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 4 integer polynomials with leading coefficient prime(n)^2, whose (complex) roots all have absolute value 1/sqrt(prime(n)).
LINKS
Robin Visser, Table of n, a(n) for n = 1..1000
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], 2020.
D. W. Farmer, S. Koutsoliotas, and S. Lemurell, Varieties via their L-functions, J. Number Theory 196 (2019), 364-380.
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) ~ (32/3) * prime(n)^(3/2).
EXAMPLE
For n = 1, the a(1) = 35 possible isogeny classes correspond to the following 35 possible Hasse-Weil zeta functions of abelian surfaces over F_2: 4x^4 - 8x^3 + 8x^2 - 4x + 1, 4x^4 - 6x^3 + 5x^2 - 3x + 1, 4x^4 - 6x^3 + 6x^2 - 3x + 1, 4x^4 - 4x^3 + 2x^2 - 2x + 1, 4x^4 - 4x^3 + 3x^2 - 2x + 1, 4x^4 - 4x^3 + 4x^2 - 2x + 1, 4x^4 - 4x^3 + 5x^2 - 2x + 1, 4x^4 - 2x^3 - x^2 - x + 1, 4x^4 - 2x^3 - x + 1, 4x^4 - 2x^3 + x^2 - x + 1, 4x^4 - 2x^3 + 2x^2 - x + 1, 4x^4 - 2x^3 + 3x^2 - x + 1, 4x^4 - 2x^3 + 4x^2 - x + 1, 4x^4 - 4x^2 + 1, 4x^4 - 3x^2 + 1, 4x^4 - 2x^2 + 1, 4x^4 - x^2 + 1, 4x^4 + 1, 4x^4 + x^2 + 1, 4x^4 + 2x^2 + 1, 4x^4 + 3x^2 + 1, 4x^4 + 4x^2 + 1, 4x^4 + 2x^3 - x^2 + x + 1, 4x^4 + 2x^3 + x + 1, 4x^4 + 2x^3 + x^2 + x + 1, 4x^4 + 2x^3 + 2x^2 + x + 1, 4x^4 + 2x^3 + 3x^2 + x + 1, 4x^4 + 2x^3 + 4x^2 + x + 1, 4x^4 + 4x^3 + 2x^2 + 2x + 1, 4x^4 + 4x^3 + 3x^2 + 2x + 1, 4x^4 + 4x^3 + 4x^2 + 2x + 1, 4x^4 + 4x^3 + 5x^2 + 2x + 1, 4x^4 + 6x^3 + 5x^2 + 3x + 1, 4x^4 + 6x^3 + 6x^2 + 3x + 1, 4x^4 + 8x^3 + 8x^2 + 4x + 1.
PROG
(Sage)
from sage.rings.polynomial.weil.weil_polynomials import WeilPolynomials
def a(n):
p = Primes()[n-1]
return len(list(WeilPolynomials(4, p)))
(Sage)
def a(n):
R.<x> = PolynomialRing(CC)
num_solutions = 0
p = Primes()[n-1]
for Cp in range(ceil(p+1-4*sqrt(p)), floor(p+1+4*sqrt(p))+1):
for Cp2 in range(ceil(p^2+1-4*p), floor(p^2+1+4*p)+1):
a2 = (Cp^2 + Cp2 + 2*p*(1-Cp) - 2*Cp)
if a2%2 != 0:
continue
L_poly = 1 + (Cp-p-1)*x + a2/2*x^2 + p*(Cp-p-1)*x^3 + p^2*x^4
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