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# Some mixed character sum identities of Katz II

- Ron Evans
^{1}Email author and - John Greene
^{2}

**Received:**12 September 2016**Accepted:**21 December 2016**Published:**3 April 2017

## Abstract

A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of *q* elements, where *q* is a power of a prime \(p >3\). His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. The first author recently gave such a proof of his identities when \(q \equiv 1 \pmod 4\), and this paper provides such a proof for the remaining case \(q \equiv 3 \pmod 4\). Our proofs are valid for all characteristics \(p>2\). Along the way we prove some elegant new character sum identities.

## Keywords

- Hypergeometric \({}_2F_1\) character sums over finite fields
- Gauss and Jacobi sums
- Norm-restricted Gauss and Jacobi sums
- Eisenstein sums
- Hasse–Davenport theorems
- Quantum physics

## Mathematics Subject Classification

- 11T24
- 33C05

## 1 Background

Let \(\mathbb {F}_q\) be a field of *q* elements, where *q* is a power of an odd prime *p*. Throughout this paper, *A*, *B*, *C*, *D*, \(\chi \), \(\lambda \), \(\nu \), \(\mu \), \(\varepsilon \), \(\phi \) denote complex multiplicative characters on \(\mathbb {F}_q^*\), extended to map 0 to 0. Here \(\varepsilon \) and \(\phi \) always denote the trivial and quadratic characters, respectively. Define \(\delta (A)\) to be 1 or 0 according as *A* is trivial or not, and let \(\delta (j,k)\) denote the Kronecker delta for \(j,k \in \mathbb {F}_q\).

Much of this paper deals with the extension field \(\mathbb {F}_{q^2}\) of \(\mathbb {F}_q\). Let \(M_4\) denote a fixed quartic character on \(\mathbb {F}_{q^2}\) and let \(M_8\) denote a fixed octic character on \(\mathbb {F}_{q^2}\) such that \(M_8^2 = M_4\).

*A*,

*V*(

*j*) [13, pp. 226–229] for which the identities

*q*-dimensional vector \((V(j))_{j \in \mathbb {F}_q} \) is a minimum uncertainty state, as described by Sussman and Wootters [17].) Katz’s proof [13, Theorem 10.2] of the identities (1.5) required the characteristic

*p*to exceed 3, in order to guarantee that various sheaves of ranks 2, 3, and 4 have geometric and arithmetic monodromy groups which are SL(2), SO(3), and SO(4), respectively.

As Katz indicated in [13, p. 223], his proof of (1.5) is quite complex, invoking the theory of Kloosterman sheaves and their rigidity properties, as well as results of Deligne [6] and Beilinson, Bernstein, Deligne [4]. Katz [13, p. 223] wrote, “It would be interesting to find direct proofs of these identities.”

The goal of this paper is to respond to Katz’s challenge by giving a direct proof of (1.5) (a “character sum proof” not involving algebraic geometry). This has the benefit of making the demonstration of his useful identities accessible to a wider audience of mathematicians and physicists. Since a direct proof for \(q \equiv 1 \pmod 4\) has been given in [8], we will assume from here on that \(q \equiv 3 \pmod 4\).

A big advantage of our proof is that it works for all odd characteristics *p*, including \(p=3\). As a bonus, we obtain some elegant new double character sum evaluations in (5.11)–(5.14).

Our method of proof is to show (in Sect. 6) that the double Mellin transforms of both sides of (1.5) are equal. The Mellin transforms of the left and right sides of (1.5) are given in Theorems 3.1 and 5.1, respectively. A key feature of our proof is a formula (Theorem 4.1) relating a norm-restricted Jacobi sum over \(\mathbb {F}_{q^2}\) to a hypergeometric \({}_2F_1\) character sum over \(\mathbb {F}_q\). Theorem 4.1 will be applied to prove Theorem 5.3, an identity for a weighted sum of hypergeometric \({}_2F_1\) character sums. Theorem 5.3 is crucial for our proof of (1.5) in Sect. 6.

Hypergeometric character sums over finite fields have had a variety of applications in number theory. For some recent examples, see [2, 3, 7, 11, 14–16].

