On character sums and complex functions over nite elds

On character sums and complex functions over nite elds N. V. Proskurin April 17, 2018 In this lecture I want to present the complex functions over nite elds theory. The theory has been developed through parallels with the classical functions theory. It includes: Gauss and Jacobi sums; other character (or trigonometric) sums; Fourier expansions, power series expansions, dierentiations; hypergeometric functions and other special functions. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 1 / 25

Notation for nite elds F q nite eld of order q = p l with prime subeld F p = Z/pZ. F q the multiplicative group of F q. tr: F q F p the trace, tr(x) = x + x p +... + x pl 1. e q : F q C xed (once for all) a non-trivial additive character (say, e q (x) = exp ( 2πi tr(x)/p ) for all x F q ). F q the group of multiplicative characters, i. e. the group of homomorphisms χ: Fq C extended to all of F q by χ(0) = 0. ɛ the trivial character, ɛ(x) = 1 for all x Fq, ɛ(0) = 0. δ(0) = 1 and δ(x) = 0 for all x Fq. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 2 / 25

Gauss sum The Gauss sum G(χ) = x F q χ(x)e q (x), χ F q, being considered as a function F q C, is quite similar to the Euler gamma function Γ (s) = 0 exp( x) x s 1 dx, s C, Re s > 0. In detail, note that x exp( x) is a character of the additive group of R, that x x s is a character of the R +, and that the integration over dx/x is invariant under multiplicative translations. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 3 / 25

Jacobi sum Similarly, we see that the Jacobi sum J(α, β) = x F q α(x)β(1 x), α, β F q, being considered as a function F q F q C, is the analogue of the Euler beta function B(a, b) = Re a > 0, Re b > 0. 1 0 x a 1 (1 x) b 1 dx, a, b C, N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 4 / 25

more identities A lot of identities satised by the gamma and beta functions have nite eld analogues. Say, the formulas G(χ) G(χ) = χ( 1) q and J(α, β) = G(α)G(β) G(αβ) with χ ɛ and αβ ɛ, are analogues of the reection formula and the formula Γ (s) Γ (1 s) = B(a, b) = π sin πs Γ (a)γ (b) Γ (a + b) with s, a, b C. The analogy observed dates back to the XIX century. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 5 / 25

more advanced analogy Given s C and integer m 1, one has the multiplication formula m 1 n=0 χ ( Γ s + n ) = (2π) (m 1)/2 m 1/2 ms Γ (ms). m Given a character ψ F q and an integer m (q 1), we have a nite eld counterpart of the multiplication formula. That is the DavenportHasse relation (1934) { G(ψχ) = ψ(m) } m G(χ) G(ψ m ), χ where the products range over χ F q under the assumption χm = ɛ. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 6 / 25

On cubic exponential and Kloosterman sums A very instructive sample of analogy between classical special functions and character sums is given by Iwaniec and Duke (1993). Consider the integral representation K 1/3 (u) = u 1/3 0 ) exp ( t u2 dt 4t (2t ) 4/3 for the BesselMacdonald function and the Nicholson formula for the Airy integral 0 cos(t 3 + tv) dt = v 1/2 which are valid at least for real u, v > 0. 3 K 1/3(2(v/3) 3/2 ), N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 7 / 25

From these two formulas, it follows exp ( i (cx 3 + x) ) dx = 3 1/2 0 ( x c ) 1/3 exp ( x 1 27cx ) dx x with any real c > 0. Assume 3 (q 1). This case there exists cubic character ψ of Fq and cubic Kloosterman sums, which one can treat as analogue of the integral in the right-hand side. Following the analogue, Iwaniec and Duke deduced very important formula which relates cubic Kloosterman sum to cubic exponential sum. That is e q (cx 3 + x) = ψ(xc 1 ) e q (x (27cx) 1 ) for c Fq. x F q x F q The proof involves the DavenportHasse relation. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 8 / 25

Elements of analysis Consider the complex vector space Ω q of all functions F q C supplied with the inner product f, g = x F q f (x)g(x) for all f, g Ω q. The additive characters x e q (hx), x, h F q form an orthogonal bases for the Ω q. The multiplicative characters χ F q together with the function δ form an orthogonal bases for the Ω q. N. V. Proskurin On character sums and complex functions over nite elds April 17, 2018 9 / 25

Additive Fourier transform Given a function F : F q C, its additive Fourier transform is the function ˆF : F q C dened by ˆF (x) = y Fq F (y)e q (yx) for all x F q. The Fourier inversion formula F (z) = 1 ˆF (x)eq ( xz) q x F q for all z F q allows one to recover F from ˆF and can be considered as the expansion of F over basis consisting of additive characters. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 10 / 25

Multiplicative Fourier transform Given a function F : F q C, its multiplicative Fourier transform is the function F : F q C dened by F (χ) = x F q F (x)χ(x) for all χ F q. The Fourier inversion formula F (z) = 1 q 1 χ F q F (χ)χ(z) for all z F q allows one to recover F from F. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 11 / 25

Power series expansion Given a function F : F q C, one has a similar expansion with one additional term F (z) = F (0) δ(z) + χ F q C χ χ(z) with C χ = 1 q 1 F (χ). One can treat the sum over χ as the sum over ɛ, ρ, ρ 2,... ρ q 1, whenever ρ generates the group F q. So, that is a nite eld analogue of power series expansion. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 12 / 25

Binomial formula By Green (1987), given ρ F q and x F q, one has the expansion ρ(1+x) = δ(x)+ q q 1 χ F q ( ) ρ χ(x) χ with ( ) ρ χ = χ( 1) J(ρ, χ), q which is a nite eld analogue of the classical binomial formula (1 + x) r = r k=0 ( ) r x k k with ( ) r = k r! (r k)! k!, x C. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 13 / 25

