# Restriction of scalars with simple endomorphism algebra

## Main Article Content

## Abstract

Suppose $L/K$ be a finite abelian extension of number fields of odd degree and suppose an abelian variety $A$ defined over $L$ is a $K$-variety. If the endomorphism algebra of $A/L$ is a field $F$, the followings are equivalent :

(1) The enodomorphiam algebra of the restriction of scalars from $L$ to $K$ is simple.

(2) There is no proper subfield of $L$ containing $L^{G_F}$ on which $A$ has a $K$-variety descent.

## Article Details

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

## References

[1] B. H. Gross, Arithmetic on Elliptic Curves with Complex Multiplications, Lecture Notes in Math. 776, Springer, 1980. Google Scholar

[2] E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284 (1989) 307–327. Google Scholar

[3] T. Nakamura, On abelian varieties associated with elliptic curves with complex multiplications, Acta Arith. 97 (2001), no. 4, 379–385. Google Scholar

[4] A. Weil, Adeles and algebraic groups, Progr. Math. 23 (1982). Google Scholar

[5] H. Yu, Idempotent relations and the conjecture of Birch and Swinnerton-Dyer, Math. Ann. 327 (2003) 67–78. Google Scholar