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

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 4 integer polynomials with leading coefficient prime(n)^2, whose (complex) roots all have absolute value 1/sqrt(prime(n)).

Examples

			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.
		

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.

Crossrefs

Cf. A362201, A362570.

Programs

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. = 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

Formula

a(n) ~ (32/3) * prime(n)^(3/2).