A362198 a(n) = number of isogeny classes of abelian surfaces over the finite field of order prime(n).

35, 63, 129, 207, 401, 513, 765, 897, 1193, 1683, 1861, 2425, 2821, 3031, 3461, 4139, 4861, 5109, 5877, 6409, 6683, 7521, 8099, 8987, 10223, 10865, 11185, 11839, 12173, 12849, 15301, 16031, 17143, 17519, 19441, 19833, 21027, 22239, 23065, 24317, 25589, 26019, 28203, 28647, 29545, 29993

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

Cf. A362201, A362570.

Sequence in context: A026064 A338244 A250764 * A357188 A350344 A245274

Adjacent sequences: A362195 A362196 A362197 * A362199 A362200 A362201

Keyword

nonn

Author

Robin Visser, Apr 10 2023

Status

approved