Characteristic polynomials of the curve y² = x⁷ + ax⁴ + bx over finite fields.pdf Introduction Let Fq be a finite field of size q = pn, p > 2. In this note, we study the hyperelliptic curves of genus g = 3 of the form C : у2 = ж2й+2 + ажй+1 + Ьж- The Jacobian JC of the curves is split [1] over certain finite field extension: JC ~ JDl x JD2, where D1 and D2 are explicitly given curves. This fact allows us to reduce the problem of point-counting on the curve C to counting points on the curves D1 and D2. For genus 2 case it was done in the works [2, 3]. The work [1] contains algorithms for g > 2 case. In this work, we give explicit formulae for the number of points on the Jacobian in the case of g = 3. The point-counting on the curve is equivalent to finding of zeta-function of the curve Z(C/Fq;T) = exp (£#C(F„,)£) = (1 -'qT), where Lc,q(T) = qgT2g + a^-1 T2g-1 + ... + aflTg + afl_iTg-1 + ... + aiT + 1 and a G Z, к I ^ (2f)qi/2 for«= i,...,g. Let xC,q (T) be a characteristic polynomial of the Frobenius endomorphism. Then LC,q(T) = T2gxC,q(1/T) and #JC(Fq) = LCq(1) = xC,q(1). Therefore, the computation of #JC (Fq) is equivalent to the computation of the characteristic polynomial. In this work, we enumerate all possible characteristic polynomials for the curve C in the case of g = 3. 1. Characteristic polynomials for genus 3 curves Let C : y2 = x7 + ax4 + bx be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3. Since, there is a map (x,y) ^ (x3,xy) from C to an elliptic curve E1 : y2 = x3 + ax2 + bx, we have JC ~ E1 x A over Fq for some abelian surface A. Therefore, XC,q (T) = XEi,q (T)XA,q (T) . The characteristic polynomial for E1 can be efficiently computed using SEA-algorithm [4]. So, we only have to determine the coefficients of xA,q(T) = T4 - b1T3 + b2T2 - b1qT + q2. From [1, Th. 2], we have Jc ~ E x Jd over Fq [^j, where E2 is an elliptic curve with equation y2 = x3 - 3 Уbx + a and D is a hyperelliptic curve with equation y2 = (x2 - 4\\\\b)(x3 - 3\\\\bx + a). Moreover, the Jacobian JD is also split, since E1 ^ E2 in general. First we describe the characteristic polynomials in the simplest case when b is a cubic residue. In this case for each cubic root, we have a map to an elliptic curve, so we obtain the following theorem. Theorem 1. Let C : y2 = x7 + ax4 + bx be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3, and let b be a cubic residue. Then 1) if q = 1 (mod 6), then JC ~ E1 x over Fq and Xc,q (T) = (T2 - t1T + q)(T2 - t2T + q)2, where E1 : y2 = x3 + ax2 + bx, E2 : y2 = x3 - 3^x + a are elliptic curves and t1, t2 are their traces of the Frobenius endomorphism; 2) if q = 5 (mod 6), then JC ~ E1 x E2 x E2 over Fq and Xc,q (T) = (T2 - t1T + q)(T2 - t2T + q)(T2 + ^T + q), where E2 is a quadratic twist of E2. In general case, we have JC ~ E1 x A, where A can be simple. Theorem 2. Let C : у2 = ж7 + аж4 + Ьж be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3. Then 1) ~ E1 x A over Fq, where E1 is an elliptic curve with equation у2 = ж3 + аж2 + Ьж and A is an abelian surface; 2) if q = 5 (mod 6), we have JC ~ E1 x E2 x E2 and Xc,q (T) = (T2 - t1T + q)(T2 - t2T + q)(T2 + ^T + q), where E1,E2,t1,t2 are the same as in Theorem 1; 3) if q = 1 (mod 6) and ^Ь Е Fq, then JC ~ E1 x over Fq and Xc,q (T) = (T2 - t1T + q)(T2 - t2T + q)2; 4) if q = 1 (mod 6), ffi Е Fq and E2 is ordinary, then xC,q(T) = (T2 - t1T + q)xA(T), where xA(T) is one of the following polynomials: • (T4 - ^T3 + (*1 - q)T2 - ^qT + q2), V Е Fq; • (T4 + t 2T3 + (t - q)T2 + ^qT + q2), Vb Е Fq; • (T4 - 2^T3 + (t + 2q)T2 - 2^qT + q2), Vb Е Fq, A is split; • (t4 + 2t2T3 + (t + 2q)T2 + 2t2qT + q2), Vb Е Fq, A is split. Here, t2 is a trace of Frobenius of elliptic curve E2 : у2 = ж3 - 3Ьж + аб; 5) if q = 1 (mod 6), Fq and E2 is supersingular, then A is supersingular and Xcq(T) = (T2 - t1T + q)xAq(T) where xAq(T) is one of the following polynomials: ' • (T4 - qT2 + q2); • (T4 + 2qT2 + q2); • (T2 + q)(T ± Vq)2, P = 7 (mod 12), n is even, A is split; • (T ± Vq)2, n is even, A is split; • (T2 ± TVq + q)2, n is even, A is simple; • (T4 + VqT3 + qT2 + q3/2T + q2), p = 1 (mod 5), n is even, A is simple; • (T4 - VqT3 + qT2 - q3/2T + q2), p = 1 (mod 10), n is even, A is simple. Conclusion In this work, we obtained the complete list of the characteristic polynomials for the genus 3 curve у2 = ж7 + аж4 + Ьж in terms of traces of Frobenius of certain elliptic curves. Since #JC (Fq) = xC, q (T), this gives us the explicit formulae for the number of points on the Jacobian.

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