#### Vol. 312, No. 1, 2021

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Relating the Tate–Shafarevich group of an elliptic curve with the class group

### Dipendra Prasad and Sudhanshu Shekhar

Vol. 312 (2021), No. 1, 203–218
##### Abstract

The paper formulates a precise relationship between the Tate–Shafarevich group of an elliptic curve $E$ over $ℚ$ with a quotient of the class group of $ℚ\left(E\left[p\right]\right)$ on which $Gal\left(ℚ\left(E\left[p\right]\right)∕ℚ\right)={GL}_{2}\left(ℤ∕p\right)$ operates by its standard 2-dimensional representation over $ℤ∕p$. We establish such a relationship in many cases.

##### Keywords
class group, Tate–Shafarevich group, Selmer group, elliptic curve, Galois cohomology
Primary: 11G05