A talk given at University of Wisconsin at Madison (April 6, 2006). Zhi-Wei Sun

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1 A tal given at University of Wisconsin at Madison April 6, RECENT PROGRESS ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing , P. R. China Abstract. In 1913 A. Flec proved that if p is a prime, and n > 0 and r are integers then r mod p n 1 0 mod p n 1/p 1. Only recently the significance of Flec s congruence was realized. It plays a fundamental role in Colmez and Wan s investigation of the ψ-operator related to Fontaine s theory and p-adic Langlands correspondence. In this tal we give a survey of the recent developments of Flec s congruence and its various extensions, as well as some important applications to Stirling numbers of the second ind and homotopy exponents of special unitary groups given by Davis and the speaer. Both number-theoretic and combinatorial approaches will be introduced. 1. Lucas and Wolstenholeme s congruences Let x 0 = 1 and x = xx 1 x +1 for Z + = {1, 2, 3,... }. For N = {0, 1, 2,... }, define the binomial coefficient x = x! 1

2 2 ZHI-WEI SUN which is an integer if x Z. It is easy to see that x x + 1 = 1. A fundamental congruence involving binomial coefficients was given by E. Lucas in Theorem 1.1 Lucas Theorem. Let p be a prime, and let n, r N and s, t {0,..., p 1}. Then pn + s pr + t n r s t mod p. In the case s = t = 0, the Lucas congruence can be further improved. Theorem 1.2. Let p 5 be a prime. i Wolstenholeme, 1862 We have 2p 1 = 1 2p 1 mod p 3. p 1 2 p ii Ljunggren, 1952 For n, r N we have pn pr n mod p 3. r iii Jacobsthal, 1952 If n r 0 then / np n 1 + p 3 nrn rz p, rp r where Z p is the ring of p-adic integers in the p-adic field Q p. For n N define the q-integer [n] q by [n] q = 1 qn 1 q = 0 r<n q r.

3 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 3 For n, r N the Gauss q-binomial coefficient [ ] { [n]q [n 1] n q [n r+1] q [r] = q [r 1] q [1] q if n r, r q 0 otherwise. is a natural q-analogue of the usual binomial coefficient n r. Note that lim q 1 [ ] n = r q n. r Both Lucas theorem and Wolstenholme s congruence have their q-analogues. 2. Flec s and Weisman s congruences For m Z +, n N and r Z, we set C m n, r = r mod m n 1. For a prime p and a p-adic number α, we let ord p α denote the p-adic order of α. In 1913 A. Flec published the following result; we don t now his motivation. Theorem 2.1 Flec, Let p be any prime, and let n Z + and r Z. Then C p n, r 0 mod p n 1/p 1 ; in other words, ord p C p n, r n 1. p 1

4 4 ZHI-WEI SUN Proof. A. Granville Let ζ p be a primitive p-th root of unity. Then pc p n, r = n =0 p 1 = j=0 as is well-nown. Clearly n p 1 1 ζ jr p n =0 j=0 ζ j r p n ζ j p = p 1 j=1 p 1 1 ζp j x p 1 = lim x 1 x 1 = p, j=1 ζp jr 1 ζp j n and each 1 ζ j p/1 ζ p with 1 j p 1 is a unit in the ring Z[ζ p ]. So ord p 1 ζ j p = 1/p 1 for all j = 1,..., p 1. Therefore ord p C p n, r n p 1 1 > n 1 p 1 1 and hence ord p C p n, r n 1/p 1. For a p-adically continuous function fx from Z p to the p-adic completion of Q p, there are two popular ways to expand fx: One is Mahler s interpolation series fx = n=0 a n x n, another is van der Put s expansion fx = n=0 b nχ n x where χ n x = [ord p x n > log p max{n, 1}]. For a proposition P we let [P ] tae 1 or 0 according as P holds or not. If 0 n < p α then we have the relation a n = C p n, 0b 0 + 0<β α p β 1 r<p β C p β n, rb r. Motivated by this and unaware of Flec s earlier result, C. S. Weisman [Michigan Math. J ] obtained the following result.

