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Parks, James

Open this publication in new window or tab >>On the Lang-Trotter conjecture for two elliptic curves### Akbary, Amir

### Parks, James

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Ramanujan Journal, ISSN 1382-4090, Vol. 49, no 3, p. 585-623Article in journal (Refereed) Published
##### Abstract [en]

##### 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)
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##### Note

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.

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).

QC 20190807

Available from: 2019-08-07 Created: 2019-08-07 Last updated: 2019-08-07Bibliographically approvedOpen this publication in new window or tab >>LOW-LYING ZEROS OF QUADRATIC DIRICHLET L-FUNCTIONS: A TRANSITION IN THE RATIOS CONJECTURE### Fiorilli, Daniel

### Parks, James

### Södergren, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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##### Note

Univ Ottawa, Dept Math & Stat, 585 King Edward, Ottawa, ON K1N 6N5, Canada..

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden.;Univ Gothenburg, SE-41296 Gothenburg, Sweden..

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.

QC 20190301

Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-03-01Bibliographically approvedOpen this publication in new window or tab >>Low-lying zeros of quadratic Dirichlet L-functions: lower order terms for extended support### Fiorilli, Daniel

### Parks, James

### Sodergren, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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##### Funder

EU, FP7, Seventh Framework Programme, DFF-1325-00058
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Univ Lethbridge, Canada.

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.

QC 20170712

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