*i*is a fixed primitive fourth root of unity in \(\mathbb {F}_{q^2}\). Write \(\overline{z}= x - iy\) and note that \(\overline{z}= z^q\). The restriction of \(M_8\) to \(\mathbb {F}_q\) equals \(\varepsilon \) or \(\phi \) according as

*q*is congruent to 7 or 3 mod 8. In particular,

*C*on \(\mathbb {F}_q\), we let

*CN*denote the character on \(\mathbb {F}_{q^2}\) obtained by composing

*C*with the norm map

*N*on \(\mathbb {F}_{q^2}\) defined by

*B*on \(\mathbb {F}_q\),

*BCN*is to be interpreted as the character (

*BC*)

*N*, i.e.,

*BNCN*.

*a*as in (1.3), define

*V*(

*j*) to be the following norm-restricted Gauss sums:

## 2 Mellin transform of the sums *V*(*j*)

*G*and

*J*over \(\mathbb {F}_q\). For any character \(\beta \) on \(\mathbb {F}_{q^2}\), we have

*C*on \(\mathbb {F}_q\), \(G_2(CN M_8)\) equals \(G_2(CN \overline{M}_8)\) or \(G_2(CN M_8^3)\) according as

*q*is congruent to 7 or 3 mod 8. The Hasse-Davenport theorem on lifted Gauss sums [5, Theorem 11.5.2] gives

*q*in place of

*p*, we can express \(E_2(\beta )\) in terms of Gauss sums when \(\beta \) is nontrivial, as follows:

The next theorem gives an evaluation of \(S(\chi )\) in terms of Gauss sums.

### Theorem 2.1

### Proof

*z*by \(z/j^2\) to get

*j*on the right equals \(q-1\) when \(C \in \{\nu , \phi \nu \}\) and it equals 0 otherwise. Since \(\phi N = M_8^4\), the result now follows from the definition of \(G_2\). \(\square \)

## 3 Double Mellin transform of *V*(*j*)*V*(*k*)

*S*in terms of Gauss and Jacobi sums.

### Theorem 3.1

### Proof

*q*is congruent to 7 or 3 mod 8, so that the bracketed expression for \(i=0\) in (3.4) is to be compared to that for \(i=1\) in (3.5) when

*q*is congruent to 3 mod 8. \(\square \)

## 4 Identity for a norm-restricted Jacobi sum in terms of a \({}_2F_1\)

*D*be a character on \(\mathbb {F}_q\). Define the norm-restricted Jacobi sums

*R*(

*D*,

*j*) in terms of a \({}_2F_1\) hypergeometric character sum.

### Theorem 4.1

### Proof

*z*in (4.1) by \(-zj^2\). By (1.6), we obtain

*z*in the sum must be a square, since

*N*(

*z*) is a square in \(\mathbb {F}_q\). Thus

*y*by

*yx*, we have

*D*, which completes the proof when \(j = \pm 1\). Thus assume for the remainder of this proof that \(j^2 \ne 1\).

*Q*(

*D*,

*j*) equals

## 5 Double Mellin transform of *P*(*j*, *k*)

The following theorem evaluates *T*.

### Theorem 5.1

*D*on \(\mathbb {F}_q\) and \(j \in \mathbb {F}_q^*\), we define

### Proof

*j*by

*jk*in (1.3), we obtain

*k*and

*x*by their negatives. Therefore,

*x*by

*ax*and employing (5.3), the desired result (5.2) readily follows. \(\square \)

We proceed to analyze *h*(*D*, *j*).

### Lemma 5.2

*D*, we have

*D*is trivial, then \(h(D,j)=0\).

### Proof

*h*(

*D*,

*j*) in (5.3). The evaluation in (5.5) is the same as that in [8, (5.21)], the proof of which is valid for

*q*congruent to either 1 or 3 mod 4. Finally, let \(j \ne \pm 1\). Then since

*x*by \(1-x(2j^2+2)/(j+1)^2\) shows that \(h(\varepsilon ,j)= -1 + 1 = 0\). \(\square \)

### Theorem 5.3

*D*on \(\mathbb {F}_q\), define

*D*,

### Proof

*D*be nontrivial. By Lemma 5.2,

*Y*(

*D*) denote this sum on

*j*. It remains to prove that

*j*on the right vanishes unless \(\chi \in \{\nu _1, \nu _1\phi \}\), and so we obtain the desired result (5.10). \(\square \)

As interesting consequences of Theorem 5.3, we record the elegant double character sum evaluations (5.11)–(5.14) below.