5 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 5 Theorem 2.2 Weisman, Let p be any prime, and let a Z +, n N and r Z. Then n p a 1 ord p C p an, r ϕp a If n p a 1 = mϕp a 1 for some m Z +, then. C p an, r p m 1 mod p m. The first part of Theorem 2.2 is a generalization of Flec s result; Weisman s proof is very complicated. Quite recently Z.W. Sun and D. Wan were able to give a proof via roots of unity on the basis of the identity C p an, r = 1 p n =0 n C p a 1, r p 1 j=0 ζ j r p a 1 ζj p an, where ζ p a is a primitive p a -th root of unity in the complex field C. 3. On Flec quotients and generalized Flec quotients Let p be a prime, and let n N and r Z. We define the Flec quotient F p n, r := p n 1/p 1 C p n, r + [n = 0]]. If a Z + then we define the generalized Flec quotient or Weisman quotient F p an, r = p n pa 1 /ϕp a C p an, r + [n < p a 1 ]. For a Z and m Z + we let {a} m be the least nonnegative residue of a mod m. Thus {a} m /m is the fractional part {a/m} of a/m.

6 6 ZHI-WEI SUN Theorem 3.1 Z. W. Sun and D. Wan, 2006, arxiv:math.nt/ Let p be a prime, and let a Z + and n N. and i F p an, r 0 mod p for some r Z. ii For all r Z, we have if n 2p a 1. F p n + p a p 1, r F p n, r mod p a, F p a n + p a p 1, r F p an, r mod p iii For any r Z we have d r + 1 F p an, r F p an +, 0 mod p, =0 where d = {p a 1 1 n} ϕpa is the least nonnegative integer with n + d p a 1 1 mod ϕp a. In view of the Kummer-type congruences in Theorem 3.1ii, the following conjecture loos reasonable. Conjecture 3.1 Z. W. Sun and D. Wan, Let p be a prime, and let a, b, n Z + and r Z. If n 2p a+b 2, then F p a n + ϕp a+b, r F p an, r mod p b. For a prime p and an integer a, we define q p a = a p 1 1/p which is called a Fermat quotient when a 0 mod p. By a number-theoretic method involving roots of unity, Gauss sums and the Sticelberger congruence, Z. W. Sun and D. Wan recently determined F p n, r modulo p explicitly.

7 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 7 Theorem 3.2 Z. W. Sun and D. Wan, arxiv:math.nt/ Let p be a prime, and let n N and r Z. Set n 0 = {n} p and n 1 = {n 0 n} p 1 = { n/p } p 1. If n 0 n 1, then F p n, r 1n 1 n 1! n 0 =0 n0 1 r n 1 mod p. If n 0 > n 1 = 0, then F p n, r 1 {r} p n0 {r} p mod p. If n 0 > n 1 > 0, then F p n, r 1n 1 1 n 1 1! n 0 =0 n0 1 r n 1 q p r mod p. Let p be an odd prime. For each a Z let ā = a+pz F p = Z/pZ. Let ω be the Teichmüller character of the multiplicative group Fp = F p \ { 0}. For a F p, ωā is just the p 1-th root of unity in the unique unramified extension of the p-adic field Q p such that ωā a mod p. If ζ p is a primitive p-th root of unity in the algebraic closure of Q p, then for n N and π = 1 ζ p we have p 1 p 1 a n ζp a a=1 a=1 ω n āζ a p πn n! mod π n +1 by Sticelberger s congruence for Gauss sums, where n = { n} p 1. Moreover, we have the following lemma which plays a ey role in the proof of Theorem 3.2.

8 8 ZHI-WEI SUN Lemma 3.1. Let p be a prime, and let n N and n = { n} p 1. Define Gn = p 1 a=1 an ζ a p and π = 1 ζ p, where ζ p is a primitive p-th root of unity in the complex field C. Then Gn 1 n 1 p 2 m=n sm, n πm m! mod p, where sm, 0,..., sm, m are Stirling numbers of the first ind defined by x m = m =0 1m sm, x. Here is a consequence of Theorem 3.2. Corollary 3.1. Let p be a prime and let n N and r Z. Then F p pn, r rn n! mod p where n = { n} p 1. Consequently, F p p p 1 { 1 h p+1/2 2, r r p mod p if p 3 & 4 p + 1, 1 hp 1/2 r p v 2 mod p if 4 p 1, where p is the Legendre symbol, and h p and hp are the class numbers of the quadratic fields Q p and Q p respectively, and for p 1 mod 4 we write the fundamental unit of Q p in the form v+u p/2 with u, v Z and u v mod 2. Let n be a positive integer and p > 2n + 1 be a prime. By the first part of Corollary 3.1 in the case r = 0, we have 2pn 1 = 1 n 1 2pn 2pn 1 n 1 mod p 2n+1. pn 1 2 pn p =0

