Fields of definition for homomorphisms of abelian varieties.

*(English)*Zbl 0808.14037From the paper: The results in this paper were motivated by questions which arise naturally from some recent results on ‘adelic representations’ attached to points on Shimura varieties [see, e.g., the author, Am. J. Math. 114, No. 6, 1221–1241 (1992; Zbl 0788.14016)]. If for \(i=1\) and 2, \(Q_ i = (A_ i, C_ i, \theta_ i, t_{i1}, \dots, t_{is})\) are polarized abelian varieties with endomorphism and level structure [see section 1.5 of G. Shimura, Ann. Math. (2) 83, 294–338 (1966; Zbl 0141.37503)], \(\operatorname{Aut} (Q_ i) = 1\), and \(\lambda\) is an isomorphism from \(Q_ 1\) onto \(Q_ 2\), then \(\lambda\) is defined over every field of definition for \(Q_ 1\) and \(Q_ 2\). The main results of the author’s paper cited above lead to the question of whether the previous sentence is true when the word ‘isomorphism’ is replaced by ‘isogeny’. In this paper we show that the answer is no.

However, the answer is yes if for some \(N\geq 3\) and for \(i=1\) and 2 the set \(\{t_{ij}\}^ s_{j = 1}\) includes a basis for the \(N\)-torsion on \(A_ i\). For example, if \(A\) and \(B\) are abelian varieties defined over a field \(F\), of dimensions \(d\) and \(e\), respectively, and \(L\) is the intersection of the fields \(F(A_ N,B_ N)\) for all integers \(N\) prime to the characteristic of \(F\) and greater than 2, then every element of \(\operatorname{Hom} (A,B)\) is defined over \(L\), \(L/F\) is unramified at the discrete places of good reduction for \(A \times B\), and \([L:F]\) divides \(H(d,e)\), where \(H(d,e)\) is a number given by an explicit formula and is less than \(4(9d)^{2d} (9e)^{2e}\).

However, the answer is yes if for some \(N\geq 3\) and for \(i=1\) and 2 the set \(\{t_{ij}\}^ s_{j = 1}\) includes a basis for the \(N\)-torsion on \(A_ i\). For example, if \(A\) and \(B\) are abelian varieties defined over a field \(F\), of dimensions \(d\) and \(e\), respectively, and \(L\) is the intersection of the fields \(F(A_ N,B_ N)\) for all integers \(N\) prime to the characteristic of \(F\) and greater than 2, then every element of \(\operatorname{Hom} (A,B)\) is defined over \(L\), \(L/F\) is unramified at the discrete places of good reduction for \(A \times B\), and \([L:F]\) divides \(H(d,e)\), where \(H(d,e)\) is a number given by an explicit formula and is less than \(4(9d)^{2d} (9e)^{2e}\).

##### MSC:

14K02 | Isogeny |

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

14G35 | Modular and Shimura varieties |

14K05 | Algebraic theory of abelian varieties |

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\textit{A. Silverberg}, J. Pure Appl. Algebra 77, No. 3, 253--262 (1992; Zbl 0808.14037)

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##### References:

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