The conjecture

The two examples computed in Section 6 show that for an abelian subvariety $A$ of $J_0(N)$ an invisible element of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ at the base level $N$ might become visible at a multiple level $NM$ . We state a general conjecture according to which any element of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ should have such a property.

theorem_type[conjecture][theorem][][plain][][] Let tex2html_wrap_inline$h = 0$ or tex2html_wrap_inline$1$. Suppose that tex2html_wrap_inline$A$ is an abelian subvariety of tex2html_wrap_inline$J_h(N)$ and tex2html_wrap_inline$c&isin#in;OT2wncyrmnSh(Q, A)$. Then there is a positive integer tex2html_wrap_inline$M$, and a homomorphism of abelian varieties tex2html_wrap_inline$&iota#iota;:A &rarr#to;J_h(NM)$ of finite degree coprime to the order of tex2html_wrap_inline$c$ such that tex2html_wrap_inline$&iota#iota;_*c=0$.

Conjecture 7.1.1   Let $h = 0$ or $1$ . Suppose $A$ is a $J_h$ -modular abelian variety and $c \in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q},A)$ . Then there is a $J_h$ -modular abelian variety $C$ and an inclusion $\iota : A \to C$ such that $\iota_* c = 0$ .

Remark 7.1.2   For any prime $\ell$ , the Jacobian $J_h(N)$ comes equipped with two morphisms $\alpha^*, \beta^* : J_h(N) \rightarrow J_h(N \ell)$ induced by the two degeneracy maps $\alpha, \beta : X_h(\ell N) \rightarrow X_h(N)$ between the modular curves of levels $\ell N$ and $N$ , and it is natural to consider visibility of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ in $J_h(N\ell)$ via morphisms $\iota$ constructed from these degeneracy maps.

Remark 7.1.3   It would be interesting to understand the set of all levels $N$ of $J_h$ -modular abelian varieties $C$ that satisfy the conclusion of the conjecture.