9 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 9 This is a new extension of Wolstenholme s congruence; in the case n = 2 it gives 4p 1 2p 1 4p 1 mod p 5. p Z. W. Sun and D. Wan also presented a combinatorial approach to Flec quotients. The following lemma provides a basis for induction arguments. Lemma 3.2. Let p be a prime, and let n N with n p. Then p 1 1 F p n, r j j=1 j 1 F p n p + 1, r i mod p. i=0 For n N the Stirling numbers Sn, N of the second ind are given by It is well nown that in other words, e x 1 = x n = N Sn, x. Sn, xn n! = ex 1 ;! n= n= Sn, x n where Sn, =! Sn,. n! For m = 1, 2,..., the m-th order Bernoulli polynomials B m n t n N are defined by and those B m n x m e tx e x 1 m = n=0 B n m t xn n!, = B n m 0 are called higher-order Bernoulli numbers. The usual Bernoulli polynomials and numbers are B n t = B n 1 t and B n = B n 0 = B 1 n respectively.

10 10 ZHI-WEI SUN By Lemma 3.2, Sun and Wan showed that if p is a prime, n N, r Z and m n mod p then e 1 n x m 1 F p n, r [x n ] e rx mod p, x where n = { n} p 1 and [x n ]fx denotes the coefficient of x n in the power series expansion of fx. This yields the following result. Theorem 3.3 Z. W. Sun and D. Wan, arxiv:math.nt/ Let p be a prime, and let n N and r Z. Set n = { n} p 1. For any integer m n mod p, if m 0 then 1 n F p n, r is congruent to n Sn + m, m r n r = Sm + n, m +! =0 =0 m m m rm+n = 1 m + n! modulo p; if m < 0 then we have F p n, r 1n n! B m n =0 r p 1 n!b m n r mod p. Let p be an odd prime, and let h p and h + p denote the class numbers of the cyclotomic field Qζ p and its maximal real subfields Qζ p + ζ 1 p respectively, where ζ p is a primitive p-th root of unity in the complex field C. It is well nown that h p = h p /h + p is an integer. If p divides none of the numerators of the Bernoulli numbers B 0, B 2,..., B p 3 Z p, then p is said to be a regular prime. In 1850 E. Kummer proved that p h p p h p p is regular = x p + y p = z p has no integral solution with xyz 0. Theorem 3.3 has the following interesting consequence.

11 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 11 Corollary 3.2 Z. W. Sun and D. Wan, arxiv:math.nt/ Let p be a prime. i For every n = 2,..., p we have n pn 1 1 p 1 n 1!B p n p n mod p n+1. p 1 =1 ii If 3 n p and r Z then Bp n+1 r F p pn 2, r n! + r + 1 B p n r n 1 n mod p. Note that Corollary 3.2i in the case n = 2 gives the Wolstenholme congruence. 4. Polynomial extensions of Flec s and Weisman s results Let p be a prime, and let A = Z p [T ] be the formal power series ring over Z p. The Z p -linear Frobenius map φ acts on the ring A by φt = 1 + T p 1. Equivalently, φ1 + T = 1 + T p. This map φ is injective and of degree p. This implies that {1, T,..., T p 1 } and {1, 1 + T,..., 1 + T p 1 } are bases of A over the subring φa. The operator ψ : A A is defined by p 1 ψ 1 + T i φx i i=0 = x 0, where x 0,..., x p 1 A. The ψ-operator plays a basic role in L-functions of F -crystals, Fontaine s theory of φ, Γ-modules, Iwasawa theory, p-adic

12 12 ZHI-WEI SUN L-functions and p-adic Langlands correspondence. Both P. Colmez and D. Wan noted that if n N and r Z then T n n ψ 1 + T r =ψ where we let = = = n =0 =0 n r mod p r mod p C l,m n, r = n 1 n 1 + T r 1 n ψ1 + T r n 1 n 1 + T r/p n 1 l N r mod m r/p l T l = n r/m 1. l T l C l,p n, r, This led Colmez 2004 and Wan 2005 to study the p-adic order of l=0 C l,p n, r. Colmez did not now Flec s congruence and just presented a weaer estimate ord p C l,p n, 0 n/p l. In May 2005, the speaer came to Univ. of California at Irvine and told Flec s result to Wan who was studying the ψ-operator then, soon Wan got that if p is a prime, l, n N and r Z then n lp 1 ord p C l,p n, r, p 1 Note that in the case l = 0 this reduces to Flec s result. In June 2005, by a combinatorial argument the speaer was able to give a common generalization of Weisman s and Wan s extensions of Flec s congruence.

