ACONJECTUREOFEVANS ON SUMS OF KLOOSTERMAN SUMS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 9, Setember 010, Pages S ( Article electronically ublished on May, 010 ACONJECTUREOFEVANS ON SUMS OF KLOOSTERMAN SUMS EVAN P. DUMMIT, ADAM W. GOLDBERG, AND ALEXANDER R. PERRY (Communicated by Jim Haglund Abstract. In a recent aer, Evans relates twisted Kloosterman sheaf sums to Gaussian hyergeometric functions, and he formulates a number of conjectures relating certain twisted Kloosterman sheaf sums to the coefficients of modular forms. Here we rove one of his conjectures for a fourth order twisted Kloosterman sheaf sum T n of the quadratic character on F. In the course of the roof we develo reductions for twisted moments of Kloosterman sums and aly these in the end to derive a congruence relation for T n with generalized Aéry numbers. 1. Introduction and statement of results Fix an odd rime, and let F be the finite field with elements. For any multilicative character χ on F, we extend χ to a function on F by setting χ(0 = 0. Let ε and φ be the trivial and quadratic characters on F. Also let ψ : F C be the additive character x e πix/. We shall write x for the multilicative inverse of x F. For a F,the classical Kloosterman sum is (1.1 K(a := ψ(x + ax. The nth twisted Kloosterman sheaf sum of φ is (1. T n := φ(a(g(a n + g(a n 1 h(a+g(a n h(a + + g(ah(a n 1 + h(a n, where g(a andh(a are the roots of the olynomial (1.3 X + K(aX +. We study a conjecture of Evans (see below that relates T to the coefficient a( of the unique normalized newform in S (Γ 0 (8, which is given by (1. a(nq n := q (1 q n (1 q n = q q 3 q 5 +q 7 11q 9. n=1 n=1 Received by the editors July, 009 and, in revised form, July 7, Mathematics Subject Classification. Primary 11L05; Secondary 33C0. c 010 Evan Dummit, Adam Goldberg, Alexander Perry 307

2 308 E. P. DUMMIT, A. W. GOLDBERG, AND A. R. PERRY To do so, we emloy the Gaussian hyergeometric series. For multilicative characters A and B on F, define the normalized Jacobi sum ( A = B( 1 J(A, B = B( 1 A(xB(1 x. B x F For multilicative characters A 0,...,A n and B 1,...,B n on F, Greene [7] defined the Gaussian hyergeometric series over F by ( A0, A 1,... A n n+1f n x = ( ( ( A0 χ A1 χ An χ χ(x, B 1,... B n 1 χ B 1 χ B n χ where the sum is over all multilicative characters χ on F (see also [11]. We will be interested in the case where the to arameters are the quadratic character φ and the bottom ones the trivial character ε. For brevity we write this function as (1.5 n+1f n (x = n+1 F n ( φ, φ,... φ ε,... ε χ x = 1 χ ( φχ χ n+1 χ(x. In [] and [3], Ahlgren and Ono rove 3 F 3 (1 = a( by showing that the coefficients a( and the hyergeometric values 3 F 3 (1 count oints on the Calabi-Yau variety {(x, y, z, w (F x + 1 x + y + 1 y + z + 1 z + w + 1 w =0}. With this result, Evans Conjecture can be stated as follows. Conjecture (Evans, [5]. If is an odd rime, then T = 3 F 3 (1 +. We rove the following theorem: Theorem 1.1. Evans Conjecture is true. Ahlgren and Ono s work in [3] was motivated by Beukers Conjecture relating the coefficient a( totheaéry number 1 A ( 1,where n ( ( n + j n (1.6 A(n :=. j j j=0 Beukers conjecture says that ( 1 (1.7 A a( mod. Ahlgren and Ono roved this conjecture using combinatorial roerties of F 3 (1 and the relation 3 F 3 (1 = a( [3]. Motivated by these results, we derive a hyergeometric exression for the sum T n and aly work of Ahlgren and Ono on hyergeometric congruences to rove the following. 1 Aéry used the numbers A(n in his famous roof that ζ(3 is irrational [].

