A364690 Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).

128, 2048, 2187, 16807, 32768, 131072, 524288, 1953125, 2097152, 8388608, 14348907, 48828125, 134217728, 536870912, 30517578125, 549755813888, 847288609443, 2199023255552, 19073486328125, 140737488355328, 562949953421312, 36028797018963968, 144115188075855872, 450283905890997363

Offset: 1

Author

Robin Visser, Aug 02 2023

Keywords

• nonn

Comments

By Hasse's theorem, every elliptic curve E over GF(q) has cardinality at most q + 1 + floor(2*sqrt(q)). Moreover, for every prime power q, there exists an elliptic curve E over GF(q) with cardinality at least q + floor(2*sqrt(q)). Thus these are the prime powers q for which A005523(n) = q + floor(2*sqrt(q)), where q = A246655(n).

By a theorem of Deuring and Waterhouse, these are exactly the prime powers q = p^k such that q is not prime, q is not a square, and p divides floor(2*sqrt(q)).

Examples

			The first few values of the sequence (factorized) are 2^7, 2^11, 3^7, 7^5, 2^15, 2^17, 2^19, 5^9, 2^21, 2^23, 3^15, 5^11, 2^27, 2^29, ...
		

Links

• Katie Ahrens and Jon Grantham, Table of n, a(n) for n = 1..146
• 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.

Crossrefs

Cf. A005523, A246655.

Subsequence of A246547.

Programs

Sage

for q in range(1, 100000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):
            print(q)