Computing the cardinality of CM elliptic curves using torsion points

François Morain

Displaying similar documents to “Computing the cardinality of CM elliptic curves using torsion points”

On a theorem of Mestre and Schoof

John E. Cremona, Andrew V. Sutherland (2010)

Journal de Théorie des Nombres de Bordeaux

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A well known theorem of Mestre and Schoof implies that the order of an elliptic curve $E$ over a prime field $𝔽_{q}$ can be uniquely determined by computing the orders of a few points on $E$ and its quadratic twist, provided that $q > 229$ . We extend this result to all finite fields with $q > 49$ , and all prime fields with $q > 29$ .

On rational torsion points of central $ℚ$ -curves

Fumio Sairaiji, Takuya Yamauchi (2008)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ be a central $ℚ$ -curve over a polyquadratic field $k$ . In this article we give an upper bound for prime divisors of the order of the $k$ -rational torsion subgroup $E_{t o r s} (k)$ (see Theorems 1.1 and 1.2). The notion of central $ℚ$ -curves is a generalization of that of elliptic curves over $ℚ$ . Our result is a generalization of Theorem 2 of Mazur [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].

Weber’s class number problem in the cyclotomic $ℤ_{2}$ -extension of $ℚ$ , II

Takashi Fukuda, Keiichi Komatsu (2010)

Journal de Théorie des Nombres de Bordeaux

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Let $h_{n}$ denote the class number of $n$ -th layer of the cyclotomic $ℤ_{2}$ -extension of $ℚ$ . Weber proved that $h_{n} (n \geq 1)$ is odd and Horie proved that $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ satisfying $ℓ \equiv 3, 5 (mod 8)$ . In a previous paper, the authors showed that $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ less than $10^{7}$ . In this paper, by investigating properties of a special unit more precisely, we show that $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ less than $1.2 \cdot 10^{8}$ . Our argument also leads to the conclusion that $h_{n} (n \geq 1)$ is not divisible by...

Almost powers in the Lucas sequence

Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek (2008)

Journal de Théorie des Nombres de Bordeaux

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The famous problem of determining all perfect powers in the Fibonacci sequence ${(F_{n})}_{n \geq 0}$ and in the Lucas sequence ${(L_{n})}_{n \geq 0}$ has recently been resolved []. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_{n} = q^{a} y^{p}$ , with $a > 0$ and $p \geq 2$ , for all primes $q < 1087$ and indeed for all but $13$ primes $q < 10^{6}$ . Here the strategy of [] is not sufficient due to the sizes...

Practical Aurifeuillian factorization

Bill Allombert, Karim Belabas (2008)

Journal de Théorie des Nombres de Bordeaux

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We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials $Φ_{d} (a)$ for integers $a$ and $d > 0$ . Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time $\tilde{O} (d^{2} L)$ , using $O (d L)$ space, where $L ≔ log (|a| + 1)$ .

Displaying similar documents to “Computing the cardinality of CM elliptic curves using torsion points”

On a theorem of Mestre and Schoof

On rational torsion points of central ℚ -curves

Weber’s class number problem in the cyclotomic ℤ 2 -extension of ℚ , II

Almost powers in the Lucas sequence

Practical Aurifeuillian factorization

On rational torsion points of central $ℚ$ -curves

Weber’s class number problem in the cyclotomic $ℤ_{2}$ -extension of $ℚ$ , II