3 A CONJECTURE OF EVANS ON SUMS OF KLOOSTERMAN SUMS 309 Theorem 1.. For λ F, define the generalized Aéry number by (1.8 A(, λ := 1 j=0 ( 1 ( 1 j + j j λ j for λ F and A(, 0 = 1. Then,ifn is a ositive integer, (1.9 ( T n+3 ( 1 n+1 A, (x x n +1(x x n +1 mod. x 1,...,x n F We briefly outline the structure of the aer. In the first section we rovide background information on Gaussian hyergeometric functions. Next we rove reductions for T n, including a urely hyergeometric exression as a sum of 3 F s. We then aly a finite field analogue of Clausen s theorem to the secific n = case. To comlete the roof of Theorem 1.1, we exress both T and 3 F 3 (1 as a trace of squares of F 1 s. In the last section we rove Theorem 1... Preliminaries on Gaussian hyergeometric functions Here we state some basic results on Gaussian hyergeometric functions. Any Gaussian hyergeometric function can be exressed as a Diohantine sum over F [7]. We recall that for x 0, F 1 (x = φ( 1 (.1 φ(yφ(1 yφ(1 xy, (. 3F (x = 1 y,z F y F φ(yφ(1 yφ(zφ(1 zφ(1 xyz. Note. We note that 3 F (0 = F 1 (0 = 0 according to Greene s conventions given in the introduction. However, it is also natural to define 3 F (0 and F 1 (0 by the right hand sides of the equalities above. We will take this convention for the rest of the aer because it simlifies the statements of Theorems 3.1 and 1.. Like their classical counterarts, Gaussian hyergeometric functions satisfy many transformation laws. In this aer we need the following two transformations, which are alications of Theorems. and. of [7]. Lemma.1. For x 0, 1, we have (.3 F 1 (x =φ( 1 F 1 (1 x, (. F 1 (x =φ(x F 1 (1/x. In what follows, we will need to exress a 3 F as a square of a F 1. Todoso, we use a secial case of a formula rovided by Evans and Greene in Theorem 1.7 of [6]. Lemma.. If t 0, ±1, then ( ( 1 3F 1 t = φ(t 1 1 ( 1 t + F 1. The following exression for 3 F 3 (1 as a trace of squares of F 1 s is roven in Lemma. of [3].

4 3050 E. P. DUMMIT, A. W. GOLDBERG, AND A. R. PERRY Lemma.3. We have that 3 F 3 (1 = φ(x F 1 (x. Finally, we will need two secial evaluations of hyergeometric functions from Corollary 11.1 and Theorem 11.1 of [11]. Lemma.. We have that { 0 if 3mod, (.5 F 1 ( 1 = x( 1 x+y+1 if 1mod,x + y =, andx odd, { 0 if 3mod, (.6 3F (1 = x if 1mod,x + y =, andx odd. 3. Reductions of twisted Kloosterman sums Using the relations g(a +h(a = K(a andg(ah(a =, which follow from (1.3, the olynomial g(a n +g(a n 1 h(a+ +g(ah(a n 1 +h(a n can be written as a olynomial in K(a with coefficients in Z. Therefore T n can be written as a linear combination of the twisted Kloosterman moments (3.1 S(m = φ(ak(a m. In this section we give general reductions for such exonential sums, relate them to hyergeometric functions, and secialize to the case T Reductions of T n. The next roosition reduces the exonential sum S(m = φ(ak(am to a symmetric sum of quadratic characters. Proosition 3.1. For an integer m 1, we have that S(m +1=φ( 1 φ(x x m +1φ(x x m +1. x 1,...,x m F Proof. By definition, we have that φ(ak(a m+1 = φ(a ψ(x x m + y + a(x x m + y = x 1,...,x m,y F ψ(x x m + y φ(aψ(a(x x m + y. x 1,...,x m,y F If x x m + y = 0, then the sum on a vanishes, so we may multily by { φ(x x m + y 0 if x1 + + x = m + y =0, 1 otherwise

5 A CONJECTURE OF EVANS ON SUMS OF KLOOSTERMAN SUMS 3051 inside the sum on x 1,...,x m,y.thuswehave φ(ak(a m+1 = φ( i x i + yψ( i x i + y x 1,...,x m,y F φ(a( i x i + yψ(a( i x i + y = G(φ φ( i x i + yψ( i x i + y, x 1,...,x m,y F where G(φ = a F φ(aψ(a is the usual Gauss sum. Making the change of variables (x 1,...,x m,y (x 1 y, x y,...,x m y, y, we have φ(ak(a m+1 = G(φ φ( i x i +1 φ(yψ(y( i x i +1. x 1,...,x m F y F As before, we can multily by φ( i x i +1 to get φ(ak(a m+1 = G(φ φ( i x i +1φ( i x i +1 x 1,...,x m F φ(y( i x i +1ψ(y( i x i +1 y F = G(φ φ( i x i +1φ( i x i +1. x 1,...,x m F Since G(φ = φ( 1 by Proosition 6.3. of [8], this finishes the roof. Remark. Note in articular that S( =. Also,thesumS(1= a F φ(ak(a is not covered by the roosition, but it is easy to evaluate S(1 = G(φ = φ( 1 directly. The form of the character sum in the roosition motivates the study of the auxiliary function (3. Q(a, b = φ(x + y + aφ(x + y + b x,y F for a, b F. The function Q(a, b serves as the connection between twisted Kloosterman sheaf sums and hyergeometric functions. ( ab Proosition 3.. If a, b F are not both zero, then Q(a, b = φ( 1 3 F + φ(ab. Additionally, Q(0, 0 = ( 1φ( 1. Proof. First, we may assume that a 0 by interchanging a and b (if necessary because the sum is clearly symmetric in a and b. Now,forc F,d F,wenote the well-known formula φ(cx + dx + e = x F { ( 1φ(c if d ce =0, φ(c if d ce 0.

