Arithmetic properties of coefficients of L-functions of elliptic curves

Abstract

Let n 1 a n n −s be the L-series of an elliptic curve E defined over the rationals without complex multiplication. In this paper, we present certain similarities between the arithmetic properties of the coefficients {a n } ∞ n=1 and Euler's totient function ϕ(n). Furthermore, we prove that both the set of n such that the regular polygon with |a n | sides is ruler-and-compass constructible, and the set of n such that n − a n + 1 = ϕ(n) have asymptotic density zero. Finally, we improve a bound of Luca and Shparlinski on the counting function of elliptic pseudoprimes.

On the prime factors of the iterates of the Ramanujan τ–function

“…so we see that if one of the above expressions is zero but τ (q) = 0, it follows that one of τ (q) 2 − bq 11…”

“…Then we believe that for most n, τ (2) (n) = 0, and we provide some heuristics below. Indeed, it is known that there is a constant c > 0 such that on a set of n of asymptotic density 1, τ (n) is a multiple of all prime powers p a < c 1 log n/ log log n. This was done in Proposition 1 in [11] for the nth coefficient of the modular form associated with an elliptic curve over Q without complex multiplication and the same argument works with the Ramanujan function τ (n). Let p be such that τ (p) = 0 and for each prime q let a q be the minimal positive integer such that τ (q aq ) is divisible by p and the exponent ν p (τ (q aq )) is odd.…”

“…However, for us, a = ±τ (p) and b = τ (p) 2 − 4p 11 . Thus, we get a certain number of equations for p 11 which we check that they have no convenient solution. For example, (a, b) = (1, −7) gives τ (p) = ±1, −7 = τ (p) 2 − 4p 11 = 1 − 4p 11 , so 4p 11 = 8, a contradiction.…”

“…The above bound is valid provided that all primes dividing the discriminant of f A (y) are in S, which is why we incorporated 11 into S. We can take L to be the field Q(A 1/11 , e 2πi/11 ), where A 1/11 is the real 11th root of A . The absolute values of the discriminant of K 1 := Q(A 1/11 ) is at most (11|A |) 11 and of the discriminant of K 2 := Q(e 2πi/11 ) is at most 11 11 , and the degrees of both fields are at most 11. So, the discriminant Δ L of L (which is the compositum of K i , for i = 1, 2) is in absolute value at most…”

“…where α p , β p are the roots of the quadratic polynomial x 2 − τ (p)x + p 11 . Since its discriminant (α p − β p ) 2 = τ (p) 2 − 4p 11 is negative, it follows that α p , β p are complex conjugates.…”