# Research projects

# 5. First explicit reciprocity law for unitary Friedberg-Jacquet periods

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.

# 4. Spherical functions on symmetric spaces of Friedberg-Jacquet type

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.

arXiv →# 3. Spherical functions of symmetric forms and a conjecture of Hironaka

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.

arXiv →# 2. On Howard's main conjecture and the Heegner point Kolyvagin system

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.

arXiv →# 1. A proof of Kolyvagin's conjecture via the BDP main conjecture

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.

arXiv →