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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References
S. Anni, S. Siksek, On Serre’s uniformity conjecture for semistable elliptic curves over totally real fields. arXiv e-prints (2016)
J.E. Cremona, Algorithms for Modular Elliptic Curves (Cambridge University Press, Cambridge, 1997)
J. González, On the division polynomials of elliptic curves. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, 377–381 (2000)
S. Howe, K. Joshi, Elliptic curves of almost-prime conductor. arXiv e-prints (2012)
M.A. Kenku, On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. Number Theory. 15, 199–202 (1982)
A. Kraus, Quelques remarques à propos des invariants c 4, c 6 et Δ d’une courbe elliptique. Acta Arith. 54, 75–80 (1989)
D.S. Kubert, Universal bounds on the torsion of elliptic curves. Compos. Math. 38, 121–128 (1979)
B. Mazur, Rational isogenies of prime degree. Invent. Math. 44, 129–162 (1978)
K.A. Ribet, S. Takahashi, Parametrizations of elliptic curves by Shimura curves and by classical modular curves. Proc. Natl. Acad. Sci. USA 94, 11110–11114 (1997)
J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972)
B. Setzer, Elliptic curves of prime conductor. J. Lond. Math. Soc. (2) 10, 367–378 (1975)
J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics (Springer, New York, 1994)
The LMFDB Collaboration: LMFDB: The L-functions and Modular Forms Database (2017). http://www.lmfdb.org
W.A. Stein et al, The Sage Development Team: Sage Mathematics Software (Version 7.3) (2017). http://www.sagemath.org
J. Vélu, Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris Sér. A-B. 273, A238–A241 (1971)
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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