arXiv subjects
Publications and source records attributed to Lazar Radicevic.
In this paper we count, ordered by naive height, the genus $2$ curves over the rationals which admit a monic Weierstrass model of odd degree and whose Jacobian has a marked rational $3$-torsion point.
In this paper we study the arithmetic and invariant theory of genus one normal curves embedded in $\mathbb{P}^{n-1}$. We generalize the notion of genus one model of degree $n$, introduced by Cremona, Fisher and Stoll for $n \leq 5$, to arbitrary odd $n$, and describe the invariant theory of a genus one curve of degree $n$ embedded in $ \mathbb{P}^{n-1}$ in terms of the minimal graded free resolution of its homogeneous ideal. We prove that everywhere locally soluble genus one curves over $ \mathbb{Q}$ admit minimal integral models, with the same invariants as those of the minimal model of their Jacobian elliptic curve. We then apply these results to study the capitulation problem for the Tate-Shafarevich group of an elliptic curve $E/\mathbb{Q}$. We prove that every element of $\text{Sha}(E/\mathbb{Q})[n]$ of odd index $n$ splits over a degree $n$ number field $K$, of absolute discriminant at most $c(n) H_E^{2n-2}$, where $H_E$ is the naive height of $E$ and $c(n)$ is a constant only depending on $n$.
We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of $(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of $E(L)$. Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of the p-torsion part of the Tate-Shafarevich group of $E$, for $p \leq 11$, and hence are counterexamples to the Hasse principle.