Expansion for the exponent For another example, let F be additive character e q. One has e q ( z) = 1 + q q 1 χ F q χ(z) G(χ) for all z F q. That is a nite eld analogue for the power series expansion for the exponent, z n e(z) = 1 + for all z C. n! n=1 N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 14 / 25

Dierentiation Given character χ F q, dene the linear operator Dχ : Ω q Ω q by D χ F (x) = 1 G(χ) t F q F (t) χ(x t) for all F : F q C and x F q. If χ ɛ, one can rewrite this as 1 G(χ) Dχ F (x) = 1 q t F q F (t) χ(t x). This should be compared with the Cauchy integral formula 1 n! f (n) (x) = 1 f (t) dt 2πi (t x) n+1 for the derivative f (n) of any order n of the analytic function f. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 15 / 25

By Evans (1986), D χ F is the derivative of order χ of F. One nds easily some standard properties. D χ takes constant functions to zero function, whenever χ ɛ. D α D β = D αβ for characters α, β F q subject to αβ ɛ. D ɛ F (x) = F (x) t F q F (t), with F and x as above. Given two functions E and F, x F q, and the character ν we have the formula for integration by parts E(x) D ν F (x) = ν( 1) F (x) D ν E(x) x F q x F q and the Leibniz rule for the ν-th derivative of the product D ν EF (x) = 1 q 1 µ F q G(µ)G(µν) G(ν) D µ E(x) D νµ F (x). N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 16 / 25

Given any character ν, let F (x) = e q ( x) for all x F q. This case we have D ν F = F. For any F : F q C and a F q, we have expansion F (x) = 1 G(ν)D ν F (a) ν(a x) q 1 ν F q for all x F q, x a. That is a nite eld analogue of the Taylor expansion. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 17 / 25

Hermite character sums Given any character ν F q and x F q, let H ν (x) = 1 ν(u)e q (u 2 + 2ux). G(ν) u F q That is given by Evans (1986) as a nite eld analogue of the classical Hermite polynomials H n, n 0. We have H n ( x) = ( 1) n H n (x) for all x R and integer n 0; H ν ( x) = ν( 1) H ν (x) for all x F q, ν F q. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 18 / 25

Also, for all x F q, one has H ν (x) = ν( 1)F (x) 1 D ν F (x), where F (x) = e q (x 2 ). That is a nite eld analogue of the Rodriguez formula H n (x) = ( 1) n exp(x 2 ) d n for the classical Hermite polynomials H n. dx n exp( x 2 ), x R, In a similar manner one can treat the Legendre polynomials, the Bessel functions and other classical polynomials and some special functions. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 19 / 25

Hypergeometric functions For the hypergeometric function 2 F 1 one has the Euler integral representation 2F 1 (a, b; c; x) = Γ(c) 1 Γ(b) Γ(c b) 0 t b (1 t) c b (1 xt) a dt t (1 t). Greene (1987) dened the hypergeometric functions on F q by 2F 1 [ α, β γ ] x = ɛ(x) βγ( 1) q for any characters α, β, γ F q and x F q. t F q β(t) βγ(1 t) α(1 xt) N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 20 / 25

With this denition and notation one has power series expansion [ α, β ] 2F 1 x = q ( )( ) αχ βχ χ(x), γ q 1 χ γχ χ F q and one has the more general denition [ α0, α 1,..., α ] n n+1f n x = q ( )( ) α0 χ α1 χ β 1,..., β n q 1 χ β 1 χ χ F q for any integer n 1 and characters α 0,... β n F q.... ( ) αn χ χ(x) β n χ N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 21 / 25

Like their classical counterparts, hypergeometric functions over nite elds satisfy many transformation identities. Say, for classical Pfa's transformation 2F 1 (a, b; c; x) = (1 x) a 2F 1 ( a, c b; c; x/(x 1) ) one has nite eld analogue 2F 1 [ α, β γ ] [ αγ, β x = β(1 x) 2 F 1 γ for any characters α, β, γ F q and x F q, x 1. x ] x 1 N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 22 / 25

Turn to the function 3 F 2. In the classical context, there are some summation formulas. We mean the formulas of Saalsch utz, Dixon, Watson, Whipple. One has nit eld analogues for each of them. Say, as analogue for Saalsch utz's formula one has [ α, β, γ ] ( )( ) γ β 3F 2 1 = βγ( 1) 1 ( ) βρ ρ, αβγρ αρ γρ q βρ( 1). α N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 23 / 25

As one more example, consider Clausen's famous identity 3F 2 (2c 2s 1, 2s, c 1/2; 2c 1, c; x) = 2 F 1 (c s 1/2, s; c; x) 2, which was utilized in de Branges' proof of the Bieberbach conjecture. By Evans and Greene (2009), one has a nite eld analogue of this formula. That is [ α 3F 2 γ 2, α 2, γφ 2 γ 2, γ ] x γ(x) φ(1 x) γ(4) J(αγ, αγ) [ αγφ, α = + 2F 1 q J(α, α) γ x ] 2, where x F q, x 1, φ is the quadratic character (φ ɛ, φ 2 = ɛ), and it is assumed that γ φ, α 2 ɛ, γ, γ 2. N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 24 / 25

A study of special function analogues over nite elds serve as a useful tool when applied to problems related to character sums. One can nd Magma code to compute the hypergeometric functions over nite elds in Lennon's dissertation (2006). N. V. Proskurin On character sums and complex functions over nite April elds17, 2018 25 / 25