OFFSET
1,1
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.
FORMULA
a(n) ~ (1024/45) * prime(n)^3.
PROG
(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.<x> = 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Apr 10 2023
STATUS
approved