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.
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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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Deines, A. (2018). Discriminant Twins. In: Bouw, I., Ozman, E., Johnson-Leung, J., Newton, R. (eds) Women in Numbers Europe II. Association for Women in Mathematics Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-74998-3_6
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DOI: https://doi.org/10.1007/978-3-319-74998-3_6
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