A conjecture of Evans on Kloosterman sums

Transcription

1 Evan P. Dummit, Adam W. Goldberg, and Alexander R. Perry 2009 Wisconsin Number Theory REU Project

2 Notation We will make use of the following notation: p odd prime Additive character ψ : F p C, x e 2πix p x = 1/x in F p ε : F p C, x x φ : F p C, x ( x p ) (Legendre)

3 Gauss and Jacobi Sums Definition For a F p, Quadratic Gauss Sum: G(φ) := a F p φ(a)ψ(a).

4 Gauss and Jacobi Sums Definition For a F p, Quadratic Gauss Sum: G(φ) := a F p φ(a)ψ(a). Definition For multiplicative characters χ 1, χ 2 J(χ 1, χ 2 ) := a F p χ 1 (a)χ 2 (1 a).

5 Kloosterman Sum Definition For a F p, the classical Kloosterman sum is K(a) := ψ(x + ax). x F p

6 Sum Definitions Definition The n-th twisted Kloosterman sheaf of φ is T n := φ(a)(g(a) n + g(a) n 1 h(a) + + h(a) n ), a F p where g(a) and h(a) are the roots of the polynomial X 2 + K(a)X + p.

7 Evans s Conjecture Define a(n) by a(n)q n := q (1 q 2n ) 4 (1 q 4n ) 4 = q 4q 3 2q 5 +24q 7 11q 9. n=1 n=1

8 Evans s Conjecture Define a(n) by a(n)q n := q (1 q 2n ) 4 (1 q 4n ) 4 = q 4q 3 2q 5 +24q 7 11q 9. n=1 n=1 Conjecture (Evans) If p is an odd prime, then T 4 p = a(p).

9 Earlier Related Work Emma and D. H. Lehmer showed φ( 1)T 3 p = { 0 p 2 mod 3 4x 2 2p p = x 2 + 3y 2

10 Earlier Related Work Emma and D. H. Lehmer showed φ( 1)T 3 p = { 0 p 2 mod 3 4x 2 2p p = x 2 + 3y 2 These are the values of the p-th coefficients of the eigenform q ( )) d (1 q 2n ) 3 (1 q 6n ) 3 S 3 (Γ 0 (12),. 3 n=1

11 Earlier Related Work, cont. Green s analogy between character sum expansion of a function f : F p C and power series expansion of an analytic function: x k χ(x).

12 Earlier Related Work, cont. Green s analogy between character sum expansion of a function f : F p C and power series expansion of an analytic function: x k χ(x). Basis was analogy between the Gauss sum G(χ) = χ(x)ψ(x) x F p

13 Earlier Related Work, cont. Green s analogy between character sum expansion of a function f : F p C and power series expansion of an analytic function: x k χ(x). Basis was analogy between the Gauss sum G(χ) = χ(x)ψ(x) and the gamma function Γ(x) = x F p 0 t x e t dt t.

14 Classical Hypergeometric functions Definition ( ) α1, α 2,... α p pf q x := β 1,... β q n=0 (α 1 ) n (α 2 ) n (α p ) n (β 1 ) n (β q ) n x n n!,

15 Classical Hypergeometric functions Definition ( ) α1, α 2,... α p pf q x := β 1,... β q n=0 where (α) n = α(α + 1) (α + n 1). (α 1 ) n (α 2 ) n (α p ) n x n (β 1 ) n (β q ) n n!,

16 Gaussian hypergeometric functions I Definition For characters A and B on F p, define the binomial coefficient ( ) A B := B( 1) J(A, B) = B( 1) p p x F p A(x)B(1 x).

17 Gaussian hypergeometric functions I Definition For characters A and B on F p, define the binomial coefficient ( ) A B Definition := B( 1) J(A, B) = B( 1) p p For n N +, and x F p, define n+1f n (x) := p p 1 χ x F p A(x)B(1 x). ( ) φχ n+1 χ(x). χ

18 Gaussian hypergeometric functions II Gaussian hypergeometric functions may in general be written as Diophantine sums over F p.

