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Discriminant Twins

  • Alyson Deines
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 11)

Abstract

The conductor and minimal discriminant are two invariants that measure the bad reduction of an elliptic curve. The conductor of an elliptic curve E over \(\mathbb {Q}\) is an arithmetic invariant. It is an integer N that measures the ramification in the extensions \(\mathbb {Q}(E[p^{\infty }])/\mathbb {Q}\). The minimal discriminant Δ is a geometric invariant. It counts the number of irreducible components of \(\tilde {E}(\mathbb {F}_p)\). When two elliptic curves have the same conductor and discriminant, we call them discriminant twins. In this paper, we explore when discriminant twins occur. In particular, we prove there are only finitely many semistable isogenous discriminant twins.

Keywords

Elliptic curves Discriminant Conductor p-Adic Uniformization Isogenies 

Notes

Acknowledgements

The author would like to thank Ben Lundell for many helpful discussions and the referee for pointing out a hole in Proposition 6.3, noting simpler Weierstrass equations, and other useful comments.

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Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  • Alyson Deines
    • 1
  1. 1.CCR La JollaSan DiegoUSA

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