Combinatorial Congruences and ψ-operators

Transcription

1 Combinatorial Congruences and ψ-operators Daqing Wan Department of Mathematics University of California, Irvine CA September 27, 2005 Abstract The ψ-operator for (ϕ, Γ-modules plays an important role in the study of Iwasawa theory via Fontaine s big rings In this note, we prove several sharp estimates for the ψ-operator in the cyclotomic case These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of which extends the classical congruences of Flec (93 and Weisman (977 Combinatorial Congruences Let p be a prime, n Z >0 Throughout this paper, let [x] denote the integer part of x if x 0 and [x] = 0 if x < 0 In the author s course lectures [4] on Fontaine s theory and p-adic L-functions given at UC Irvine (spring 2005 and at the Morningside Center of Mathematics (summer 2005, the following two congruences were discovered Theorem For integers r Z, 0, we have r(modp ( n ( n ( r p 0 (modp [ n p p ] We shall see that the theorem comes from a simple estimate of ψ(π n for the cyclotomic ϕ-module Partially supported by NSF The author thans ZW Sun for helpful discussions on combinatorial congruences

2 Theorem 2 For integer 0, we have i 0 + +i p =n i +2i 2 + r(modp ( ( n i +2i 2 + r p i 0 i i p 0 (modp [ n(p p p ] As we shall see, this theorem comes from a simple estimate of ψ(π n for the cyclotomic ϕ-module Note that when p = 2, Theorem 2 is equivalent to Theorem The above two congruences can be extended from p to q = p a, where a is a positive integer To do so, it suffices to estimate the a-th iterate ψ a (π n This can be done by induction The estimate of ψ a (π n for n > 0 leads to Theorem 3 For integers r Z, 0 and a > 0, we have r(modp a ( n ( n ( r p a The estimate of ψ a (π n for n < 0 leads to Theorem 4 Let S (n, r, p a = i 0 + +i (p a =n i +2i 2 + r(modp a Then for integer 0, we have ( n i 0 i (p a 0(modp [ n p a p a p a ] (p ( (i + 2i 2 + r/p a S (n, r, p a 0(modp [(an a+(p (ap a+ p ] As ZW Sun informed me, the special case = 0 of Theorem was first proved by Flec [] in 93, and the special case of Theorem 3 for = 0 was first proved by Weisman [5] in 977 A different extension of Theorem and Weisman s congruence has been obtained by ZW Sun [2] using different combinatorial arguments Motivated by applications in algebraic topology, Sun-Davis [3] proved yet another extension: r(modp a ( n ( n ( r p a 0(modp (ordp([n/pa ]! ord p(! 2

3 2 The operator ψ Let p be a fixed prime Let π be a formal variable Let A + = Z p [[π]] be the formal power series ring over the ring of p-adic integers Let A be the p-adic completion of A + [ π ], and let B = A[ p ] be the fraction field of A The rings A +, A and B correspond to A + Q p, A Qp and B Qp in Fontaine s theory We shall not discuss the Galois action on A, which is not needed for our present purpose The Frobenius map ϕ acts on the above rings by ϕ(π = ( + π p If we let [ε] = + π, then ϕ([ε] = [ǫ] p The map ϕ is inective of degree p This gives Proposition 2 {, π,, π p } (and {, [ε],, [ε] p } is a basis of A over the subring ϕ(a Definition 22 The operator ψ : A A is defined by ( p ψ(x = ψ [ε] i ϕ(x i = x 0 = p ϕ (Tr A/ϕ(A (x, i=0 where x : A A denotes the multiplication by x as ϕ(a-linear map Example 23 ψ([ε] n = It is clear that ψ is ϕ -linear: { [ε] n/p, if p n; 0, if p n ψ(ϕ(ax = aψ(x a, x A Example 24 Let a be a positive integer relatively prime to p Then ψ( ( + π a = ( + π a In fact, ( ψ [ε] a ( = ψ [ε] ap [ε]ap [ε] ( = [ε] a ψ + [ε] a + + [ε] (p a = [ε] a = ( + π a 3