13 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 13 Theorem 4.1 [Z. W. Sun, Acta Arith ]. Let p be a prime, and let l, α, β N and α β. Then we have r mod p β n r 1 p α p l n p α 1 l ϕp α l 1α β Z p for all integers n p α 1 and r; moreover, we can substitute [β = 0] for the first l in the exponent if α is greater than one. In the case α = β = a > 1, Theorem 3.1 gives which implies that n p a 1 ord p C l,p an, r ϕp a la, ord p r mod p a n m n p a 1 ϕp a for any m 1 mod p. This is also true for a = 1 as pointed out by Flec. For a positive integer a, let ψ a be the a-th iteration of ψ acting on the ring A = Z p [T ]. D. Wan noted that for any n N and r Z we have T ψ a n 1 + T r = 1 n T l C l,p an, r. To understand the ψ a -action, it is thus essential to understand the p-adic property of the cyclotomic ψ-coefficients C l,p an, r l = 0, 1,.... Theorem 4.2 D. Wan, Finite Fields Appl., in press. Let p be a prime, and let a, l, n N and r Z. Then l=0 n lp ord p C l,p an, a p a 1 r ϕp a.

14 14 ZHI-WEI SUN To understand how sharp the congruence in Theorem 4.2 is, we define the normalized cyclotomic ψ-coefficient { } n r l,p a := p n p a 1 lp a φp a r mod p a 1 n r/p a Surprisingly it has many properties similar to properties of the usual binomial coefficients. In 2005 Sun and Wan proved the following new analogue of Lucas theorem. l. Theorem 4.3 Sun and Wan, arxiv:math.nt/ Let p be any prime, and let r Z and a, l, n, s, t N with a 2 and s, t < p. Then we have the congruence { } pn + s pr + t l,p a+1 1 t { } s n t r l,p a mod p; in other words, p n p p n p a 1 lp a φp a a 1 lp a φp a r mod p a r mod p a pn + s r/p 1 p a p + t l n s r/p 1 a mod p. t l This theorem in the case l = 0 was first obtained by Z. W. Sun and D. M. Davis [Trans. Amer. Math. Soc., arxiv:math.nt/ ]. Note that a 2 is assumed in Theorem 4.3. The case a = 1 is more subtle.

15 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 15 Theorem 4.4 Sun and Wan, arxiv:math.nt/ Let p be a prime, l, n N, r Z and s, t {0,..., p 1}. If p n, or p 1 n l 1, or s = 2t and p 2, then { } pn + s 1 t pr + t l,p 2 s t { n r } l,p mod p. When p n, p 1 n l 1 and t s, p 1], we have { } { n l 1 n l 1 pn + s s+ 1 p 1 n p 1 1 t l / t 1 s mod p if n > l + 1, pr + t l,p 2 0 mod p if n l Further results related to Stirling numbers of the second ind and homotopy exponents of SUn Let p be a prime. If n Z + then n ord p n! = p i < i=1 i=1 n p i = n p 1 and hence ord p n! n 1/p 1. Thus, when a Z +, n p a 1 and r Z, by Weisman s result we have n p a 1 n/p ord p C p an, a 1 1 r ϕp a = ord p p 1 n p a 1!. For a topological purpose, few years ago a topologist D. M. Davis conjectured that if n > l 0 are integers then n l + 1 ord 2 l ord 2!. 2 2 N This and his other related conjectures are indeed very sophisticated! In July 2005 he made his conjectures public and wrote to the speaer the following comments:

16 16 ZHI-WEI SUN I have wored very hard off and on during the past two years trying to prove these conjectures. In the past, I have communicated them privately to others at least 5 experts without any significant progress. Two wees later the above conjecture was confirmed by the speaer, and the following general theorem was soon established. Theorem 5.1 [Davis and Sun, arxiv:math.at/ ]. Let p be a prime, a, n N and r Z. Then, for any polynomial fx Z[x], we have ord p r mod p a n 1 f r p a n ord p! p a + τ p {r} p a, {n r} p a, where τ p s, t = ord s+t p s is the number of carries occurring in the addition of s and t in base p. For, n N it is nown that!sn, = j=0 p 1 1 j j n = 1 j r=0 j r mod p 1 j j n. j Theorem 5.1 in the case r = 0 has the following consequence on p-adic orders of Stirling numbers of the second ind. Corollary 5.1 [Davis and Sun, arxiv:math.at/ ]. Let p be any prime and n be a positive integer. If L n 1 + n/pp 1, then for all m n we have ord p m!sp 1p L + n 1, m n 1 + ord p n p!.

