The Lang–Trotter conjecture for products of non-CM elliptic curves

Chen, Hao; Jones, Nathan; Serban, Vlad

doi:10.1007/s11139-021-00543-3

Cited by 4 publications

(5 citation statements)

References 24 publications

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“…This modification relies on a variation of [27, Lemma 5.1, p. 246] applied to 𝐸 𝑗 defined over 𝐾 by counting only degree one primes. We obtain the upper bound 𝜅 1 (𝐸 𝑗 , 𝐾) 𝑥(log log 𝑥) 2 (log 𝑥) 2 for some positive constant 𝜅 1 (𝐸 𝑗 , 𝐾), which depends on 𝐸 𝑗 and 𝐾. Putting everything together gives part (i) of Theorem 1.…”

Section: Elliptic Curves With Shared Frobenius Fieldsmentioning

confidence: 88%

“…Remark 5. Theorem 3 may be viewed under the general Lang-Trotter theme of results about the number of primes for which the Frobenius trace of an abelian variety is fixed, such as those proven in [2,5,6,16,17,19,22,23], and [27]. The connection between Theorem 3, and thus Theorem (𝑥) for 𝐸 1 , 𝐸 2 , 𝛼 1 , and 𝛼 2 as in the setting of Theorem 3.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative upper bounds related to an isogeny criterion for elliptic curves

Cojocaru,

Hinz,

Wang

2024

Bulletin of London Math Soc

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For and elliptic curves defined over a number field , without complex multiplication, we consider the function counting nonzero prime ideals of the ring of integers of , of good reduction for and , of norm at most , and for which the Frobenius fields and are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that and are not potentially isogenous if and only if , we investigate the growth in of . We prove that if and are not potentially isogenous, then there exist positive constants , , and such that the following bounds hold: (i) ; (ii) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) under GRH, Artin's Holomorphy Conjecture for the Artin ‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin ‐functions of number field extensions.

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Section: Elliptic Curves With Shared Frobenius Fieldsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Quantitative upper bounds related to an isogeny criterion for elliptic curves

Cojocaru,

Hinz,

Wang

2024

Bulletin of London Math Soc

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“…This description may then be generalized to prime distribution problems related to compatible systems of Galois representations, such as those associated to modular forms and those associated to higher dimensional abelian varieties. For example, such generalizations were proposed in [AkPa19], [BaGo97], [ChJoSe20], [CoDaSiSt17], [Ka09], and [Mu99]. The goal of our paper is to prove results related to the generalization of the Lang-Trotter Conjecture on Frobenius traces formulated in [CoDaSiSt17] in the setting of generic abelian varieties defined over Q, as explained below.…”

Section: Introductionmentioning

confidence: 98%

“…A prominent open problem in arithmetic geometry, formulated by Lang and Trotter in the 1970s [LaTr76, Part I], concerns the distribution of the Frobenius traces associated to the reductions modulo primes of an elliptic curve defined over Q and without complex multiplication. In recent years, this problem has been formulated in broader settings, such as that of abelian varieties (e.g., [CoDaSiSt17,ChJoSe20,Ka09]). The goal of the present article is to provide upper bounds related to the distribution of the Frobenius traces defined by the product of non-isogenous elliptic curves defined over Q and having no complex multiplication, as explained below.…”

Section: Introductionmentioning

confidence: 99%

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

Cojocaru

2022

Can. J. Math.-J. Can. Math.

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Let g ≥ 1 be an integer and let A/Q be an abelian variety that is isogenous over Q to a product of g elliptic curves defined over Q, pairwise non-isogenous over Q and each without complex multiplication. For an integer t and a positive real number x, denote by π A (x, t) the number of primes p ≤ x, of good reduction for A, for which the Frobenius trace a 1,p (A) associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove thatThese bounds largely improve upon recent ones obtained for g = 2 by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for g = 1 by M.R. Murty, V.K. Murty, and N. Saradha, combined with a refinement in the power of log x by D. Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying |a 1,p (A)| > p 1 3g+1 −ε for any fixed ε > 0.the Frobenius traces defined by the product of non-isogenous elliptic curves defined over Q and having no complex multiplication, as explained below.Let g ≥ 1 be an integer and let A/Q be an abelian variety that is isogenous over Q to a product of g elliptic curves defined over Q, pairwise non-isogenous over Q and each without complex multiplication.

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“…A prominent open problem in arithmetic geometry, formulated by S. Lang and H. Trotter in the 1970s ([LaTr76, Part I]), concerns the distribution of the Frobenius traces associated to the reductions modulo primes of an elliptic curve defined over Q and without complex multiplication. In recent years, this problem has been formulated in broader settings, such as that of abelian varieties (e.g., [CoDaSiSt17], [ChJoSe20],…”

Section: Introductionmentioning

confidence: 99%

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

Cojocaru,

Wang

2022

Preprint

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Let g ≥ 1 be an integer and let A/Q be an abelian variety that is isogenous over Q to a product of g elliptic curves defined over Q, pairwise non-isogenous over Q and each without complex multiplication. For an integer t and a positive real number x, denote by π A (x, t) the number of primes p ≤ x, of good reduction for A, for which the Frobenius trace a 1,p (A) associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that1− 2 3g+2 if t = 0. These bounds largely improve upon recent ones obtained for g = 2 by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for g = 1 by M.R. Murty, V.K. Murty, and N. Saradha, combined with a refinement in the power of log x by D. Zywina. Under the same assumptions, we also prove the existence of a density one set of primes p satisfying |a 1,p (A)| > p 1 3g+1 −ε for any fixed ε > 0.

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The Lang–Trotter conjecture for products of non-CM elliptic curves

Cited by 4 publications

References 24 publications

Quantitative upper bounds related to an isogeny criterion for elliptic curves

Quantitative upper bounds related to an isogeny criterion for elliptic curves

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

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