4 By p-adic continuity, the above example holds for any p-adic unit a Z p In the general theory of (ϕ, Γ-modules, it is important to find the fix points of ψ for applications to p-adic L-functions and Iwasawa theory In the simplest cyclotomic case, we have the following description for the fixed points (see [4] Proposition 25 { A ψ= = } π Z p Z p ϕ p p (x x [ε] i ϕ(a i + πz p [[π]], a i = 0, where a i Z p =0 For example, if a is a positive integer relatively prime to p, then the element a ( + π a π (A+ ψ= gives the cyclotomic units and the Euler system This element is the Amice transform of a p-adic measure which produces the p-adic zeta function of Q This type of connections is conectured to be a general phenomenon for (ϕ, Γ-modules coming from global p-adic Galois representations 3 Sharp estimates for ψ The ring A is a topological ring with respect to the (p, π-topology A basis of neighborhoods of 0 is the sets p A + π n A +, where Z and n N The operator ψ is uniformly continuous This continuity will give rise to combinatorial congruences For s A +, one checs that i= i= ψ(π p s = ψ(([ε] p s = ψ(([ε] p s + pss = πψ(s + pψ(ss (p, πψ(sa + In particular, Thus, by iteration, we get ψ(π p A + (p, πa + 4

5 Proposition 3 (Wea Estimate Let n 0 Then ψ(π n A + (p, π [n/p] A + = [n/p] π p [n/p] A + Since the exponent [(n p/p] is decreasing in, this proposition implies that for x π n A +, we have ψ(x = a π, a Z p, ord p (a [(n p/p] This already gives a non-trivial combinatorial congruence Let r be an integer Let us calculate ψ(π n [ε] r in a different way Lemma 32 ψ(π n [ε] r = 0 π r(modp ( n ( n Proof Since π = [ε] and [ε] = + π, we have ( ( r/p ψ(π n [ε] r = ψ(([ε] n [ε] r ( n ( n = ψ ( [ε] n r =0 ( n = ( n [ε] ( r/p r(modp ( n ( ( r/p = ( n r(modp 0 = ( ( n ( r/p π ( n 0 r(modp π Comparing the coefficients of π in this equation and the wea estimate, we get Corollary 33 (Wea Congruence Let n 0 We have ( ( n ( r/p ( n 0(modp [(n p/p] r(modp 5

6 The above simple estimate is crude and certainly not optimal since we ignored a factor of π We now improve on it Theorem 34 (Sharp Estimate I For n 0, we have ψ(π n A + [n/p] π p [ n p p ] A + Proof We prove the theorem by induction The theorem is trivial if n p Write Then, ϕ(π = ( + π p = π p pπs, s A + ψ(π p s = ψ((ϕ(π + pπs s = πψ(s + pψ(πs s This proves that the theorem is true for n = p Let n > p Assume the theorem holds for n It follows that ψ(π n A + = ψ(π p π n p A + πψ(π n p A + + pψ(π n+ p A + By the induction hypothesis, the right side is contained in = [(n p/p] π [n/p] = [(n+ p/p] π p [ n p p p ] A + + p [(n+ p/p] π p [ n p p ] A + + π p [ n p p ] A + π p [ n p p p ] A + The function [(n p/(p ] is decreasing in and vanishes for [n/p] Comparing the coefficients of π in the lemma and the above sharp estimate, we deduce Corollary 35 (Sharp Congruence I Let r Z ( ( n ( r/p ( n 0(modp [ n p p ], r(modp where 0 is a non-negative integer 6

7 Theorem 36 (Sharp Estimate II For n > 0, we have Proof Note that ( [n(p /p] n(p p ψ π na+ p[ p ] A + πn ϕ(π/π = π p + ( ( p p π p (π p, p, p so (ϕ(π/π n (π p, p n Then ( ( ( ϕ(π n ψ π na+ = ψ ϕ(π n A + π = (( ϕ(π n π nψ A + π n π n p n i ψ(π i(p A + Then, By Sharp Estimate I, we have ( ψ π na+ ψ(π i(p A + [n(p /p] [n(p /p] i=0 [i(p /p] π n π p [ i(p p p ] A + [p/(p ] i n n(p p p[ p ] A + πn i(p p n i+[ p p ] A + Corollary 37 (Sharp Congruence II Let ( ( n (i + 2i 2 + r/p S (n, r, p = i 0 i p i 0 + +i p =n i +2i 2 + r(modp Then integer 0, we have S (n, r, p 0(modp [ n(p p p ] 7

