# On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives

@article{Liu2022OnTB,
title={On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives},
author={Yifeng Liu and Yichao Tian and Liang Xiao and Wei Zhang and Xinwen Zhu},
journal={Inventiones mathematicae},
year={2022}
}
• Published 26 December 2019
• Mathematics
• Inventiones mathematicae
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group…

### Arithmetic level raising for certain quaternionic unitary Shimura variety

. In this article we prove an arithmetic level raising theorem for the symplectic group of degree four in the ramiﬁed case. This result is a key step towards the Beilinson–Bloch–Kato conjecture for

### Chow groups and L-derivatives of automorphic motives for unitary groups, II.

• Mathematics
Forum of Mathematics, Pi
• 2022
Abstract In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical

### INDIVISIBILITY OF HEEGNER CYCLES OVER SHIMURA CURVES AND SELMER GROUPS

• Haining Wang
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2022
In this article, we show that the Abel–Jacobi images of the Heegner cycles over the Shimura curves constructed by Nekovar, Besser and the theta elements contructed by Chida–Hsieh form a bipartite

### Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are

### Arithmetic level raising on triple product of Shimura curves and Gross-Schoen diagonal cycles II: Bipartite Euler system

In this article, we study the Gross-Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction. We prove the unramified arithmetic level raising theorem for triple

### Chow groups and $L$-derivatives of automorphic motives for unitary groups

• Mathematics
Annals of Mathematics
• 2021
In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is

### Arithmetic level raising on triple product of Shimura curves and Gross-Schoen Diagonal cycles I: Ramified case

In this article we study the Gross-Schoen diagonal cycle on a triple product of Shimura curves at a place of bad reduction. We relate the image of the diagonal cycle under the Abel-Jacobi map to

### Isolation of the cuspidal spectrum: The function field case

• Mathematics
Science China Mathematics
• 2021
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercuspidal representations

### Level raising and Diagonal cycles on triple product of Shimura curves: Ramified case

In this article we study the diagonal cycle on a triple product of Shimura curves at places of bad reduction. We relate the image of the diagonal cycle under the Abel-Jacobi map to certain period

### Unitary moduli schemes: smooth case

We will start by discussing a ‘baby version’ of section 4 of [2], and hope that this will provide some intuition for what in the paper. Let E/Q be a quadratic imaginary field, let V/E be an hermitian

## References

SHOWING 1-10 OF 72 REFERENCES

### Weight-monodromy conjecture for p-adically uniformized varieties

The aim of this paper is to prove the weight-monodromy conjecture (Deligne’s conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces

### Hirzebruch–Zagier cycles and twisted triple product Selmer groups

Let E be an elliptic curve over $${{\mathbb {Q}}}$$Q and A another elliptic curve over a real quadratic number field. We construct a $${{\mathbb {Q}}}$$Q-motive of rank 8, together with a

### The Sato-Tate conjecture for Hilbert modular forms

• Mathematics
• 2009
We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations

### L-Functions and Tamagawa Numbers of Motives

• Mathematics
• 2007
The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture

### On the p-adic cohomology of the Lubin-Tate tower

We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F)

### Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

• Mathematics
Algebra & Number Theory
• 2020
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally

### Le lemme fondamental pour les groupes unitaires

• Mathematics
• 2004
Let G be an unramified reductive group over a non archimedian local field F. The so-called "Langlands Fundamental Lemma" is a family of conjectural identities between orbital integrals for G(F) and

### Whittaker rational structures and special values of the Asai $L$-function

• Mathematics
• 2014
Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of \$GL_n(\mathbb

### Automorphy for some l-adic lifts of automorphic mod l Galois representations

• Mathematics
• 2008
We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate

### Special cycles on unitary Shimura varieties II: global theory

• Mathematics
• 2009
We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and