A362201 a(n) = number of isogeny classes of dimension 3 abelian varieties over the finite field of order prime(n).

215, 677, 2953, 7979, 30543, 50371, 112283, 156589, 277517, 555843, 678957, 1153875, 1569637, 1810805, 2364089, 3389675, 4675707, 5167277, 6846631, 8147047, 8855295, 11222313, 13014767, 16045439, 20772343, 23449327, 24870063, 27880975, 29473619, 32839031, 46617799, 51162221

Offset: 1

Author

Robin Visser, Apr 10 2023

Keywords

• nonn

Comments

Two abelian varieties over a finite field are isogenous if and only if their Hasse-Weil zeta functions coincide.

Thus a(n) is the number of degree 6 integer polynomials with leading coefficient prime(n)^3 and whose (complex) roots all have absolute value 1/sqrt(prime(n)).

Links

• Robin Visser, Table of n, a(n) for n = 1..50
• S. A. DiPippo and E. W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, arXiv:math/9803097 [math.NT], 1998-2000.
• T. Dupuy, K. Kedlaya, D. Roe, and C. Vincent, Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB, arXiv:2003.05380 [math.NT].
• S. Haloui, The characteristic polynomials of abelian varieties of dimensions 3 over finite fields, arXiv:1003.0374 [math.AG], 2020.
• LMFDB, Abelian variety count results.
• W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521-560.

Crossrefs

Cf. A362198, A362570.

Programs

Sage

from sage.rings.polynomial.weil.weil_polynomials import WeilPolynomials
def a(n):
    p = Primes()[n-1]
    return len(list(WeilPolynomials(6, p)))

Sage

def a(n):
    R. = PolynomialRing(CC)
    num_solutions = 0
    p = Primes()[n-1]
    for a1 in range(ceil(-6*sqrt(p)), floor(6*sqrt(p))+1):
        for a2 in range(ceil(-15*p), floor(15*p)+1):
            for a3 in range(ceil(-20*p*sqrt(p)), floor(20*p*sqrt(p))+1):
                L_poly = 1+a1*x+a2*x^2+a3*x^3+p*a2*x^4+p^2*a1*x^5+p^3*x^6
                for r in L_poly.roots():
                    if (abs(abs(r[0]) - 1/sqrt(p)) > 1e-12):
                        break
                else:
                    num_solutions += 1
    return num_solutions

Formula

a(n) ~ (1024/45) * prime(n)^3.