A262587 - OEIS

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A262587

"Special" prime powers in Serre's sense.

2, 3, 5, 7, 8, 13, 17, 31, 32, 37, 43, 73, 101, 128, 157, 197, 211, 241, 257, 307, 343, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2048, 2187, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See Hirschfeld, pp. 49-50 for precise definition.

By a theorem of Hasse-Weil and Serre, every (absolutely irreducible, smooth) genus 2 curve over GF(q) has cardinality at most q + 1 + 2*floor(2*sqrt(q)). This sequence consists exactly of the prime powers q (excluding 4 and 9) for which there does not exist any genus 2 curve over GF(q) with cardinality equal to q + 1 + 2*floor(2*sqrt(q)). - Robin Visser, Aug 26 2023

REFERENCES

J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.

LINKS

Robin Visser, Table of n, a(n) for n = 1..10000

Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.

Jean-Pierre Serre, Nombres de points des courbe algebriques sur F_q, Sémin. Théorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.

PROG

(Sage)

for q in range(1, 1000):

if Integer(q).is_prime_power():

p = Integer(q).prime_factors()[0]

if (not Integer(q).is_square()):

if ((floor(2*sqrt(q))%p == 0) or (q-1).is_square() or