On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$\sum_{k=-n}^n\binom{2n}{n-k}k^{2r}$$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.Comment: 6 page

Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3), {equation} where $x \in \mathbb R$, $t \in \mathbb R$, $u=u(x,t)$ is a real-valued function and $\alpha, \ c_1, \ c_2, \ c_3$ are real constants with $\alpha \neq 0$, is locally well-posed in $H^s(\mathbb R)$ for $s \ge 2$. In the Hamiltonian case (\textit i.e. when $c_1=c_2$), the IVP associated to \eqref{05KdV} is then globally well-posed in the energy space $H^2(\mathbb R)$.Comment: We corrected a few typos and fixed a technical mistake in the proof of Lemma 6.3. We also changed a comment on the work of Guo, Kwak and Kwon on the same subject according to the new version they posted recently on the arXiv (arXiv:1205.0850v2

On the quantitative variation of congruence ideals and integral periods of modular forms

We prove the conjecture of Pollack and Weston on the quantitative analysis of the level lowering congruence \`{a} la Ribet for modular forms of higher weight. It was formulated and studied in the context of the integral Jacquet-Langlands correspondence and anticyclotomic Iwasawa theory for modular forms of weight two and square-free level for the first time. We use a completely different method based on the $R=\mathbb{T}$ theorem proved by Diamond-Flach-Guo and Dimitrov and an explicit comparison of adjoint $L$-values. As applications, we discuss the comparison of various integral canonical periods, the $\mu$-part of the anticyclotomic main conjecture for modular forms, and the primitivity of Kato's Euler systems