Cancellation in algebraic twisted sums on GL_ m

Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

An exceptional Siegel–Weil formula and poles of the Spin L-function of

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Generating functions on covering groups

In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation $\unicode[STIX]{x1D6E9}_{n}^{(n)}$. This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial $L$ function $L^{S}(s,\unicode[STIX]{x1D70B}^{(n)})$. In fact we prove that this partial $L$ function has at most a simple pole at $s=1$. Here, $\unicode[STIX]{x1D70B}^{(n)}$ is a genuine irreducible cuspidal representation of the group $\text{GL}_{r}^{(n)}(\mathbf{A})$.

Integral moments of automorphic L-functions

Let [Formula: see text] be a self-contragredient irreducible unitary cuspidal representation of [Formula: see text] with [Formula: see text], and [Formula: see text] be the automorphic [Formula: see text]-function attached to [Formula: see text]. Assume that [Formula: see text] is self-contragredient. Under the Generalized Ramanujan Conjecture and Generalized Riemann Hypothesis for [Formula: see text], the estimate [Formula: see text] holds for all real number [Formula: see text] as [Formula: see text].

LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

Zero correlation with lower-order terms for automorphic L-functions

Let [Formula: see text] be a self-contragredient automorphic cuspidal representation of [Formula: see text] for [Formula: see text]. Using a refined version of the Selberg orthogonality, we recompute the [Formula: see text]-level correlation of high non-trivial zeros of the product [Formula: see text]. In the process, we are able to extract certain low-order terms which suggest the asymptotics of these statistics are not necessarily universal, but depend upon the conductors of the representations and hence the ramification properties of the local components coming from each [Formula: see text]. The computation of these lower-order terms is unconditional as long as all [Formula: see text].

Endoscopy and the transfer from GSp(4) to GL(4)

We prove Jiang's conjecture (1996) to show the existence of an L-function of a cuspidal representation of GSp(4,𝔸) × GSp(4,𝔸) which has a pole of order 2 at

FACTORIZATION OF AUTOMORPHIC L-FUNCTIONS AND THEIR ZERO STATISTICS

In this paper we define a Rankin–Selberg L-function attached to two Galois invariant automorphic cuspidal representations of GL m(𝔸E) and GL m′(𝔸F) over cyclic Galois extensions E and F of prime degree. This differs from the classical case in that the two extension fields E and F could be completely unrelated to one another, and we exploit the existence of the automorphic induction functor over cyclic extensions (see [J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, No. 120 (Princeton University Press, Princeton, NJ, 1989)]) to define the L-function. Using a result proved by C. S. Rajan, we prove a prime number theorem for this L-function, and proceed to calculate the n-level correlation function of high nontrivial zeros of a product L(s, π1)L(s, π2)…L(s, πk) where πi is a Galois invariant cuspidal representation of GL ni(𝔸Fi) with Fi a cyclic Galois extension of prime degree ℓi for i = 1,…,k, thus generalizing the results of Liu and Ye [Functoriality of automorphic L-functions through their zeros, Sci. China Ser. A51(1) (2008) 1–16].

Two Linnik-type problems for automorphic L-functions

Let m ≥ 2 be an integer, and π an irreducible unitary cuspidal representation for GLm(), whose attached automorphic L-function is denoted by L(s, π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s, π) in the half-plane ℜs > 1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then the first sign change in the sequence {λπ(n)}n=1∞ occurs at some n ≪ Qπ1 + ϵ, where Qπ is the conductor of π, and the implied constant depends only on m and ϵ. This improves the previous bound with the above exponent 1 + ϵ replaced by m/2 + ϵ. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −(L′/L)(s, π) in the half-plane ℜs > 1.