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On the Lang-Trotter conjecture for two elliptic curves
Univ Lethbridge, Dept Math & Comp Sci, 4401 Univ Dr, Lethbridge, AB T1K 3M4, Canada..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Univ Lethbridge, Dept Math & Comp Sci, 4401 Univ Dr, Lethbridge, AB T1K 3M4, Canada.
2019 (English)In: Ramanujan Journal, ISSN 1382-4090, Vol. 49, no 3, p. 585-623Article in journal (Refereed) Published
Abstract [en]

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 Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239-268, 2004), following the work of David et al. (Math Ann 368(1-2):685-752, 2017).

Place, publisher, year, edition, pages
Springer, 2019. Vol. 49, no 3, p. 585-623
Keywords [en]
Frobenius distributions, Lang-Trotter conjecture for two elliptic curves, Lang-Trotter constant for two elliptic curves, Hurwitz-Kronecker class number
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-255554DOI: 10.1007/s11139-018-0050-7ISI: 000475734300010Scopus ID: 2-s2.0-85053495291OAI: oai:DiVA.org:kth-255554DiVA, id: diva2:1341064
Note

QC 20190807

Available from: 2019-08-07 Created: 2019-08-07 Last updated: 2019-08-07Bibliographically approved

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Parks, James

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  • apa
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