Torsion on elliptic curves in isogeny classes

Fujita, Yasutsugu; Nakamura, Tetsuo

doi:10.1090/S0002-9947-07-04212-2

Torsion on elliptic curves in isogeny classes
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by Yasutsugu Fujita and Tetsuo Nakamura PDF

Trans. Amer. Math. Soc. 359 (2007), 5505-5515 Request permission

Abstract:

Let $E$ be an elliptic curve over a number field $K$ and $\mathcal C$ its $K$-isogeny class. We are interested in determining the orders and the types of torsion groups $E(K)_{\textrm {tors}}$ in $\mathcal C$. For a prime $l$, we give the range of possible types of $l$-primary parts $E(K)_{(l)}$ of $E(K)_{\textrm {tors}}$ when $E$ runs over $\mathcal C$. One of our results immediately gives a simple proof of a theorem of Katz on the order $\sup _{E \in \mathcal C}|E(K)_{(l)}|$ of maximal $l$-primary torsion in $\mathcal C$.

References

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