Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

We study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

Twisted Linear Periods and a New Relative Trace Formula

We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formula appearing in the theory of automorphic representations. We verify a fundamental lemma for U2×U2↪U4 via an explicit calculation, giving the first known example of endoscopy for symmetric spaces and showing strong evidence that there is a general theory of endoscopy lurking in this situation. AMS subject classification Primary 20G05

We elaborate an explicit version of the relative trace formula on ${\rm PGL}(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution resul

On the relative trace formula

The exterior square L-function attached to an automorphic cuspidal representation of GL 2n has a pole if and only if a certain period integral does not vanish on the space of the representation.We conjecture, in the "if" direction, a similar result is true for representations of GL 2 ( D), where D is a division algebra.We prove a partial result which provides evidence for the conjecture.The proof is based on a relative trace formula.

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non-degenerate character. This gives a smooth function on the torus. We prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas of this work. However we have to face additional difficulties that appear only in the Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.

For many applications of the trace formula to problems in the theory of automorphic forms one needs to stabilize the trace formula. Such a stabilization has been achieved by Arthur, building on work of Langlands and Kottwitz, subject to conjectures for orbital integrals known as fundamental lemmas. For the group Sp4 the standard invariant fundamental lemma has been established by Hales. However, for the stabilization of the full trace formula one needs to prove a fundamental lemma for weighted orbital integrals on Sp4. We prove this weighted fundamental lemma and thereby make Arthur's stabilization of the Sp4 trace formula unconditional.

A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained.

We continue the analysis of the elliptic part of the trace formula for $GL(2)$ initiated in the earlier paper of the author with the same title. In that paper Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result that the trace of the $p$th Hecke operator on cuspidal automorphic representations is bounded by $p^{\frac14}$.

It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write will, have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the l-adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such pseudo-stabilization include Kottwitz' analysis in [K2] of the zeta functions of certain Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of l-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1]. Conditional results also suffice for important local applications, such as the local Langlands conjecture for GL(n) [HT, He]. The present article develops a technique for obtaining conditional base change and functorial transfer. Let Un be a unitary group over a number field F attached to a quadratic extension E/F. The technique applies to quadratic base change from Un to GL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, if p is an automorphic representation of U which is locally supercuspidal at two places of F split in E, then the expected consequences of the stable trace formula hold for p; in particular p admits a base change to a cuspidal automorphic representation of GL(n)E (Theorem 2.2.2). Slightly more general results are available when F is totally real and E is totally imaginary, and when p is of cohomological type. Automorphic descent from GL(n)E to Un can be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.

We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L 2[0,∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.