19 Gaussian hypergeometric functions II Gaussian hypergeometric functions may in general be written as Diophantine sums over F p. Examples: 2F 1 (x) = ε(x) φ( 1) p 3F 2 (x) = ε(x) 1 p 2 y,z F p y F p φ(y)φ(1 y)φ(1 xy), φ(y)φ(1 y)φ(z)φ(1 z)φ(1 xyz).

20 Gaussian hypergeometric functions II Gaussian hypergeometric functions may in general be written as Diophantine sums over F p. Examples: 2F 1 (x) = ε(x) φ( 1) p 3F 2 (x) = ε(x) 1 p 2 y,z F p y F p Transformation laws for x 0, 1: φ(y)φ(1 y)φ(1 xy), φ(y)φ(1 y)φ(z)φ(1 z)φ(1 xyz). 2F 1 (x) = φ( 1) 2 F 1 (1 x), 2F 1 (x) = φ(x) 2 F 1 (x).

21 Gaussian hypergeometric functions III Two other relations:

22 Gaussian hypergeometric functions III Two other relations: Lemma If t 0, ±1, then ( ) 1 3F 2 1 t 2 = φ(t 2 1) ( 1 p + 2F 1 ( 1 t 2 ) 2 ).

23 Gaussian hypergeometric functions III Two other relations: Lemma If t 0, ±1, then and Lemma ( ) 1 3F 2 1 t 2 = φ(t 2 1) ( 1 p + 2F 1 ( 1 t 2 ) 2 ). p 3 4F 3 (1) = p 2 φ(x) 2 F 1 (x) 2. x F p

24 Relating geometric objects and counting functions Ahlgren and Ono proved that p 3 4F 3 (1) = a(p) p

25 Relating geometric objects and counting functions Ahlgren and Ono proved that p 3 4F 3 (1) = a(p) p by showing that the coefficients a(p) and the hypergeometric values p 3 4F 3 (1) count points on the Calabi-Yau variety {(x, y, z, w) (F p ) 4 x + 1 x + y + 1 y + z + 1 z + w + 1 w = 0}.

26 Evans s Conjecture reformulated With this, Evans s conjecture may be rephrased as follows:

27 Evans s Conjecture reformulated With this, Evans s conjecture may be rephrased as follows: Conjecture (Evans) If p is an odd prime, then T 4 p = p3 4F 3 (1) + p.

28 Twisted moments Note g(a) n + g(a) n 1 h(a) + + g(a)h(a) n 1 + h(a) n Z[K(a)].

29 Twisted moments Note g(a) n + g(a) n 1 h(a) + + g(a)h(a) n 1 + h(a) n Z[K(a)]. Therefore the m-th moments S(m) = φ(a)k(a) m a F p govern the behavior of the Kloosterman sheaves.

30 Reductions of S(m) Proposition For an integer m 1, we have that S(m+1) = pφ( 1) φ(x 1 + +x m +1)φ(x 1 + +x m +1). x 1,...,x m F p

31 Character Sums I Definition For a, b F p, Q(a, b) := x,y F p φ(x + y + a)φ(x + y + b)

32 Character Sums I Definition For a, b F p, Q(a, b) := x,y F p φ(x + y + a)φ(x + y + b) Proposition If a, b F p, then Q(a, b) = p 2 φ( 1) 3 F 2 ( ab 4 ) + pφ(a/b).

33 Character Sums I Definition For a, b F p, Q(a, b) := x,y F p φ(x + y + a)φ(x + y + b) Proposition If a, b F p, then Q(a, b) = p 2 φ( 1) 3 F 2 ( ab 4 ) + pφ(a/b). The evaluation of Q(1, 1) reduces to yield the Lehmers result.