8 Proof ( ψ = π nψ π (( n[ε] r [ε] p [ε] n [ε] r = π nψ(( + [ε] + + [ε]p n [ε] r = ( π n [ε] (i +2i 2 + r/p = π n = i 0 + +i p =n i +2i 2 + r(modp i 0 + +i p =n 0 i +2i 2 + r(modp π n S (n, r, p ( π n i 0 i p n i 0 i p ( (i + 2i 2 + r/p The function [(n(p p /(p ] is decreasing in and vanishes for [n(p /p] Comparing the coefficients of, the congruence π n follows 4 Sharp estimates for ψ a Let a be a positive integer In this section, we extend the sharp estimates for ψ to ψ a Theorem 4 (Sharp Estimate I For n 0, we have ψ a (π n A + [n/p a ] π p [ n p a p a p a ] (p A + Proof We prove the theorem by induction on a The theorem is true if a = Assume now a 2 and assume that the theorem holds for a 8

9 Then, ψ a (π n A + = ψ(ψ a π n A + [n/p a ] ψ( π i p [ n p i=0 [n/p a ] i=0 [n/p a ] π [i/p] π p [ n p a 2 ip a p a 2 (p p i [n/p a ] a 2 ip a p a 2 (p ] A + ]+[ i p p ] A + p [ n p a 2 ip a p a 2 (p ]+[ i p p ] A + The exponent of p for a fixed is decreasing in i and hence the minimun exponent of p is attained when i = [n/p a ] The minimun exponent is computed to be [ n pa 2 [n/p a ]p a p a p a 2 ] + [ [n/pa ] p p ] = [ n pa p a p a ] (p The proof of the lemma gives more general Lemma 42 ψ a (π n [ε] r = 0 π r(modp a ( n ( n ( ( r/p a Comparing the coefficients of π in the lemma and the sharp estimate for ψ a, we get Corollary 43 (Sharp Congruence I Let r Z Then ( ( n ( r/p ( n a 0(modp [ n p a p a p a ] (p, r(modp a where 0 is a non-negative integer Theorem 44 (Sharp Estimate II For n > 0 and a > 0, we have ( ψ a π na+ [ (an a+(p ap a+ ] (an a+(p (ap a+ p[ p ] A + πn 9

10 Proof The theorem is true for a = Assume now that a > and assume that the theorem is true for a Then ( ( ( ψ a π na+ = ψ ψ a π na+ [ (a n a+2(p ] (a p a+2 ((a n a+2(p ((a p a+2 ψ p[ p ] A + πn i ((a n a+2(p ((a p a+2 ]+[ (n (p ip π n ip[ p p ] A +, where the indices i and satisfy 0 [ (a n a + 2(p ], 0 i [(n (p /p] (a p a + 2 For fixed i + =, the exponent of p is decreasing in and the minimun value is attained when = and i = 0 It follows that ( ψ a ((a n a+2(p ((a p a+2 π na+ p[ p ]+[n ] A + πn 0 [ (an a+(p ap a+ ] =0 π n p[ (an a+(p (ap a+ p ] A +, where we stop at = [ (an a+(p ap a+ ] in the summation as the exponent of p is zero if [ (an a+(p ap a+ ] Corollary 45 (Sharp Congruence II Let ( ( S (n, r, p a n (i + 2i 2 + r/p a = i 0 i (p a i 0 + +i (p a =n i +2i 2 + r(modp a Then for integer 0, we have S (n, r, p a 0(modp [(an a+(p (ap a+ p ] 0

11 References [] LE Dicson, History of the Theory of Numbers, Vol I, AMS Chelsea Publ, 999, Chapter XI, pp [2] ZW Sun, Polynomial extension of Flec s congruence, preprint, 2005, arxiv:mathnt/ [3] ZW Sun and DM Davis, A combinatorial congruence for polynomials, arxiv:mathnt/ [4] D Wan, Fontaine s Rings and p-adic L-Functions, Lecture Notes at the Morningside Center of Mathematics, 2005 [5] CS Weisman, Some congruences for binomial coefficients, Michigan Math J, 24(977, 4-5