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Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich–Tate groups of elliptic curves. In this paper we prove analogous results for (p-primary) Shafarevich–Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a “Heegner hypothesis”. More precisely, we show that the structure of Shafarevich–Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga–Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the order of these groups. As a second contribution, we prove a result on the structure of (p-primary) Selmer groups of modular forms in the sense of Bloch–Kato. [ABSTRACT FROM AUTHOR]

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus $ 3$ It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with $ 8$, uses the semistable degenerated point configurations in $ (P^1)^8$ $ Y=\overline{\mathcal{B}/\Gamma[1-{\textrm i}]}.$ We use the standard notation $ \bar M_{0,8}$}[rr]& &X\;.} \end{displaymath} --> $\displaystyle \xymatrix{ &\bar M_{0,8}\ar[dl]\ar[dr]&\\ Y\ar@{-->}[rr]& &X\;.}$ The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) $ A,B$ $ X=\mathop{\rm proj}\nolimits(A), Y=\mathop{\rm proj}\nolimits(B).$ [ABSTRACT FROM AUTHOR]