ALMOST ALL ELLIPTIC CURVES ARE SERRE CURVES

By N. Jones. 2006

89 citations recorded by the source corpus.

Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/b9083c95a6a0010e64e16e168da50c59dda19129

How researchers see this paper

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In Algebra and Number Theory

1. Bjorn Poonen
2. Eric M. Rains
3. David Zywina
4. Peter Stevenhagen
5. V. Spiridonov
6. Alexei Borodin
7. Qing Liu
8. Alice Fialowski

In Geometry and Topology

1. Bjorn Poonen
2. Eric M. Rains
3. David Zywina
4. Peter Stevenhagen
5. Manjul Bhargava
6. Alexei Borodin
7. A. Shankar
8. V. Spiridonov

In Information Systems

1. Bjorn Poonen
2. Eric M. Rains
3. David Zywina
4. Manjul Bhargava

Citations in the corpus

• Character sums for primitive root densities (2011, background): “Nathan Jones [5] recently proved that, in an asymptotic sense that is easily made precise, almost all elliptic curves over Q are Serre curves.”
• Random maximal isotropic subspaces and Selmer groups (2010, methodology): “o heuristics for Xand rank The Hilbert irreducibility theorem shows that asymptotically 100% of elliptic curves (ordered by naïve height) have E(Q)[p] = 0. (For much stronger results, see [Duk97] and [Jon10].) So for statistical purposes, when letting E run over all elliptic curves, we may ignore co”
• Hilbert's irreducibility theorem and the larger sieve (2010, background): “∈Q, one has ρEt,ℓ(Gal(Q/Q)) = GL2(Z/ℓZ) for all ℓ≥17. This will be reproved when we generalize to higher dimensions and number fields and is essentially Theorem 1.15. Building on Duke’s theorem, Jones [Jon10] was able to show that [GL2(Zb) : ρE(Gal(Q/Q))] = 2 for “most” elliptic curve Eover Q(such cu”