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(This is inspired by http://mathoverflow.net/questions/34110/algebraic-geometry-examples.)

I want to collect here (counter)examples in arithmetic geometry.

1. Curves violating the Hasse principle: The Selmer curwecurve $3X^3 + 4Y^3 + 5Z^3 = 0$. It is a nontrivial element of the Tate-Shafarevich Tate–Shafarevich group of the elliptic curve $3\cdot4\cdot5\cdot X^3 + Y^3 + Z^3 = 0$. It is also an example of an abelian variety for which finiteness of Sha is known. In fact, $|\mathrm{III}(E/\mathbf{Q})| = 3^2$.

2. Non-isogeneous elliptic curves having the same Hasse-Weil Hasse–Weil $L$-series: $y^2 = x^3 \pm ix + 3$ over $K = \mathbf{Q}(i)$ (cf. Cornell-Silverman-StevensCornell–Silverman–Stevens, p. 32).

3. Counterexample to the Hasse norm theorem for non-cyclic extensions: $L = \mathbf{Q}(\sqrt{13},\sqrt{17})$ Galois with $G =\mathbf{Z}/2 \times \mathbf{Z}/2$, see Cassels-FrölichCassels–Fröhlich, p. 360, Exercise 5.3.

tbc

3 added 206 characters in body

(This is inspired by http://mathoverflow.net/questions/34110/algebraic-geometry-examples.)

I want to collect here (counter)examples in arithmetic geometry.

1. Curves violating the Hasse principle: The Selmer curwe $3X^3 + 4Y^3 + 5Z^3 = 0$. It is a nontrivial element of the Tate-Shafarevich group of the elliptic curve $3\cdot4\cdot5\cdot X^3 + Y^3 + Z^3 = 0$. It is also an example of an abelian variety for which finiteness of Sha is known. In fact, $|\mathrm{III}(E/\mathbf{Q})| = 3^2$.

2. Non-isogeneous elliptic curves having the same Hasse-Weil $L$-series: $y^2 = x^3 \pm ix + 3$ over $K = \mathbf{Q}(i)$ (cf. Cornell-Silverman-Stevens, p. 32).

3. Counterexample to the Hasse norm theorem for non-cyclic extensions: $L = \mathbf{Q}(\sqrt{13},\sqrt{17})$ Galois with $G =\mathbf{Z}/2 \times \mathbf{Z}/2$, see Cassels-Frölich, p. 360, Exercise 5.3.

tbc