×

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 \(M(\mathfrak p)\).

MSC:

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

Citations:

Zbl 1484.11135

Software:

endomorphisms
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Borel, A., Linear Algebraic Groups. Graduate Texts in Mathematics (1991), New York: Springer, New York · Zbl 0726.20030 · doi:10.1007/978-1-4612-0941-6
[2] Cohen, H., Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics (2000), New York: Springer, New York · Zbl 0977.11056 · doi:10.1007/978-1-4419-8489-0
[3] Costa, E.; Mascot, N.; Sijsling, J.; Voight, J., Rigorous computation of the endomorphism ring of a Jacobian, Math. Comput., 88, 317, 1303-1339 (2019) · Zbl 1484.11135 · doi:10.1090/mcom/3373
[4] Deligne, P., Milne, J.S., Ogus, A., Shih, K.: Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900. Springer, Berlin (1982) · Zbl 0465.00010
[5] Jouve, F.; Kowalski, E.; Zywina, D., Splitting fields of characteristic polynomials of random elements in arithmetic groups, Israel J. Math., 193, 1, 263-307 (2013) · Zbl 1332.11098 · doi:10.1007/s11856-012-0117-x
[6] Kedlaya, KS, Quantum computation of zeta functions of curves, Comput. Complex., 15, 1, 1-19 (2006) · Zbl 1118.14062 · doi:10.1007/s00037-006-0204-7
[7] Klüners, J., On polynomial decompositions, J. Symbolic Comput., 27, 3, 261-269 (1999) · Zbl 0967.12002 · doi:10.1006/jsco.1998.0252
[8] Lombardo, D., Computing the geometric endomorphism ring of a genus-2 Jacobian, Math. Comput., 88, 316, 889-929 (2019) · Zbl 1410.11043 · doi:10.1090/mcom/3358
[9] Milne, J.S.: Abelian varieties (v2.00) (2008). www.jmilne.org/math/ · Zbl 0604.14028
[10] Noot, R., Classe de conjugaison du Frobenius d’une variété abélienne sur un corps de nombres, J. Lond. Math. Soc., 79, 1, 53-71 (2009) · Zbl 1177.14084 · doi:10.1112/jlms/jdn049
[11] Serre, J-P.: Oeuvres/collected papers. IV. 1985-1998. Springer Collected Works in Mathematics. Springer, Heidelberg (2013). Reprint of the 2000 edition
[12] Szutkoski, J., van Hoeij. M.: The complexity of computing all subfields of an algebraic number field. preprint (2017). arXiv:1606.01140 · Zbl 1419.11143
[13] van Geemen, B.: Kuga-Satake varieties and the Hodge conjecture. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), vol. 548 of NATO Sci. Ser. C Math. Phys. Sci., pp. 51-82. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0987.14008
[14] van Hoeij, M.; Klüners, J.; Novocin, A., Generating subfields, J. Symbolic Comput., 52, 17-34 (2013) · Zbl 1278.11111 · doi:10.1016/j.jsc.2012.05.010
[15] Zywina, D.: Determining monodromy groups of abelian varieties. preprint (2020). arXiv:2009.07441v1
[16] Zywina, D., The splitting of reductions of an abelian variety, Int. Math. Res. Notices, 2014, 18, 5042-5083 (2014) · Zbl 1318.14040 · doi:10.1093/imrn/rnt113
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.