-
On the arithmetic of Shalika models and the critical values of L-functions for GL2n
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 3, June 2014
- pp. 675-728
- 10.1353/ajm.2014.0021
- Article
- Additional Information
Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\Bbb{A})$ over a totally
real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika
model of $\Pi_f$. Comparing it with a rational structure on a realization of $\Pi_f$ in cuspidal cohomology in
top-degree, we define certain periods $\omega^{\epsilon}(\Pi_f)$. We describe the behavior of such top-degree
periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result
for all the critical values of the standard $L$-functions $L(s,\Pi\otimes\chi)$; here we use the recent work
of B. Sun on the non-vanishing of a certain quantity attached to $\Pi_\infty$. As applications, we obtain
algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to
holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly,
for certain (self-dual of symplectic type) Rankin-Selberg $L$-functions for ${\rm GL}_3\times{\rm GL}_2$; assuming
Langlands Functoriality, this generalizes to certain Rankin-Selberg $L$-functions of ${\rm GL}_n\times{\rm GL}_{n-1}$.
Thirdly, for the degree four $L$-functions attached to Siegel modular forms of genus $2$ and full level.
Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our
main theorem is compatible with conjectures of Deligne and Gross.