Abelian Varieties over $\mathbf{Q}(\sqrt{6})$ with Good Reduction Everywhere

Abstract

The elliptic curve with Weierstrass equation $Y^2 + \sqrt{6}XY - Y = X^3 - (2 + \sqrt{6})X^2$ has good reduction modulo every prime of the ring of integers of $\mathbf{Q}(\sqrt{6})$. We show that every abelian variety over $\mathbf{Q}(\sqrt{6})$ that has good reduction everywhere is isogenous to a power of this elliptic curve.