Galois structure on integral valued polynomials

We characterize finite Galois extensions $K$ of the field of rational numbers in terms of the rings ${\rm Int}_{\mathbb{Q}}(\mathcal O_K)$, recently introduced by Loper and Werner, consisting of those polynomials which have coefficients in $\mathbb{Q}$ and such that $f(\mathcal O_K)$ is contained in $\mathcal O_K$. We also address the problem of constructing a basis for ${\rm Int}_{\mathbb{Q}}(\mathcal O_K)$ as a $\mathbb{Z}$-module.Comment: final version, accepted for publication in J. Number Theory (2016). any comment is welcom

Variation of anticyclotomic Iwasawa invariants in Hida families

Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.Comment: Essentially final version, to appear in Algebra & Number Theor