Fields of definition of building blocks

We investigate the fields of definition up to isogeny of the abelian varieties known as building blocks. These varieties are defined as the $\mathbb{Q}$-varieties admitting real or quaternionic multiplications of the maximal possible degree allowed by their dimensions (cf. Pyle (2004)). The Shimura-Taniyama conjecture predicts that every such variety is isogenous to a non-CM simple factor of a modular Jacobian $J_1(N)$.

The obstruction to descend the field of definition of a building block up to isogeny is given by Ribet in 1994 as an element in a Galois cohomology group. In this paper we begin by studying these elements from an abstract Galois-cohomological point of view, and obtain results and formulas for the computation of invariants related to them. When considered for the element attached to a building block, these invariants give the structure of its endomorphism algebra, and also complete information on the possible fields of definition up to isogeny of this building block.

We implemented these computations in Magma for building blocks given as $\overline{\mathbb{Q}}$-simple factors up to isogeny of the Jacobian of the modular curve $X_1(N)$. Using this implementation we computed a table for conductors $N\leq500$, which is described in the last section. This table is a source of examples of building blocks with different behaviors and of statistical information; in particular, it contains many examples that answer a question posed by Ribet in 1994 on the existence of a smallest field of definition up to isogeny for RM-building blocks of even dimension.