### Theorem 5.4

### Proof

This follows by putting \(D=\phi \) in (5.7). \(\square \)

### Theorem 5.5

*u*|, |

*v*| is the unique pair of positive integers with \(p \not \mid u\) for which \(q^2=u^2 + 2v^2\), and where the sign of

*u*is determined by the congruence \(u \equiv -1 \pmod 8\). In particular, when \(q=p \equiv 3 \pmod 8\), we have

### Proof

*q*, which proves (5.12).

Now suppose that \(q \equiv 3 \pmod 8\). An application of (2.1) shows that \(J_2(M_8^5, \phi N)\) is the complex conjugate of \(J_2(M_8, \phi N)\), so that the sum in (5.13) equals \(2 \mathfrak {R}J_2(M_8, \phi N)\).

*q*is prime, i.e., \(q=p\). Then \(G_2(\phi N) = p\) and by [5, Theorems 12.1.1 and 12.7.1(b)],

*u*,

*v*such that \(q^2 = u^2 + 2v^2\). Since \(u_1 \equiv -1 \pmod 8\), it is easily seen using the binomial theorem that \(u \equiv -1 \pmod 8\). If \(p=\pi \overline{\pi }\) divided

*u*, then

*p*would divide

*v*, so that the prime \(\overline{\pi }\) would divide \(\pi ^{2t}\), which is impossible. Thus \(p \not \mid u\). For an elementary proof of the uniqueness of |

*u*|, |

*v*|, see [5, Lemma 3.0.1]. \(\square \)

### Remark

*D*, and then replacing

*j*by

*jI*, where

*I*is a primitive fourth root of unity in \(\mathbb {F}_q\). More work is needed to evaluate

*Z*in the remaining case where \(q \equiv 1 \pmod 8\). In this case

*Z*is equal to the sum \(R_2\) in [8, (5.44)] with \(\nu _1= B_8\) and \(A_4 = B_8^2\) for an octic character \(B_8\) on \(\mathbb {F}_q\). The proof of [5, Theorem 3.3.1] shows that

*c*and

*d*are the unique pair of integers up to sign for which

First suppose that \(p \equiv 7 \pmod 8\). Then \(q = p^{2t}\) for some \(t \ge 1\). If \(t=1\), then \(J(B_8, \phi )=p\) by [5, Theorem 11.6.1]. For general *t*, the Hasse-Davenport lifting theorem thus yields \(J(B_8, \phi )=(-1)^{t-1}p^t\), so that \(J(B_8, \phi )^2=q\). Thus \(Z=4q\) by (5.16).

Now suppose that \(p \equiv 5 \pmod 8\). Since \(G(B_8)=G(B_8^p)=G(B_8^5)\) by [5, Theorem 1.1.4(d)], \(J(B_8, \phi ) = G(\phi )\). Thus \(J(B_8, \phi )^2=q\), so again \(Z=4q\). This completes the proof of (5.17).

*p*ramifies in the cyclotomic field \(\mathbb {Q}(\exp (2\pi i/8))\). In view of (5.20) and unique factorization in \(\mathbb {Q}(i\sqrt{2})\), we may suppose without loss of generality that \(J(B_8, \phi ) = \pi ^2\) when \(t=1\). For general

*t*,

*c*and

*d*such \(c^2 + 2d^2 = q\). Note that

*p*cannot divide

*c*, for otherwise

*p*also divides

*d*(since \(c^2 + 2d^2 = q\)), so that

*p*divides \(\pi ^{2t}\), yielding the contradiction that the prime \(\overline{\pi }\) divides \(\pi \). By (5.21),

*t*,

*c*and

*d*such \(c^2 + 2d^2 = q\). Arguing as in the case \(p \equiv 3 \pmod 8\), we again obtain \(Z=4c^2\) for

*c*as in (5.19). This completes the proof of (5.18).

## 6 Proof of Katz’s identities (1.5)

*S*and

*T*are given in Theorems 3.1 and 5.1, respectively. These theorems show that

*S*and

*T*both vanish when \(\chi _1\) or \(\chi _2\) is even, so we may assume that (3.2) and (3.3) hold. For brevity, write \(D = \mu \phi ^i\), where \(i \in \{0,1\}\). Then the equality \(S=T\) is equivalent to

*W*(

*D*) in Theorem 5.3, we easily see that (6.1) holds. This completes the proof that \(S=T\).

## Declarations

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## Authors’ Affiliations

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