arXiv subjects

Song Wang

Publications and source records attributed to Song Wang.

471 records · Page 27Linked to original sources

arXiv2003

A cuspidality criterion for the functorial product on GL(2) x GL(3), with a cohomological application

This paper was motivated by a question of Avner Ash, asking if it is possible to construct non-selfdual, non-monomial, cuspidal cohomology classes for suitable congruence subgroups Γof SL(n,\Z). Such a construction, in special examples, has been known for some time for n=3; it is of course impossible for n=2. We show in this paper the existence of many such examples for n=6, which are primitive, by making use of the functorial product on GL(2) x GL(3), which was recently shown to be automorphic by Kim and Shahidi. We establish a general cuspidality criterion for this product, which is essential to the construction. We also show that there exist non-selfdual, monomial (cuspidal) classes for any n=2m > 3, and non-selfdual, non-monomial (but imprimitive) classes for n=4.

math.NT↗

arXiv2003

On the symmetric powers of cusp forms on $GL (2)$ of icosahedral type

In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this paper, which may be of independent interest, for the Kim--Shahidi automorphic tensor product $π\boxtimes {\rm sym}^{2} (π')$, where $π$ and $π'$ are cuspidal automorphic representations of $GL (2) / F$. We also give a complete structure theory of modular icosahedral representations. As a result, we prove that $L (s, {\rm sym}^{m} (ρ) \otimes χ)$ does not admit any Landau--Siegel zero when it is not divisible by $L$--functions of quadratic characters. In general, there is no such divisibility and and there are no Landau--Siegel zeros for such $L$--functions.

math.NT↗

arXiv2001

On the exceptional zeros of Rankin-Selberg L-functions

The main objects of study in this article are two classes of Rankin-Selberg L-unctions, namely L(s, f \times g) and L(s, sym^2(g) \times sym^2(g)), where f, g are newforms, holomorphic or of Maass type, on the upper half plane, and sym^2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any Landau-Siegel zeros. These zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym^2(g).

math.NT↗