On Periods and $L$-functions for $\mathbf{GL}_4 \times \mathbf{GL}_2$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918508615434240 |
|---|---|
| author | Cauchi, Antonio Terradillos, Armando Gutierrez |
| author_facet | Cauchi, Antonio Terradillos, Armando Gutierrez |
| contents | We give a new integral representation of the $\wedge^2 \otimes \mathrm{std}_2$ $L$-function of generic cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$. In the former case, we use it to prove a relation between its central $L$-value and the generalized Shalika period. Exploiting the theta correspondence for $(\mathbf{GL}_4,\mathbf{GL}_4)$, we further establish a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $(\mathbf{GL}_4 \times \mathbf{GL}_2,\mathbf{GL}_2 \times \mathbf{GL}_2)$ and its corresponding period. In the case of cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ that are unramified everywhere, our formulas give new evidence towards conjectures of Wan-Zhang and of Gan-Gross-Prasad for $\mathbf{GSpin}_6 \times \mathbf{GSpin}_3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14586 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Periods and $L$-functions for $\mathbf{GL}_4 \times \mathbf{GL}_2$ Cauchi, Antonio Terradillos, Armando Gutierrez Number Theory Representation Theory We give a new integral representation of the $\wedge^2 \otimes \mathrm{std}_2$ $L$-function of generic cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$. In the former case, we use it to prove a relation between its central $L$-value and the generalized Shalika period. Exploiting the theta correspondence for $(\mathbf{GL}_4,\mathbf{GL}_4)$, we further establish a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $(\mathbf{GL}_4 \times \mathbf{GL}_2,\mathbf{GL}_2 \times \mathbf{GL}_2)$ and its corresponding period. In the case of cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ that are unramified everywhere, our formulas give new evidence towards conjectures of Wan-Zhang and of Gan-Gross-Prasad for $\mathbf{GSpin}_6 \times \mathbf{GSpin}_3$. |
| title | On Periods and $L$-functions for $\mathbf{GL}_4 \times \mathbf{GL}_2$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2602.14586 |