Twisted Linear Periods and a New Relative Trace Formula

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.

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.

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.

Local Orbital Integrals of a Relative Trace Formula

Following the work of Harris and Kudla, we prove a general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain L-function. As a consequence, we deduce a theorem relating the existence of GL2 (K) -invariant linear forms on irreducible, admissible representations of GL2() for a commutative semi-simple cubic algebra over a non-archimedean local field k in terms of local epsilon factors which was proved only in some cases by the first author in his earlier work in [D. Prasad, Invariant forms for representations of GL2 over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363.]. This has been achieved by globalising a locally distinguished supercuspidal representation to a globally distinguished representation, a result of independent interest which is proved by an application of the relative trace formula.

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idele class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ⊗ η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula. We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.

The workshop on Automorphic Forms: New Directions provided a nice glimpse of the many streams of current research activity in this very active area. Topics included the relative trace formula and periods of automorphic forms, Arthur packets and locally/globally generic representations, Eisenstein cohomology, special values of L-functions, algebraic modular forms, p-adic modular forms, arithmetic theta functions, endoscopy and CAP representations, and proofs of the Gross-Prasad conjecture and the local Langlands conjecture for GL(n). The group of participants was notably broad in terms of nationality and age and the meeting confirmed the continued vigor of research in the field of automorphic representations.

Energy level statistics of quantized chaotic systems have been evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller and related trace formulae. Here we evaluate a spectral statistic of chaotic 4-regular quantum graphs from their periodic orbits without the semiclassical limit. The variance of the n-th coefficient of the characteristic polynomial is determined by the sizes of the sets of distinct primitive periodic orbits with n bonds which have no self-intersections, and the sizes of the sets with a given number of self-intersections which all consist of two sections of the pseudo orbit crossing at a single vertex. Using this result we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.