— Given an abelian variety over a field of zero characteristic, we give an optimal explicit upper bound depending only on the dimension for the degree of the smallest extension of the base field over which all endomorphisms of the abelian variety are defined. For each dimension, the bound is achieved over the rationals by twisting a power of a CM elliptic curve. This complements a result of Guralnick and Kedlaya giving the exact value of the least common multiple of all these degrees. We also provide a similar statement for the homomorphisms between two distinct abelian varieties. The proof relies on divisibility bounds obtained by Minkowski’s method but, in some cases, we need more precise facts on finite linear groups, including a theorem of Feit whose proof has not been published : we therefore include one based on work by Collins on Jordan’s theorem.

中文翻译：

---

Degré de définition desendomorphismes d'une variété abélienne

— 给定一个零特征域上的阿贝尔变体，我们给出了一个最优的显式上限，仅取决于定义阿贝尔变体的所有自同态的基域的最小扩展度的维数。对于每个维度，通过扭曲 CM 椭圆曲线的幂来实现有理数上的界限。这补充了 Guralnick 和 Kedlaya 给出的所有这些度数的最小公倍数的精确值的结果。我们还为两个不同的阿贝尔变体之间的同态提供了类似的陈述。证明依赖于 Minkowski 方法获得的可分界，但在某些情况下，我们需要关于有限线性群的更精确的事实，包括 Feit 定理，其证明尚未发表：