Automorphic Representations Of GL Research Articles

A Product of Tensor Product L-functions of Quasi-split Classical Groups of Hermitian Type

A family of global zeta integrals representing a product of tensor product (partial) L-functions: $$L^S(s, \pi \times \tau_1)L^S(s,\pi \times \tau_2)\cdots L^S(s, \pi \times \tau_r)$$ is established in this paper, where π is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $${\tau_1,\ldots,\tau_r}$$ are irreducible unitary cuspidal automorphic representations of $${{\rm GL}_{a_1},\ldots,{\rm GL}_{a_r}}$$ , respectively. When r = 1 and the classical group is an orthogonal group, this family was studied by Ginzburg et al. (Mem Am Math Soc 128: viii+218, 1997). When π is generic and $${\tau_1,\ldots,\tau_r}$$ are not isomorphic to each other, such a product of tensor product (partial) L-functions is considered by Ginzburg et al. (The descent map from automorphic representations of GL(n) to classical groups, World Scientific, Singapore, 2011) in with different kind of global zeta integrals. In this paper, we prove that the global integrals are eulerian and finish the explicit calculation of unramified local zeta integrals in a certain case (see Section 4 for detail), which is enough to represent the product of unramified tensor product local L-functions. The remaining local and global theory for this family of global integrals will be considered in our future work.

The Theta Period of a Cuspidal Automorphic Representation of GL(n)

The Theta Period of a Cuspidal Automorphic Representation of GL(n)

ALGEBRAIC CYCLES AND TATE CLASSES ON HILBERT MODULAR VARIETIES

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL 2(𝔸E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M ét (π∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M ét (π∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M ét (π∞) is spanned by algebraic cycles.

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

Let E/F be a quadratic extension of number fields and D a quaternion algebra over F containing E. Let πD be a cuspidal automorphic representation of GL(n,D) and π its Jacquet–Langlands transfer to GL(2n). Guo and Jacquet conjectured that if πD is distinguished by GL(n,E), then π is symplectic and L(1/2,πE)≠0, where πE is the base change of π to E. When n is odd, Guo and Jacquet also conjectured a converse. The converse does not always hold when n is even, but we conjecture it holds if and only if certain local root number conditions are satisfied, which is if and only if the corresponding generic representation of the split special orthogonal group SO(2n+1) has a special E-Bessel model. We use the theta correspondence to relate E-Bessel periods on SO(5) with GL(2,E)-periods on GL(2,D), and deduce part of our conjecture when n=2.

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

Abstract 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 s = 1, even for globally generic representations. Moreover, we show that if a cuspidal representation π of GSp(4,𝔸) comes from GSO(4,𝔸), then π is the Weil transfer of Π1 ⊗ Π2 realized as a representation of GSO(4,𝔸) where the Π i are irreducible automorphic cuspidal representations of GL(2,𝔸). This agrees with the Langlands functoriality principle as GSO(4,𝔸) is an endoscopic group for GSp(4,𝔸).

Siegel modular forms of degree two attached to Hilbert modular forms

Let E / Q be a real quadratic field and π 0 a cuspidal, irreducible, automorphic representation of GL ( 2 , A E ) with trivial central character and infinity type ( 2 , 2 n + 2 ) for some non-negative integer n . We show that there exists a non-zero Siegel paramodular newform F : H 2 → C with weight, level, Hecke eigenvalues, epsilon factor and L -function determined explicitly by π 0 . We tabulate these invariants in terms of those of π 0 for every prime p of Q .

Selberg’s integral for Ramanujan automorphic representation

Let π Δ be the automorphic representation of GL(2,ℚA) associated with Ramanujan modular form Δ and L(s, π Δ) the global L-function attached to π Δ. We study Selberg’s integral for the automorphic L-function L(s, π Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.

Integral representation for L-functions for [formula omitted

Let π be a cuspidal, automorphic representation of GSp 4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [M. Furusawa, On L-functions for GSp ( 4 ) × GL ( 2 ) and their special values, J. Reine Angew. Math. 438 (1993) 187–218] to obtain an integral representation for the degree-8 L-function L ( s , π × τ ) , where τ runs through certain cuspidal, automorphic representation of GL 2 . Our calculations include the case of any representation with unramified central character for the p-adic components of τ, and a wide class of archimedean types including Maaß forms. As an application we obtain a special value result for L ( s , π × τ ) .

