Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms. (English) Zbl 1481.11062

The authors of the paper under review prove the following theorem, which can be viewed as a local-to-global principle on determining the center of the endomorphism algebra of an abelian variety: Let \(A\) be an abelian variety over a number field \(F\) such that the base extension \(A^{\text{al}}\) to the algebraic closure of \(F\) is isogenous to a power of a simple abelian variety. Let \(L\) be the center of \(B:=\mathrm{End}(A^{\text{al}})\otimes \mathbb Q\), i.e., \(L:=Z(B)\), and let \(m\) be a positive integer such that \(m^2 = \dim_L(B)\). Suppose that the Mumford-Tate conjecture for \(A\) holds. Then, there is a set \(S\) of primes of \(F\) of positive density for which the following properties hold: (a) For each \(\mathfrak p\in S\), \(A\) has good reduction, the reduction \(A_{\mathfrak p}\) is isogenous to the \(m\)th power of a geometrically simple abelian variety over \(\mathbb F_{\mathfrak p}\), the \(\mathbb Q\)-algebra \(M(\mathfrak p) := Z(\mathrm{End}(A_{\mathfrak p}) \otimes \mathbb Q)\) is a field, generated by the \(\mathfrak p\)-Frobenius endomorphism, and there is an embedding \(L \hookrightarrow M(\mathfrak p)\) of number fields; (b) Given any \(\mathfrak q\in S\), for each \(\mathfrak p\in S\) outside of a set \(S_{\mathfrak q}\) of density \(0\), if \(M'\) is a number field that embeds into \(M(\mathfrak q)\) and into \(M(\mathfrak p)\), then \(M'\) embeds into \(L\).
Via the result of the first author et al. [Math. Comput. 88, No. 317, 1303–1339 (2019; Zbl 1484.11135)], the theorem can be used to determine a sharp upper bound on the rank of \(\mathrm{End}(A^{\text{al}})\), conditional on the Mumford-Tate conjecture, and thereby \(\mathrm{End}(A^{\text{al}})\) itself when \(A\) is the Jacobian of a curve. The authors of the paper under review also introduce an alternative algorithm of computing the center of the endomorphism algebra, using the notion of normic polynomials. The algorithm relies on the Mumford-Tate conjecture, but also they introduce an extent of the algorithm that is valid not assuming the conjecture.
The authors also introduce an analogous result for the embedding the splitting field of the Mumford-Tate group of \(A\) into the normal closure of \(M(\mathfrak p)\).


11G10 Abelian varieties of dimension \(> 1\)


Zbl 1484.11135


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