The Smallest Invariant Factor of Elliptic Curves, and Coincidences
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917431193108480 |
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| author | Milner, Alexander Shotton, Jack |
| author_facet | Milner, Alexander Shotton, Jack |
| contents | For an elliptic curve E over Q and a natural number j, Cojocaru has shown that there is an explicit constant C_E,j giving (under GRH) the density of primes p of good reduction such that the smallest invariant factor of E(F_p) is j. For E without complex multiplication, we study the question of when C_E,j is positive (a necessary and, on GRH, sufficient condition for there to be infinitely many such p), strengthening a result by Kim. Our arguments are group-theoretic using the image of the adelic Galois representation of E. Experimentally, C_E,j appears to vanish only when there is a coincidence of division fields; we document a number of families of such coincidences arising from abelian division fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Smallest Invariant Factor of Elliptic Curves, and Coincidences Milner, Alexander Shotton, Jack Number Theory 11G05 (Primary), 11F80, 11G25, 11A41, 11N05 For an elliptic curve E over Q and a natural number j, Cojocaru has shown that there is an explicit constant C_E,j giving (under GRH) the density of primes p of good reduction such that the smallest invariant factor of E(F_p) is j. For E without complex multiplication, we study the question of when C_E,j is positive (a necessary and, on GRH, sufficient condition for there to be infinitely many such p), strengthening a result by Kim. Our arguments are group-theoretic using the image of the adelic Galois representation of E. Experimentally, C_E,j appears to vanish only when there is a coincidence of division fields; we document a number of families of such coincidences arising from abelian division fields. |
| title | The Smallest Invariant Factor of Elliptic Curves, and Coincidences |
| topic | Number Theory 11G05 (Primary), 11F80, 11G25, 11A41, 11N05 |
| url | https://arxiv.org/abs/2604.21601 |