0 for j < t . Write M ′′ = ( a ′′ ij ) ≤ i,j ≤ s , M = (( i + s )[ s − t ]) ≤ i,t ≤ s and M = ( c j − t ( j )) ≤ t,j ≤ s . Then M ′′ = M M . (4)Write x [ n ] = n X t =0 c ′ t ( n ) x t , then c ′ n ( n ) = 1 and ( i + s )[ s − j ] = s − j X t =0 c ′ t ( s − j )( i + s ) t . Define c ′ t ( n ) := 0 for t > n . Write M = (( i + s ) t − ) ≤ i,t ≤ s and M = ( c ′ t − ( s − j )) ≤ t,j ≤ s . Then M = M M . (5)Notice that M is a Vandermonde matrix with determinant det M = Q st =1 t s − t .One can also easily finddet M = ( − [ s/ and det M = s Y i =1 c ( i ) . (6)Now by (3), (4), (5) and (6),det M ( s ) = a s ( s +1) ( − [ s/ s Y i =1 i s − i c ( i )( ki − i − i + s )! . Hence v (det M ( s )) = 0. (cid:3) Denote O ( x ) a number in Q p with valuation ≥ v ( x ) for x ∈ Q p . EWTON POLYGONS 5 Lemma 6. ( i ) For i + j < d , F pi − j = γ ki ( a ij + O ( γ )) . ( ii ) For i + j ≥ d , v ( F pi − j ) = ki − and F pi − ( d − i ) = γ ki − (1 + O ( γ ))( ki − . Proof. Let m = ( ki − i − j, if j < d − i ; ki − i − j + d − , if j ≥ d − i,n = ( i + j, if j < d − i ; i + j − d, if j ≥ d − i. Then pi − j = dm + ( d − n and 0 ≤ n ≤ d − 1. This lemma follows from F pi − j = X l ≥ γ m − ( d − l γ n + dl a n + dl = γ m γ n a n (1 + O ( γ )) = γ m + n a n m ! n ! (1 + O ( γ )) . (cid:3) Proposition 7. For ≤ s ≤ d − , the valuation of det A [ s +1] is w + w + · · · + w s .Proof. Note that the first row of A is (1 , , , . . . ). Let A be the matrix by deletingthe first row and column of A [ s + 1]. Then det A [ s + 1] = det A .Let D = diag { γ /d , γ /d , . . . , γ s/d } , D = diag { γ k − , γ k − , . . . , γ ( d − k − } and B [ s ] = ( γ − ki F pi − j ) ≤ i,j ≤ s . Then A = D − D B [ s ] D . It suffices to compute v (det B [ s ]).Note that for s = d − B [ d − 1] = γa + O ( γ ) · · · γa ,d − + O ( γ ) O ( γ )( k − ... . . . O ( γ )(2 k − b ,d − γa d − , + O ( γ ) . . . . . . ... O ( γ )(( d − k − b d − , · · · b d − ,d − with v ( b ij ) = 0. If 1 ≤ s ≤ d − , then B [ s ] = γa + O ( γ ) · · · γa s + O ( γ )... . . . ... γa s + O ( γ ) · · · γa ss + O ( γ ) has determinant det B [ s ] = γ s (det M ( s ) + O ( γ )) . The valuation of det B [ s ] is sv ( γ ).If d ≤ s ≤ d − 1, then B [ s ] = (cid:18) B [ d − − s ] P P Q (cid:19) . The valuation of any entry of B [ d − − s ] , P , P is v ( γ ) and Q ≡ k − . . . d − k − * mod γ. YI OUYANG, SHENXING ZHANG Thus Q is invertible over the ring of integers of Q p ( γ ). The determinantdet B [ s ] = det Q det( B [ d − − s ] − P Q − P ) = det Q det B [ d − − s ](1 + O ( γ ))has valuation ( d − − s ) v ( γ ).Finally, A = D − D B [ s ] D has valuation( s X i =1 ( ki − 1) + min { s, d − − s } ) v ( γ ) = w + w + · · · + w s . (cid:3) Proof of Theorem 1. For 1 ≤ s ≤ d − 1, we have v (det A [ s + 1]) = X i ≤ s w i = ( s ( s +1)2 d + s ( s +1) d ( p − , if s ≤ ( d − / s ( s +1)2 d + ( d − s )( d − s − d ( p − , if s ≥ d/ ≤ s ( s + 1)2 d + d − d ( p − . If p > d +34 , then d − d ( p − < /d . For 0 ≤ t < t < · · · < t s , assume t s = s . Since v ( F pi − j γ ( j − i ) /d ) ≥ i/d, we have v (det A [ t , . . . , t s ]) ≥ s + s + 22 d > v (det A [ s + 1]) . Thus v ( c s +1 ) = v (det A [ s + 1]) = P i ≤ s w s and { w , w , . . . , w d − } are slopes ofNP p (det( I − tA )).If moreover p > d , then p ≥ d +12 and d − d ( p − ≤ d . Choose β i = i/d inTheorem 4, we have v ( C s +1 ) = h ( w + w + · · · + w s )and NP q (det( I − tA h [ d ])) = NP p (det( I − tA [ d ])) . Thus w , w , . . . , w d − are q -adic slopes of NP q (det ϕ − ( I − tA h )).By Theorem 3, det ϕ − ( I − tA h ) = L ∗ ( f, t ) det ϕ − ( I − tqA h ) . Since the valuation of any entry of A h is ≥ 0, the q -adic slopes of det ϕ − ( I − tA h )are ≥ q -adic slopes of det ϕ − ( I − tqA h ) are ≥ 1. Thus any q -adic slopeof det ϕ − ( I − tA h ) less than 1 must be a q -adic slope of L ∗ ( f, t ). But L ∗ ( f, t ) hasdegree d , hence w , . . . , w d − are all slopes of L ∗ ( f, t ). (cid:3) The case for general χ . Let f ( x ) ∈ F q [ x ] be a polynomial with degree d .Assume p ∤ d . Let NP( f, x ) be the piecewise linear function whose graph is the q -adic Newton polygon of det( I − tA h ). Let HP( f, x ) be the piecewise linear functionwhose graph is the polygon with vertices( k, k ( k − d ) , k = 0 , , , . . . . EWTON POLYGONS 7 Then NP( f, x ) ≥ HP( f, x ) (cf. [LW, OY]). Setgap( f ) = max x ≥ { NP( f, x ) − HP( f, x ) } . Theorem 8. (See [OY, Theorem 4.3] .) Let α < α < · · · < α d − < denotethe slopes of the q -adic Newton polygon of L ∗ ( f, t ) . If gap( f ) < /h , then the q -adicNewton polygon of L ∗ ( f, χ, t ) has slopes { p − m χ ( i + α j ) : 0 ≤ i ≤ p m χ − − , ≤ j ≤ d − } for any non-trivial finite character χ .Proof of Theorem 2. The slopes of NP( f, x ) are { i + w j : i ≥ , ≤ j ≤ d − } . Notice that d − X i =0 w i = d − X i =0 id , NP( f, x ) − HP( f, x ) is a periodic function with period d . For 0 ≤ k < d ,NP( f, x ) − HP( f, x ) ≤ X i ≤ ( d − / id ( p − ≤ d − d ( p − . If p > h ( d − d + 1, then gap( f ) < /h and this concludes the proof. (cid:3) Acknowledgements. This paper was prepared when the authors were visitingthe Academy of Mathematics and Systems Science and the Morningside Center ofMathematics of Chinese Academy of Sciences. We would like to thank Professor YeTian for his hospitality. We also would like to thank Jinbang Yang for many helpfuldiscussions. This work was partially supported by NKBRPC(2013CB834202) andNSFC(11171317 and 11571328). References [AS] A. Adolphson, S. Sperber, Newton polyhedra and the degree of the L -function associatedto an exponential sum , Invent. Math., 1987, 88(3): 555–569.[DWX] C. Davis, D. Wan, L. Xiao, Newton slopes for Artin-Schreier-Witt towers , Math. Ann.,DOI: 10.1007/s00208-015-1262-4.[LN1] C. Liu, C. Niu, Generic twisted T -adic exponential sums of binomials , Sci. China. Math.,2011, 54(5): 865–875.[LN2] C. Liu, C. Niu, Generic twisted T -adic exponential sums of polynomials , J. NumberTheory, 2014, 140(7): 38-C59.[LW] C. Liu, D. Wan, T -adic exponential sums over finite fields , Algebra Number Theory, 2009,3(5): 489–509.[LWe] C. Liu, D. Wei, The L -function of Witt coverings , Math. Z., 2007, 255(1): 95–115.[N] C. Niu, The p -adic Riemann hypothesis of exponential sums associated to certain bino-mials over finite fields , (in Chinese) Journal of Liaocheng University: Natural ScienceEdition, 2011, 24(4): 89–94.[OY] Y. Ouyang, J. Yang, Newton polygons of L -functions of polynomials x d + ax , J. NumberTheory.[S] J. P. Serre, Endomorphismes compl´etement continus des espaces de Banach p -adiques ,(in French) Inst. Hautes ´Etudes Sci. Publ. Math. No., 1962, 12: 69–85.[Y] R. Yang, Newton polygons of L -functions of polynomials of the form x d + λx , Finite FieldsAppl., 2003, 9(1): 59–88.[Z1] H. J. Zhu, Asymptotic variations of L-functions of exponential sums , arXiv:1211.5875.[Z2] H. J. Zhu, Generic A -family of exponential sums , J. Number Theory, 2014, 143: 82–101. YI OUYANG, SHENXING ZHANG Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences,University of Science and Technology of China, Hefei, Anhui 230026, PR China E-mail address ::