A family of semistable elliptic curves with large Tate-Shafarevitch groups

Kramer, Kenneth

doi:10.1090/S0002-9939-1983-0715850-1

A family of semistable elliptic curves with large Tate-Shafarevitch groups
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Proc. Amer. Math. Soc. 89 (1983), 379-386 Request permission

Abstract:

We present a family of elliptic curves defined over the rationals ${\mathbf {Q}}$ such that each curve admits only good or multiplicative reduction and for every integer $n$ there is a curve whose Tate-Shafarevitch group over ${\mathbf {Q}}$ has more than $n$ elements of order 2. Previously known examples of large Tate-Shafarevitch groups were constructed by forcing many places of additive reduction.

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