On the arithmetic of Shalika models and the critical values of L-functions for GL2n

By H. Grobner, A. Raghuram, W. Gan. 2011

62 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17e8a09e856b10feca465109d9ddfb7c5f2daa7d

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In Discrete Mathematics and Combinatorics

1. Abhishek Saha
2. Ameya Pitale
3. Ralf Schmidt
4. Michael Harris
5. H. Grobner

In Geometry and Topology

1. Abhishek Saha
2. Ameya Pitale
3. H. Grobner
4. Ralf Schmidt
5. Michael Harris
6. T. Gee
7. W. Gan
8. Peter Scholze

In Mathematical Physics

1. Abhishek Saha
2. Ameya Pitale
3. H. Grobner
4. Ralf Schmidt
5. Michael Harris
6. T. Gee
7. W. Gan
8. Peter Scholze

Citations in the corpus

• Transfer of Siegel Cusp Forms of Degree 2 (2011, methodology): “Critical value result for GSp4 ×GL1 In [23], Grobner and Raghuram define certain periods of automorphic forms on GL2n by comparing cohomologies in top degree.”
• WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT $L$ -FUNCTIONS (2013, background): “Not., Vol. 2010, No. 2, (2009) pp. 334 – 372 [34] A. Raghuram, F. Shahidi, On certain period relations for cusp forms on GLn, Int.”