Sums of Euler products and statistics of elliptic curves

David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan

doi:10.1007/s00208-016-1482-2

Cited by 16 publications

(20 citation statements)

References 40 publications

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“…satisfies the general framework of Theorem 4.2 of [6] and thus, by [6, Formula (6.10)], we have ∆ ℓ ≪ 1/ℓ 3/2 for sufficiently large ℓ. Therefore c t 1 ,t 2 is given by the convergent Euler product in Theorem 1.4.…”

Section: The Law Of Large Numbersmentioning

confidence: 80%

See 1 more Smart Citation

On the Lang–Trotter conjecture for two elliptic curves

Akbary

Parks

2018

Ramanujan J

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Following Lang and Trotter we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over Q. In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the ℓ-adic limits determining the ℓ-factors of the universal constant, unlike the Lang-Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of [11] and [1], following the work of David, Koukoulopoulos, and Smith [6].Here we use the notion → as introduced in [5, Section 2.3]. More precisely, for a sequence (s n ), we set lim m →∞ s m := lim n→∞ s m n with m n := ℓ≤n ℓ n .We note that a similar conjecture has also been proposed for the case that E has complex multiplication (CM) and t 0 (see [2, Conjecture 1]). However, the analysis of the constant is different in the CM and non-CM cases. In this paper, we are only interested in the non-CM case.The constant c E,t can be zero in certain cases. For example for E : y 2 = (x − 1)(x − 2)(x − 3) and p > 2, one can show that a p (E) is even (see [17, p. 420]). Thus, by considering a p (E) as the trace of the Frobenius at p in the division field extension Q(E[m])/Q and applying the Chebotarev density theorem we conclude that |G E (m) t | = 0, for odd t, as m →∞. Therefore, c E,t = 0 for odd t.Lang and Trotter expressed the constant c E,t as a product of a non-negative rational number r E,t , depending on E and t, and a positive universal constant c t , depending only on t. Moreover, in [18, Theorem 4.2], they provide explicit expressions for r E,t and c t . A celebrated theorem of Serre [22] states that, for a non-CM elliptic curve E, the image of ρ E is open in ℓ GL 2 (Z ℓ ). Therefore there exists a positive integer m such that ρ E (Gal(Q/Q)) = φ −1 m (G E (m)). Let m E be the least such m. Then for m = m 1 m 2 with m 1 | m ∞ E (i.e. the prime divisors of m 1 are among the prime divisors of m E ) and (m 2 , m E ) = 1, we obtainUsing this fact, we may then write c E,t = r E,t · c t , whereThe Lang-Trotter Conjecture has been studied extensively in the literature. The best known unconditional upper bound for π E,t (x) for t = 0 is x 3 4 , obtained by Elkies [8] and Ram Murty, and is x(log log x) 2 /(log x) 2 for t 0 (see [20, Theorem 5.1] and [25, Theorem 1.4]). Under GRH, Zywina [26] has recently obtained an upper bound for π E,t (x) of size x 4 5 /(log x) 3 5 for t 0, and an upper bound of size x 3 4 /(log x) 1 2 for t = 0. The Lang-Trotter Conjecture was first shown to hold on average over a family of elliptic curves in the case t = 0 by Fouvry and Ram Murty [10]. This result was then extended to the case of non-zero integers by David and Pappalardi [7].In [18, Remark 2, p. 37] it is mentioned that, by employing a probabilistic model, one can state an analogous conject...

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Section: The Law Of Large Numbersmentioning

confidence: 80%

“…, 6 where the superscript (p) means that the matrices A ∈ GL 2 (Z/ℓ k Z) satisfy the extra condition det(A) = p. Note that, for ℓ p, f ℓ (t, p) is similar to the ℓ-factor of the universal constant c t in (1) with the imposed extra condition det(A) = p. Gekeler has shown that in (8) the limit exists. More precisely,…”

Section: Introductionmentioning

confidence: 99%

On the Lang–Trotter conjecture for two elliptic curves

Akbary

Parks

2018

Ramanujan J

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“…We note that the result of Banks and Shparlinski (and that in Theorem 1.1) gives a true error term only when the size of the interval I is greater that p −1/4+ε . This was improved (on average) by Baier and Zhao in [1] and recently by David et al [4], where the effective version of Birch's theorem is shown to hold for intervals I of length greater than p −1/2+ε , although in these cases the saving is only a power of a logarithm over the main term.…”

Section: Introductionmentioning

confidence: 94%

The Error Term in the Sato–tate Theorem Of birch

Murty¹,

Prabhu²

2019

Bull. Aust. Math. Soc.

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We establish an error term in the Sato–Tate theorem of Birch. That is, for$p$prime,$q=p^{r}$and an elliptic curve$E:y^{2}=x^{3}+ax+b$, we show that$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$for any interval$I\subseteq [0,\unicode[STIX]{x1D70B}]$, where the quantity$\unicode[STIX]{x1D703}_{a,b}$is defined by$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$and$\unicode[STIX]{x1D707}_{ST}(I)$denotes the Sato–Tate measure of the interval$I$.

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“…Framework. As in [6], we begin with an arbitrary framework. For every prime Q suppose we have a function δ(u; Q) such that Here the product is an infinite product over all the monic prime polymials in F q [X].…”

Section: Summing the Primesmentioning

confidence: 99%

“…Recently David, Koukoulopoulos and Smith [6] studied sums of Euler products over rational primes and their applications to statistics of elliptic curves. We will use similar techniques as them and apply it to summing Euler products in the function field setting.…”

Section: Summing the Primesmentioning

confidence: 99%

Expected value of high powers of trace of frobenius of biquadratic curves over a finite field

Meisner¹

2018

Math. Proc. Camb. Phil. Soc.

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Denote Θ C as the Frobenius class of a curve C over the finite field Fq. In this paper we determine the expected value of Tr(Θ n C ) where C runs over all biquadratic curves when q is fixed and g tends to infinity. This extends work done by Rudnick [14] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.

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Sums of Euler products and statistics of elliptic curves

Cited by 16 publications

References 40 publications

On the Lang–Trotter conjecture for two elliptic curves

On the Lang–Trotter conjecture for two elliptic curves

The Error Term in the Sato–tate Theorem Of birch

Expected value of high powers of trace of frobenius of biquadratic curves over a finite field

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