• Corpus ID: 119319037

Deligne's conjecture for automorphic motives over CM-fields, Part I: factorization

@article{Grobner2018DelignesCF,
  title={Deligne's conjecture for automorphic motives over CM-fields, Part I: factorization},
  author={Harald Grobner and Michael Harris and Jiezhu Lin},
  journal={arXiv: Number Theory},
  year={2018}
}

• H. Grobner, M. Harris, Jiezhu Lin
• Published 8 February 2018
• Mathematics
• arXiv: Number Theory

This is the first of two papers devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. The present paper combines the Ichino-Ikeda-Neal Harris (IINH) formula with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $GL(n)\times GL(m)$ over CM fields. The second paper reinterprets these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups. As a consequence, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are conditional on the IINH formula (which is still partly conjectural), as well as a conjecture on non-vanishing of twists of automorphic $L$-functions of $GL(n)$ by anticyclotomic characters of finite order.