The Visibility Dimension for Elliptic Curves

We now consider the case when $A=E$ is an elliptic curve over a number field $K$. Mazur proved in [Maz99] that every nonzero $c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[3]$ has visibility dimension $2$ (note that Proposition 2.3 only implies that the visibility dimension is $\leq 3$). Mazur's result is particularly nice because it shows that $c$ is visible in an abelian variety that is isogenous to the product of two elliptic curves. Using similar techniques, T. Klenke proved in [Kle01] that every nonzero $c\in H^1(K,E)[2]$ has visibility dimension $2$ (note that Proposition 2.3 only implies that the visibility dimension of any $c\in H^1(K,E)[2]$ is $\leq 4$). It is unknown whether the visibility dimension of every nonzero element of $H^1(K,E)[3]$ is $2$, and it is not known whether elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[5]$ must have visibility dimension $2$.

When $c$ lies in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ we use a classical result of Cassels to strengthen the conclusion of Proposition 2.3.

Proposition 2.4   Let $E$ be an elliptic curve over a number field $K$ and let $c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$. Then the visibility dimension of $c$ is at most the order of $c$.

Proof. Let $n$ be the order of $c$. In view of the restriction of scalars construction in the proof of Proposition 1.3, it suffices to show that there is an extension $L$ of $K$ of degree $n$ such that $\res_L(c)=0$. Without the hypothesis that $c$ lies in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$, such an extension $L$ might not exist, as Cassels observed in [Cas63]. However, in that same paper, Cassels proved that such an $L$ exists when $c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ (see also [O'N01] for another proof). Let tex2html_wrap_inline$X$ be a genus one curve in the torsor class corresponding to tex2html_wrap_inline$c$. The long exact sequence associated to displaymath0&rarr#rightarrow;^0(X_K) &rarr#rightarrow;(X_K) deg Z&rarr#rightarrow;0begins displaymath0&rarr#rightarrow;H^0(K,^0(X_K)) &rarr#rightarrow;H^0(K,(X_K)) deg Z &delta#delta; H^1(K,E) &rarr#rightarrow; &cdots#cdots;,and tex2html_wrap_inline$&delta#delta;(1)=c$ has order tex2html_wrap_inline$n$. Letting tex2html_wrap_inline$(X_K)$ denote the principal divisors on tex2html_wrap_inline$X_K$, we have an exact sequence displaymath0&rarr#rightarrow;(X_K) &rarr#rightarrow;(X_K) &rarr#rightarrow;(X_K)&rarr#rightarrow;0,from which we obtain the exact sequence displaymath(X) &rarr#rightarrow;H^0(K,(X_K)) &rarr#rightarrow;H^1(K,(X_K)).Since tex2html_wrap_inline$(X_K)=K(X)^*/K^*$, Hilbert's theorem 90 produces an injection displaymathH^1(K,(X_K))&rarrhk#hookrightarrow;H^2(K,K^*)=(K),so tex2html_wrap_inline$((X)&rarr#rightarrow;H^0(K,(X_K)))$ is isomorphic to the image of tex2html_wrap_inline$H^0(K,(X_K))$ in tex2html_wrap_inline$(K)$. Because tex2html_wrap_inline$X$ has a point everywhere locally, this image is locally zero; hence, by the local-to-global principle for the Brauer group, this image is globally zero. In other words, every tex2html_wrap_inline$K$-rational divisor class on tex2html_wrap_inline$X$ contains a tex2html_wrap_inline$K$-rational divisor.

We now show that there is a point on tex2html_wrap_inline$X$ defined over an extension of degree at most tex2html_wrap_inline$n$. Since tex2html_wrap_inline$n&isin#in;(&delta#delta;)$, there exists tex2html_wrap_inline$D&isin#in;(X)$ which maps to tex2html_wrap_inline$n &isin#in;Z$ under the degree map. By the Riemann-Roch theorem, there is an effective divisor linearly equivalent to tex2html_wrap_inline$D$. Since this divisor is effective and of degree tex2html_wrap_inline$n$, each point in the support of tex2html_wrap_inline$D$ is defined over an extension tex2html_wrap_inline$L$ of tex2html_wrap_inline$K$of degree at most tex2html_wrap_inline$n$ (alternatively, the residue field of each scheme-theoretic point is of degree at most tex2html_wrap_inline$n$). Thus the index of tex2html_wrap_inline$c$ is at most tex2html_wrap_inline$n$(recall that tex2html_wrap_inline$X(L)&ne#neq;&empty#emptyset;$ if and only if tex2html_wrap_inline$_L(c)=0$). This completes the proof because the order of tex2html_wrap_inline$c$, which is tex2html_wrap_inline$n$, divides the index of tex2html_wrap_inline$c$, which is at most tex2html_wrap_inline$n$. $\qedsymbol$

Remark 2.5   In contrast to the case of dimension $1$, it seems to be an open problem to determine whether or not elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)[n]$ split over an extension of degree $n$.