Corps de nombres peu ramifiés et formes automorphes autoduales

Let S S be a finite set of primes, p p in S S , and Q S \mathbb {Q}_S a maximal algebraic extension of Q \mathbb {Q} unramified outside S S and ∞ \infty . Assume that | S | ≥ 2 |S|\geq 2 . We show that the natural maps \[ Gal ⁡ ( Q p ¯ / Q p ) → Gal ⁡ ( Q S / Q ) \operatorname {Gal}(\overline {\mathbb {Q}_p}/\mathbb {Q}_p) \rightarrow \operatorname {Gal}(\mathbb {Q}_S/\mathbb {Q}) \] are injective. Much of the paper is devoted to the problem of constructing self-dual automorphic cuspidal representations of GL ⁡ ( 2 n , A Q ) \operatorname {GL}(2n,\mathbb {A}_{\mathbb {Q}}) with prescribed properties at all places, which we study via Arthur’s twisted trace formula. The techniques we develop also shed some light on the orthogonal/symplectic alternative for self-dual representations of GL ⁡ ( 2 n ) \operatorname {GL}(2n) .

On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of GL n( A K) . Let S be a set of finite places of K, such that the sum ∑ v∈S Nv −2/(n 2+1) is convergent. Then π is uniquely determined by the collection of the local components {π v|v∉S, v finite} of π. Combining this theorem with base change, it is possible to consider sets S of positive density, having appropriate splitting behavior with respect to a solvable extension L of K, and where π is determined up to twisting by a character of the Galois group of L over K.

Uniform approximate functional equation for principal L-functions

We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field with unitary central character. We investigate the decay rate of the terms involved using the analytic conductor of Iwaniec and Sarnak as a guideline. Straightforward extensions of the results exist for products of central values. We hope that these formulae will help further understanding of the central values of principal L-functions, such as finding good bounds on their various power means, or establishing subconvexity or nonvanishing results in certain families. A crucial role in the proofs is played by recent progress on the Ramanujan--Selberg conjectures achieved by Luo, Rudnick and Sarnak. The bounds at the non-Archimedean places enter through the work of Molteni.

Period relations and p-adic measures

Let (π,σ) be a pair of cuspidal automorphic representations of GL n × GL n −1 over the adele ring of ℚ having non-vanishing cohomology with constant coefficients. The p-adic distribution interpolating the critical values of the twisted corresponding Rankin–Selberg convolution is shown to be p-adically bounded thus leading to an associated p-adic L-function.

Distinguished representations and quadratic base change for 𝐺𝐿(3)

Let E / F E/F be a quadratic extension of number fields. Suppose that every real place of F F splits in E E and let H H be the unitary group in 3 variables. Suppose that Π \Pi is an automorphic cuspidal representation of G L ( 3 , E A ) GL(3,E_{\mathbb {A}}) . We prove that there is a form ϕ \phi in the space of Π \Pi such that the integral of ϕ \phi over H ( F ) ∖ H ( F A ) H(F)\setminus H(F_{\mathbb {A}}) is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

The symmetric-squareL-function attached to a cuspidal automorphic representation ofGL 3

The symmetric-squareL-function attached to a cuspidal automorphic representation ofGL 3

P-adic measures attached to automorphic representations ofGL(3)

P-adic measures attached to automorphic representations ofGL(3)

On the metaplectic analog of Kazhdan’s “endoscopic” lifting

A simple “twisted” form of the Selberg trace formula, due to Kazhdan, is used to prove a metaplectic analog of Kazhdan’s lifting of grossencharakters of a cyclic extension of degreer to automorphic representations of GL(r).

Strong multiplicity theorems for 𝐺𝐿(𝑛)

Let π = ⊗ π υ \pi = \otimes {\pi _\upsilon } be a cuspidal automorphic representation of G L ( n , F A ) GL(n,{F_A}) , where F A {F_A} denotes the adeles of a number field F F . Let E E be a Galois extension of F F and let { g } \{ g\} denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components π υ {\pi _\upsilon } whenever the Frobenius of the prime υ \upsilon is { g } \{ g\} , showing that such representations are often easily described and finite in number. This generalizes a result of Moreno [Bull. Amer. Math. Soc. 11 (1984), pp. 180-182].