6 305 E. P. DUMMIT, A. W. GOLDBERG, AND A. R. PERRY Alying the change of variables (x + y, xy (s, r yields Q(a, b = φ(s + aφ(s/r + b[1 + φ(s r] = s F,r F s F,r F + r,s F φ(s + aφ(s/r + b+ φ(s + aφ(s/r + bφ(s r. r F φ(aφ(bφ( r The first sum equals φ(ab by the note above, and the second sum clearly vanishes. Substituting (r, s (ra s /, as in the third sum yields φ(rφ(s + aφ(s + brφ(s r r,s F = φ(rφ( as + aφ( as + bra s /φ(1 r r,s F = φ( 1 φ(sφ(rφ(1 sφ(1 absr/φ(1 r r,s F = φ( 1 3F ( ab. For the last statement of the roosition, we have Q(0, 0 = φ(x + y φ(xy = φ( x = ( 1φ( 1. x,y F Remark. The secific evaluation of Q(a, binthecasea = b = 1 is the subject of two notable early aers. The Lehmers gave the first evaluation of the sum, directly rewriting Q(1, 1 as a Jacobsthal sum via a comlicated change of variables [9]. A later aer by Mordell revisits their method and finds a simler ath that also eventually leads to Jacobsthal sums [10]. The more general method here also allows for direct evaluation of Q(1, 1 = φ( 1 3F (1/ +, as 3 F (1/ has an exlicit evaluation as given in Theorem of [11]. If we now define for ositive integers m the sum (3.3 G(m := x 1,...,x m F we have the following result. Theorem 3.1. For m 1, there is a recurrence 3F ( (x1 + x + + x m +1(x x m +1 (3. S(m +3= 3 G(m+S(m +1 C(m, where C(m is the number of simultaneous solutions to the congruences x x m = 1 and x x m = 1.,

7 A CONJECTURE OF EVANS ON SUMS OF KLOOSTERMAN SUMS 3053 Proof. Alying Proositions 3.1 and 3., we have S(m +3=φ( 1 φ(x x m+ +1φ(x x m+ +1 = φ( 1 x 1,...,x m+ F Q(x x m +1, x x m +1 x 1,...,x m F = 3 G(m+ φ( 1 φ ((x x m +1(x x m +1 x 1,...,x m F + φ( 1(Q(0, 0 φ( 1 3 F (0C(m = 3 G(m+S(m +1 C(m, where the last term is a correction stemming from the fact that Q(0, 0 φ( 1 3 F (0 = φ( 1, and this correction must be made exactly C(m times. Remark. This result allows us to exress T n as a linear combination of 3 F s, S(1, S(, S(3 and the constants C(m. 3.. Simlifications of T. A calculation using the relations g(a+h(a = K(a and g(ah(a = shows that g(a + g(a 3 h(a+g(a h(a + g(ah(a 3 + h(a = K(a 3K(a +. Therefore we have that (3.5 T = φ(a(k(a 3K(a + = φ(ak(a 3 φ(ak(a. We can now aly the results from the revious section to reduce T. notation, we define a function F : F C by (3.6 F (a :=Q(a +1, a +1. Lemma 3.1. We have T = φ( 1 F (x+3. To ease Proof. Write T = S( 3S( as in (3.5 and aly Proosition 3.1. Lemma 3.. For a 0, ±1, we have that F (a = φ(a F 1 ( a. Proof. By Proosition 3., for a 0, 1, we have that ( (a +1 (3.7 F (a =φ(a+ φ( 1 3 F. a Taking t = a 1 a+1 in Lemma., we get ( ( ( (a +1 1 φ( 1 3 F = φ(a + F 1, a 1+a and so F (a = φ(a F 1 (1/(1 + a. Alying the two transformations in Lemma.1 comletes the roof.

