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 $π$ of $G(\mathbb{A}_{F^+}).$ We prove that if $π$ is distinguished by the unitary Friedberg–Jacquet period, then the Bloch–Kato Selmer group (with coefficients in a favorable field) of the motive of $Π=\mathrm{BC}(π)$ vanishes.

💡 Research Summary

The paper studies the Beilinson–Bloch–Kato conjecture for motives attached to cuspidal automorphic representations of a compact-at‑infinity unitary group U₂ᵣ over a CM extension F/F⁺. The authors focus on representations π that are distinguished by a unitary Friedberg–Jacquet period, i.e. the period integral over the subgroup H ≅ Uᵣ × Uᵣ. The main result (Theorem 1.1.2) asserts that, when F⁺ = ℚ and π has weight (0,…,0), the Bloch–Kato Selmer group H¹_f(F, ρ_{Π,λ}(r)) vanishes for every “admissible” prime λ of the coefficient field E attached to the base‑change Π = BC(π). The admissibility conditions (L1–L6) are technical but are expected to hold for all but finitely many primes provided Π is not a lift from a smaller group.

The proof proceeds via a novel “first explicit reciprocity law” linking the Friedberg–Jacquet period to a cohomology class coming from a special cycle on a Shimura variety. The overall strategy follows the level‑raising/Euler‑system paradigm pioneered by Bertolini–Darmon for Heegner points, but several new obstacles arise in the unitary Friedberg–Jacquet setting:

1. Multiplicity of the spherical Hecke module. For r > 1 the space of H‑invariant functions on G has rank 2r − 1 over the Hecke algebra, unlike the rank‑one situation in earlier bipartite Euler‑system constructions. Consequently, the naive special cycle (coming from an embedding U_{r‑1,1} × U_{r,0} ↪ U_{2r‑1,1}) does not map directly to the period under the arithmetic level‑raising map. The authors resolve this by considering a larger family of Hecke translates of the naive cycle, thereby generating the required Hecke‑multiple of the basic spherical function.

2. Non‑equidimensional integral models. The natural integral model of the special cycle is not equidimensional, making reduction modulo p and intersection‑number calculations difficult. To overcome this, the authors introduce derived integral models L Z, which are objects in K‑theory rather than classical cycles. These derived models retain enough information to compute the arithmetic level‑raising map while allowing the use of K‑theoretic tools.

The paper is organized as follows:

• Section 2 extends Liu’s weight‑spectral‑sequence machinery to accommodate derived integral models, defining a “potential map” in K‑theory.
• Section 3 computes intersection numbers in unitary Rapopport–Zink spaces, providing the local contribution needed for the level‑raising map.
• Section 4 constructs integral models of (RSZ) unitary Shimura varieties and defines derived special cycles, studying their generic fibers and p‑adic uniformizations.
• Section 5 carries out a detailed local harmonic analysis. By analyzing spherical functions on the double coset spaces H\G and (U₁×U₂)\U, the authors identify a Cartan‑type decomposition and an “inverse Satake transform” (in the sense of Sakellaridis) that yields an explicit isomorphism φₚ between a parameter algebra and the space of compactly supported H‑invariant functions. This isomorphism shows that the image of the derived cycle under the arithmetic level‑raising map contains a suitable Hecke translate of the basic spherical function.
• Section 6 assembles the previous ingredients to prove the first explicit reciprocity law (Theorem 6.1.4). The law states that the Friedberg–Jacquet period P_H is, up to a known Hecke factor, the image of the derived special cycle under the arithmetic level‑raising map.
• Section 7 uses the reciprocity law to build an Euler system for the Galois representation ρ_{Π,λ}. By a standard Euler‑system argument (including admissibility of primes, big‑image hypotheses, and an R = T theorem from the authors’ earlier work), the authors deduce the vanishing of the Bloch–Kato Selmer group for all admissible λ, establishing Theorem 1.1.2.

The paper also discusses broader implications. The first reciprocity law is one of two ingredients needed for an Iwasawa main conjecture in the unitary Friedberg–Jacquet setting, analogous to the work of Bertolini–Darmon for Heegner points. The authors indicate that a forthcoming “second reciprocity law” will complete the picture.

Overall, the work represents a significant advance in the arithmetic of unitary groups, extending the reach of the Beilinson–Bloch–Kato conjecture to a new class of motives and providing new tools—derived integral models and a refined local harmonic analysis—that are likely to be useful in other contexts involving bipartite Euler systems.