Primes in the Chebotarev density theorem for all number fields (with an Appendix by Andrew Fiori)

Kadiri, Habiba; Wong, Peng‐Jie

doi:10.1016/j.jnt.2022.03.012

Cited by 4 publications

(2 citation statements)

References 34 publications

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“…The proof of Theorem 1.3 involves an explicit Chebotarev bound due to Bach and Sorenson [BS96] that is dependent on GRH. An unconditional version of Theorem 1.3 can be given using an unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential in q. In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a version of Theorem 1.3 holds with the improved asymptotic bound B ≫ q 11 log 2 (qN A ), but without an explicit constant.…”

Section: Introductionmentioning

confidence: 99%

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

S.¹,

Brumer²,

Kim³

et al. 2023

Preprint

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For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes for which the Galois action on -torsion points of A is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of PSp 4 (F ), sampling of the characteristic polynomials of Frobenius, and the Khare-Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve on a publicly-accessible branch of the LMFDB.

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Section: Introductionmentioning

confidence: 99%

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

S.¹,

Brumer²,

Kim³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Ahn and Kwon is also not crucial; the existence of such a constant is enough for our purposes. A recent preprint of Kadiri and Wong [9] improves the constant to 310.…”

mentioning

confidence: 99%

An elliptic curve analogue of Pillai’s lower bound on primitive roots

Jin¹,

Washington²

2021

Can. Math. Bull.

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Let /Q be an elliptic curve. For a prime of good reduction, let ( , ) be the smallest non-negative integer that gives the -coordinate of a point of maximal order in the group (F ). We prove unconditionally that ( , ) > 0.72 log log for infinitely many , and ( , ) > 0.36 log under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime. IntroductionLet /F be an elliptic curve. Recall that there exist unique positive integers , such that (F ) Z/ Z × Z/ Z and | . Here is the maximal order of a point of (F ). In order to find a point on /F of maximal order, a natural strategy is to compute the orders of points with -coordinates 0, 1, 2, . . . and continue until the desired point is found. In practice, this works fairly well. A natural question is how long this process takes in the worst case.Along these lines, fix an elliptic curve /Q and let be a prime of good reduction. Let ( , ) denote the minimal -coordinate of a point of maximal order in the reduction /F . The goal of this note is to prove the following two lower bounds on ( , ). Theorem 1.1 Let /Q be an elliptic curve. There are infinitely many primes such that ( , ) > 0.72 log log . Theorem 1.2 Let /Q be an elliptic curve. Under GRH, there are infinitely many primes such that ( , ) > 0.36 log .These results can be viewed as elliptic curve analogues of lower bounds on the least primitive root ( ) of a prime . Pillai [14] proved that there is a positive constant such that ( ) > log log for infinitely many . Using Linnik's theorem in Pillai's proof, Fridlender [6] and Salié [15] improved the result to the following. We include a proof since it inspired the result of the present paper.

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Computing nonsurjective primes associated to Galois representations of genus 2 curves

Banwait,

Brumer,

Kim

et al. 2024

LuCaNT: LMFDB, Computation, and Number Theory

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For a genus 2 2 curve C C over Q \mathbb {Q} whose Jacobian A A admits only trivial geometric endomorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely many primes ℓ \ell for which the Galois action on ℓ \ell -torsion points of A A is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell’s classification of maximal subgroups of P S p 4 ( F ℓ ) PSp_4(\mathbb {F}_\ell ) , sampling of the characteristic polynomials of Frobenius, and the Khare–Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus 2 2 curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.

show abstract

Primes in the Chebotarev density theorem for all number fields (with an Appendix by Andrew Fiori)

Cited by 4 publications

References 34 publications

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

An elliptic curve analogue of Pillai’s lower bound on primitive roots

Computing nonsurjective primes associated to Galois representations of genus 2 curves

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