8 305 E. P. DUMMIT, A. W. GOLDBERG, AND A. R. PERRY We will need two secial evaluations of F at ±1, the oints excluded from the revious lemma. Lemma 3.3. We have that (3.8 φ( 1F (1 = φ( 1 F 1 ( 1, (3.9 φ( 1F ( 1 = F 1 (1. Proof. By Proosition 3. we have F (1 = + φ( 1 3 F (1, and so the first identity follows directly from Lemma.. Note that φ( 1F ( 1 = φ( 1 φ(x + yφ(x + y =φ( 1 φ(xyφ(x + y x,y F = φ( 1 x+y=0 φ(xy = ( 1, x,y F and the Diohantine reresentation in (.1 for F 1 imlies that F 1 (1 =1. This roves (3.9.. Proof of Theorem 1.1 By successively alying Lemmas 3.1, 3., and 3.3, we have T =3 + φ( 1 F (x =3 + φ( 1F (1 + φ( 1F ( 1 + \{±1} φ( x F 1 ( x =3 +(φ( 1F (1 φ( 1 F 1 ( 1 +(φ( 1F ( 1 F 1 (1 + φ(x F 1 (x = + φ(x F 1 (x. Alying Lemma.3 yields T = 3 F 3 ( Aéry-tye congruences For the roof of Theorem 1., we recall a result of Ahlgren [1]. Theorem 5.1 (Ahlgren. If is an odd rime and 1 λ 1, then define A(, λ := B(, λ := 1 j=0 1 j=0 ( 1 ( 1 j + j j λ j, ( 1 ( 1 + j λ j j j j i= +1 Then we have (5.1 3F (λ A(, λ+b(, λ mod j i +3j i=1+j 1 i.

9 A CONJECTURE OF EVANS ON SUMS OF KLOOSTERMAN SUMS 3055 Remark. If we define A(, 0 = 1 and B(, 0 = 0, then the results self-evidently hold for λ = 0 as well. We will use this definition in our roof below. Proof of Theorem 1.. By Proosition 3.1, S(m 0mod. Therefore, Theorems 3.1 and 5.1 together show that S(m +3 x 1,...,x m F x 1,...,x m F ( 3 (x1 + x + + x m +1(x x m +1 3F ( A, (x x m +1(x x m +1 mod. Proosition 3.1 shows that the moments S(k vanish modulo, hence writing T n as a linear combination of moments gives T n ( 1 n S(n mod. Remark. The number of simultaneous solutions to the congruences x 1 + x + + x = a and x 1 + x + + x = b is equal to 0 mod for all (a, b excetfor a = b = 0, where the number of solutions is 1 mod. Alying this fact yields that T m+ T m mod. Acknowledgements The authors would like to thank Ken Ono, Amanda Folsom, and Riad Masri for running the UW-Madison REU in Number Theory. We would also like to thank Ken Ono in articular for his guidance in rearing this manuscrit and his ersistent encouragement, and also the anonymous referee for their helful comments. References 1. S. Ahlgren, Gaussian hyergeometric series and combinatorial congruences, Symbolic comutation, number theory, secial functions, hysics and combinatorics [Eds. F. Garvan and M. E. H. Ismail], Kluwer Acad. Publ. Dordrecht (001, 1-1. MR (003i:3305. S. Ahlgren and K. Ono, Modularity of a certain Calabi-Yau threefold, Monatsh. Math. 19 (000, MR (001b: S. Ahlgren and K. Ono, A Gaussian hyergeometric series evaluation and Aéry number congruences, J. Reine Angew. Math 518 (000, MR17390 (001c: R. Aéry, Irrationalité deζ( et ζ(3, Astérisque 61 (1979, R. J. Evans, Hyergeometric 3 F (1/ evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. 138 (010, MR R. J. Evans and J. Greene, Clausen s theorem and hyergeometric functions over finite fields, Finite Fields Al. 15 (009, MR68995 (010a: J. Greene, Hyergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987, MR87956 (88e: K. Ireland and M. Rosen, A classical introduction to modern number theory, Sringer-Verlag, New York, MR (9e: D. H. Lehmer and Emma Lehmer, On the cubes of Kloosterman sums, Acta Arith. 6 (1960, 15-. MR (:6773

10 3056 E. P. DUMMIT, A. W. GOLDBERG, AND A. R. PERRY 10. L. J. Mordell, On Lehmer s congruence associated with cubes of Kloosterman s sums, Journal London Math. Soc. 36 (1961, MR0161 (3:A K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and q-series, CBMS, vol. 10, Amer. Math. Soc., Providence, RI, 00. MR0089 (005c:11053 Deartment of Mathematics, University of Wisconsin-Madison, 80 Lincoln Drive, Madison, Wisconsin address: Dummit@math.wisc.edu 617 Logan Lane, Danville, California 956 address: AdamWGoldberg@gmail.com Deartment of Mathematics, 517 Lerner Hall, Columbia University, 90 Broadway, New York, New York address: ar15@columbia.edu