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Parks, James
Publications (3 of 3) Show all publications
Akbary, A. & Parks, J. (2019). On the Lang-Trotter conjecture for two elliptic curves. Ramanujan Journal, 49(3), 585-623
Open this publication in new window or tab >>On the Lang-Trotter conjecture for two elliptic curves
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
Keywords
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:nbn:se:kth:diva-255554 (URN)10.1007/s11139-018-0050-7 (DOI)000475734300010 ()2-s2.0-85053495291 (Scopus ID)
Note

QC 20190807

Available from: 2019-08-07 Created: 2019-08-07 Last updated: 2019-08-07Bibliographically approved
Fiorilli, D., Parks, J. & Södergren, A. (2018). LOW-LYING ZEROS OF QUADRATIC DIRICHLET L-FUNCTIONS: A TRANSITION IN THE RATIOS CONJECTURE. Quarterly Journal of Mathematics, 69(4), 1129-1149
Open this publication in new window or tab >>LOW-LYING ZEROS OF QUADRATIC DIRICHLET L-FUNCTIONS: A TRANSITION IN THE RATIOS CONJECTURE
2018 (English)In: Quarterly Journal of Mathematics, ISSN 0033-5606, E-ISSN 1464-3847, Vol. 69, no 4, p. 1129-1149Article in journal (Refereed) Published
Abstract [en]

We study the 1-level density of low-lying zeros of quadratic Dirichlet L-functions by applying the L-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower-order terms when the support of the Fourier transform of the corresponding test function reaches the point 1. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.

Place, publisher, year, edition, pages
OXFORD UNIV PRESS, 2018
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-243969 (URN)10.1093/qmath/hay018 (DOI)000456696100002 ()
Note

QC 20190301

Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-03-01Bibliographically approved
Fiorilli, D., Parks, J. & Sodergren, A. (2017). Low-lying zeros of quadratic Dirichlet L-functions: lower order terms for extended support. Compositio Mathematica, 153(6), 1196-1216
Open this publication in new window or tab >>Low-lying zeros of quadratic Dirichlet L-functions: lower order terms for extended support
2017 (English)In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 153, no 6, p. 1196-1216Article in journal (Refereed) Published
Abstract [en]

We study the 1-level density of low-lying zeros of Dirichlet L-functions attached to real primitive characters of conductor at most X. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of log X, which is valid when the support of the Fourier transform of the corresponding even test function phi is contained in (-2, 2). We uncover a phase transition when the supremum sigma of the support of (phi) over cap reaches 1, both in the main term and in the lower order terms. A new lower order term appearing at sigma = 1 involves the quantity (phi) over cap (1), and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Place, publisher, year, edition, pages
Cambridge University Press, 2017
Keywords
zeros of L-functions, Katz-Sarnak heuristics, quadratic Dirichlet L-functions, 1-level density
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-211030 (URN)10.1112/S0010437X17007059 (DOI)000403416600003 ()
Funder
EU, FP7, Seventh Framework Programme, DFF-1325-00058
Note

QC 20170712

Available from: 2017-07-12 Created: 2017-07-12 Last updated: 2017-07-12Bibliographically approved
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