17 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 17 The special unitary group SUn of degree n is the space of all n n unitary matrices the conjugate transpose of such a complex matrix equals its inverse with determinant one. It plays important roles in many areas of mathematics and physics. Here is an application of Corollary 5.1 in algebraic topology. Theorem 5.2 [Davis and Sun, arxiv:math.at/ ]. For any prime p and n 2, some homotopy group π i SUn contains an element of order p n 1+ordp n/p!. Numerical examples indicate that Theorem 5.2 is very strong. The inequality in Theorem 5.1 can be improved when deg f < n/p a. Theorem 5.3 [Sun and Davis, Trans. AMS, arxiv:math.nt/ ]. Let p be a prime, and let a, n N and r Z. Then, for any polynomial fx Z[x], we have ord p ord p r mod p a n p a 1! n 1 f where {x} p a 1 is regarded as 0 if a = 0. r p a deg f + τ p {r} p a 1, {n r} p a 1. A particular case of this theorem is the following conjecture of Davis: For l, n N we have n n ord 2 ord l 2 N! l ord 2 l!. The following result is a vast generalization of Lucas theorem.

18 18 ZHI-WEI SUN Theorem 5.4. Let p be any prime. And and let l, n N, r, s, t Z and 0 s, t < p. i [Sun and Davis, Trans. AMS, arxiv:math.nt/ ] For every a = 2, 3,..., we have 1 n/p a 1! 1 n/p a 1! r mod p a r mod p a l pn + s r 1 p + t p a 1 n s t 1 r p a 1 l mod p. ii [Sun and Wan, arxiv:math.nt/ ] The congruence in the first part also holds for a = 1 as conjectured by Sun and Davis. When a > log p max{n, p} and l = 0, Theorem 5.4i yields the classical congruence of Lucas. Theorem 5.5 Z. W. Sun, arxiv:math.nt/ Let p be any prime, and let a Z +, l, n N and r Z. Then 1 n/p a 1! 1 n/p a 1! r mod p a r mod p a pn r 1 p 1 n p a 1 l l r p a 1 mod p α p, where 1 if p = 2, α p = 2 if p = 3, 3 if p > 3. Let p be a prime, a, n N and r Z. When p a > n and l = 0 r n, the congruence in Theorem 5.5 reduces to Ljunggren s congruence pn pr

19 ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS 19 n r mod p α p. Note also that the congruence holds for every l N if and only if we have pn pr f, p a+1 n mod p α p r f, p a for all fx Z[x], where n = r f, pdeg f n/p a 1! pa r mod p a 1 n f r Z p. The following theorem shows that the inequality in Theorem 5.1 is sharp for infinitely many values of l provided that n 2p a 1. Theorem 5.6 Z. W. Sun, arxiv:math.nt/ Let p be a prime, p a and let a, l, n N and r Z. Set r = {r} p a, n = r + {n r} p a m = n n r p a = p a + n r p a. and Suppose that l m > 0 and l m mod p 1p log p m [ r/pα pz & log p m Z + ]. Then we have 1 n/p α! n r In particular, 1 n/p a! r mod p a 0 mod p a 1 n l r 1 l+r mod p. p a n 1 p n/p a a 1 mod p. The following result follows from Theorem 5.3.

20 20 ZHI-WEI SUN Theorem 5.7 Z. W. Sun, arxiv:math.nt/ Let p be any prime. Let a Z, l, l, m Z +, l l > m/p and j=0 l l Then we have l l Sj, ma l j j mod p 1p log p m [p a & log p m Z + ]. l j=0 l Sj, ma l j mod p. j Corollary 5.2 L. Carlitz, Let p be any prime. Suppose that a, m N, m p and p a < m p a+1. Then p a p 1 is a period of the sequence {Sl, m} l m modulo p. In 1899 J.W.L. Glaisher proved that l j j r mod p 1 j r mod p 1 l j mod p whenever p is a prime, r Z, l, l Z + and l l mod p 1. Clearly Glaisher s congruence is our following result in the case m = 1. Corollary 5.3. Let p be a prime, m Z + and r Z. For any l, l Z + with l l > m/p and l l mod p 1p log p m, we have j r mod p 1 l j Sj, m j r mod p 1 l Sj, m mod p. j A different extension of the Glaisher congruence was given by the speaer and R. Tauraso [arxiv:math.nt/ ] in Feb