We study the local harmonic analysis underlying the first explicit reciprocity law between unitary Friedberg--Jacquet periods and cycles. We obtain a purely local description of the arithmetic level raising of such cycles as a map between spaces of spherical functions on the corresponding spherical varieties. This map admits a simple spectral description in terms of the relative Satake transform of Hironaka--Komori and a new relative Satake transform for a special parahoric subgroup.

Let $X$ be a regular scheme over a henselian discrete valuation ring $\mathcal{O}_K,$ with smooth generic fiber $X_K$. For a cohomologically trivial cycle $Z\in\mathrm{CH}^r(X_K,\Lambda)^0$ the restriction to inertia of its Abel--Jacobi image defines a singular Abel--Jacobi class $\partial\mathrm{AJ}(Z)\in H^1_{\text{sing}}(K,H^{2r-1}_{\text{\'et}}(X_{\bar{K}},\Lambda(r))).$ In the strictly semistable case, Yifeng Liu's theory of \emph{potential maps} affords a description of this class in terms of the geometry of the special fiber. We generalize this theory to regular models whose nearby cycle complex has tame unipotent inertia action. As an example, we analyze the case of products of strictly semistable schemes and simplify previous arguments in the literature by bypassing the need of considering blow-ups to strictly semistable models. (Draft available upon request)

We establish an explicit formula for the Kostka--Foulkes polynomial of type $C_n$ with unequal parameters at fundamental weights. As an application, we deduce an explicit Lusztig--Kato formula for the Hecke algebra of the special (but not hyperspecial) maximal compact subgroup of a quasi-split but non-split special orthogonal group over a $p$-adic field.

Consider a unitary group $G(\mathbb{A}_{F^+})=U_{2r}(\mathbb{A}_{F^+})$ over a CM extension $F/F^+$ with $G(\mathbb{A}_\infty)$ compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G(\mathbb{A}_{F^+}).$ We prove that if $\pi$ is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of $\Pi=\mathrm{BC}(\pi)$ vanishes.

We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\backslash G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{Sp}(4r),\mathrm{Sp}(2r)\times\mathrm{Sp}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $ (\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r)).$ As an application, we completely describe their Hecke module structure.

For all $r\ge1,$ we verify the following conjecture of Hironaka: for a $p$-adic field $F$ with $p$ odd, the space of spherical functions of $\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over the Hecke algebra.

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. We do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture. The main ingredient is an improvement of Howard's Kolyvagin system formalism. As another consequence of it, we establish the equivalence between this main conjecture and the primitivity of the Kolyvagin system in certain cases, by also exploiting a explicit reciprocity law for Heegner points.

We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang's reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang's to extend to modular abelian varieties over totally real fields.