Weierstrass
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Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography
Announce Type: replace-cross Abstract: In this work we study quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell--Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. We use geometric techniques to show that asymptotically the Jacobians of almost all integral models with two rational points at infinity have rank $r \geq Since there are $\asymp X^{\frac{13}{2}}$ such models among the $X^7$...
Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of H\"older or Sobolev regularity using a Weierstra{\ss} scale
arXiv:2504.20728v2 Announce Type: replace-cross Abstract: We study strong approximation of solutions of SDEs with bounded $\alpha$-H\"older continuous drift coefficient and constant diffusion coefficient at time point $1$. Recently, it was shown in [arXiv:1909.07961v4 (2021)] that for such SDEs the equidistant Euler scheme achieves an $L^p$-error rate of at least $(1+\alpha)/2$, up to an arbitrary small $\varepsilon$, for all $p\geq 1$ and $\alpha\in (0,1]$, in terms of the number of...