A362570 -id:A362570 - cp's OEIS frontend

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

Robin Visser, Apr 10 2023

nonn

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)).

			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.

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.

Cf. A362201, A362570.

from sage.rings.polynomial.weil.weil_polynomials import WeilPolynomials
def a(n):
    p = Primes()[n-1]
    return len(list(WeilPolynomials(4,p)))

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

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

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

Original entry on oeis.org

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

Robin Visser, Apr 10 2023

nonn

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)).

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.

Cf. A362198, A362570.

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

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

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

A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

Original entry on oeis.org

5, 7, 9, 9, 11, 9, 13, 13, 15, 13, 17, 17, 19, 20, 17, 21, 23, 15, 25, 25, 27, 27, 27, 29, 31, 31, 21, 33, 33, 35, 35, 29, 37, 37, 39, 41, 41, 41, 41, 43, 45, 37, 45, 25, 45, 47, 47, 49, 49, 51, 51, 51, 50, 53, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 61, 61, 63, 45, 63, 37, 65, 65

Offset: 1

Robin Visser, Aug 02 2023

nonn

Two elliptic curves over a finite field F_q are isogenous if and only if they have the same trace of Frobenius, or equivalently, have the same number of points over F_q.

Thus a(n) is the number of integers k such that there exists an elliptic curve over GF(q) with trace k, where q = A246655(n).

			For n = 1, the a(1) = 5 isogeny classes of elliptic curves over GF(2) are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
For n = 2, the a(2) = 7 isogeny classes of elliptic curves over GF(3) are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.

Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A005523, A362570.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p, ans = Integer(q).prime_factors()[0], 0
        for a in range(-floor(2*sqrt(q)), floor(2*sqrt(q))+1):
            if (a%p != 0) or (Integer(q).is_square() and ((abs(a) == 2*sqrt(q))
                  or ((p%3 != 1) and (abs(a) == sqrt(q))) or ((p%4 != 1) and
                  (a==0)))) or ((not Integer(q).is_square()) and
                  (((p in [2,3]) and (abs(a) == sqrt(p*q))) or (a==0))):
                ans += 1
        print(ans)

a(n) = 2*floor(2*sqrt(q)) + 1 if q is prime, where q = A246655(n).

cp's OEIS Frontend

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

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

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A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Formula