34 Character Sums II We may now write T n as sums of 3 F 2 s. For m N +, define G(m) := x 1,...,x m F p 3F 2 ( (x1 + x x m + 1)(x x m + 1) 4 ).

35 Character Sums II We may now write T n as sums of 3 F 2 s. For m N +, define G(m) := x 1,...,x m F p 3F 2 ( (x1 + x x m + 1)(x x m + 1) 4 ). Theorem If n N +, then T n is an explicit linear combination of 3 F 2 s. For m 1, there is a recurrence S(m + 3) = p 3 G(m) + ps(m + 1).

36 Character Sums II We may now write T n as sums of 3 F 2 s. For m N +, define G(m) := x 1,...,x m F p 3F 2 ( (x1 + x x m + 1)(x x m + 1) 4 ). Theorem If n N +, then T n is an explicit linear combination of 3 F 2 s. For m 1, there is a recurrence S(m + 3) = p 3 G(m) + ps(m + 1).

37 Character Sums III Lemma For F : F p C defined by F (a) := Q(a + 1, a + 1), we have T 4 = pφ( 1) F (x) + 3p 2. x F p

38 Character Sums IV: A New Hope Lemma For a ±1, we have that F (a) = p 2 φ(a) 2 F 1 ( a) 2.

39 Character Sums IV: A New Hope Lemma For a ±1, we have that Lemma We have that F (a) = p 2 φ(a) 2 F 1 ( a) 2. φ( 1)F (1) = p 2 φ( 1) 2 F 1 ( 1) 2 p φ( 1)F ( 1) = p 2 2F 1 (1) 2 p

40 Apéry numbers I Definition The Apéry numbers are defined by A(n) := n ( ) n + j 2 ( ) n 2. j j j=0

41 Apéry numbers I Definition The Apéry numbers are defined by A(n) := n ( ) n + j 2 ( ) n 2. j j j=0 They were used in the famous proof that ζ(3) is irrational.

42 Apéry numbers II Let a(n)q n := q (1 q 2n ) 4 (1 q 4n ) 4 = q 4q 3 2q 5 +24q 7 11q 9. n=1 n=1

43 Apéry numbers II Let a(n)q n := q (1 q 2n ) 4 (1 q 4n ) 4 = q 4q 3 2q 5 +24q 7 11q 9. n=1 n=1 Conjecture (Beukers) ( ) p 1 A a(p) mod p 2. 2

44 Apéry numbers II Let a(n)q n := q (1 q 2n ) 4 (1 q 4n ) 4 = q 4q 3 2q 5 +24q 7 11q 9. n=1 n=1 Conjecture (Beukers) ( ) p 1 A a(p) mod p 2. 2 This was proven by Ahlgren and Ono using the combinatorial properties of 4 F 3 (1) and the relation p 3 4F 3 (1) = a(p) p.

45 Beukers-type congruences for T n Definition For λ F p, define the generalized Apéry number by A(p, λ) := p 1 2 j=0 ( p 1 2 j )2( p j j ) λ pj.

46 Beukers-type congruences for T n Definition For λ F p, define the generalized Apéry number by Theorem A(p, λ) := x 1,...,x n F p p 1 2 j=0 ( p 1 2 j )2( p j j ) λ pj. If n is a positive integer, then modulo p 2, T n+3 equals ( ( 1) n+1 p A p, (x ) x n + 1)(x x n + 1). 4

47 Conclusion Proved Evans s Conjecture Expanded on the work of the Lehmer s with Q(a,b) Apéry number observations

48 Conclusion Theorem (DGP) Evans s Conjecture is true, namely If p is an odd prime, then T 4 p = a(p).

49 Conclusion Theorem (DGP) Evans s Conjecture is true, namely If p is an odd prime, then Theorem x 1,...,x n F p T 4 p = a(p). If n is a positive integer, then modulo p 2, T n+3 equals ( ( 1) n+1 p A p, (x ) x n + 1)(x x n + 1). 4