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}
}
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… 

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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

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

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

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

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

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

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

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
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