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Register a new userA Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)
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For a cuspidal automorphic representation Pi of GL(4,A), H. Kim proved that the exterior square transfer wedge^2Pi is an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Pi for which wedge^2Pi is cuspidal.
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We establish Langlands functoriality for the generic spectrum of GSp(4) and describe its transfer on GL(4). We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].
Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburgs global integral for the adjoint representation of $mathrm{GL}_3$. Via the local functional equation, the local gamma factor $Gamma(s,pi,mathrm{Ad},psi)$ can be defined. In a forthcoming paper, we will compute the local gamma factor $Gamma(s,pi,mathrm{Ad},psi)$ explicitly, which fills in some blanks in the archimedean local theory of Ginzburgs global integral.
If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a question of Byott and Elder, we first prove that $VC(L/K)$ holds if and only if the tamely ramified part of the extension $L/K$ is trivial and every non-zero $K[G]$-submodule of $L$ contains a unit. Moreover, the integer $d$ can take one value modulo $[L:K]$ only, namely $-d_{L/K}-1$, where $d_{L/K}$ is the valuation of the different of $L/K$. When $K$ has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that $VC(L/K)$ is valid for all extensions $L/K$ in this context. When $char{;K}=0$, we identify all abelian extensions $L/K$ for which $VC(L/K)$ is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.
Let $p$ be a prime number and $K$ a finite extension of $mathbb{Q}_p$. We state conjectures on the smooth representations of $mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular, when $K$ is unramified, we conjecture that they are of finite length and predict their internal structure (extensions, form of subquotients) from the structure of a certain algebraic representation of $mathrm{GL}_n$. When $n=2$ and $K$ is unramified, we prove several cases of our conjectures, including new finite length results.
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