Relative Trace Formula: Theory & Applications

• Relative trace formula is a collection of identities equating integrated automorphic kernels with spectral expansions via subgroup periods and orbital integrals.
• It decomposes into a geometric side, featuring orbital integrals and truncation techniques, and a spectral side capturing periods, relative characters, and scattering data.
• Applications span analytic number theory, scattering theory, and operator approaches, providing tools for bounding L-functions and establishing period identities.

Relative trace formula denotes a family of identities in which an automorphic or operator-valued kernel is integrated against subgroup periods, characters, or boundary data and then expanded in two different ways. In the automorphic setting, a recurrent starting point is an automorphic kernel

$K_f(x,y)=\sum_{\gamma\in G(F)}f(x^{-1}\gamma y),$

followed by integration over quotients attached to subgroups $H_1$ and $H_2$; in scattering-theoretic variants, one compares functional calculi or resolvents of operators relative to simpler reference configurations. The resulting formula relates a geometric side, built from relative orbital integrals or boundary-layer determinants, to a spectral side, built from periods, relative characters, or scattering data (Getz et al., 2014, Blomer, 2019, Strohmaier et al., 2021).

1. General form and basic objects

In a broad automorphic formulation, one fixes a reductive group $G$ over a number field $F$, a connected subgroup $H\subset G\times G$, and a quasi-character $\chi:H(\mathbb A_F)\to\mathbb C^\times$. The subgroup $H$ acts on $G$ by

$(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$

and a relative class is an element of $H_1$0. For $H_1$1, the stabilizer is

$H_1$2

and the relative orbital integral is

$H_1$3

On the spectral side, if $H_1$4 is a cuspidal automorphic representation, one forms a relative trace $H_1$5 by integrating the kernel $H_1$6 against $H_1$7. Under the hypotheses that $H_1$8 has cuspidal image and that the relevant classes intersecting $H_1$9 are elliptic, unimodular, and closed, one has

$H_2$0

This is the “general simple relative trace formula” proved by Getz–Hahn (Getz et al., 2014).

Compact-quotient models make the same structure especially transparent. For $H_2$1 compact, $H_2$2 unimodular, and $H_2$3, one defines

$H_2$4

The spectral decomposition of $H_2$5 then yields a sum over irreducible $H_2$6, while the geometric side becomes a sum over double cosets $H_2$7 with relative orbital integrals $H_2$8 (Delorme et al., 2018).

A common misconception is that “the” relative trace formula is a single fixed identity. The sources instead exhibit a framework whose kernel, subgroup data, truncation, and normalization vary substantially with the period problem under study, from Whittaker and toric periods to symmetric-space, twisted, and operator-theoretic settings (Getz et al., 2014, Getz et al., 2017).

2. Geometric side: double cosets, truncation, germs, and orbital integrals

On the geometric side, the kernel is grouped by relative orbits, typically double cosets such as $H_2$9 or their twisted analogues. For relatively regular semisimple elements $G$0, one encounters integrals of the form

$G$1

with $G$2 as the geometric coefficient (Getz et al., 2017).

In the Jacquet–Rallis setting on

$G$3

the geometric expansion is

$G$4

For test functions that are regular-supported at one place, the regularized geometric terms reduce to naive regular orbital integrals: $G$5 Locally, the regular orbital integral $G$6 on the general linear side can be compared with semisimple orbital integrals $G$7 on unitary groups, providing a singular transfer theorem beyond the regular semisimple range (Lu, 2024).

Local relative trace formulas often require Arthur-style truncation and a refined analysis of weighted orbital integrals. In the Ginzburg–Rallis model, the limit of truncated distributions $G$8 is expressed as

$G$9

where $F$0 is a Shalika-germ coefficient of a local quasi-character $F$1 and $F$2 is a transfer factor (Wan, 2016). For $F$3-adic symmetric spaces $F$4, the geometric side likewise arises from a Weyl integration formula adapted to the symmetric space and from explicit truncation weights $F$5 attached to $F$6-orthogonal sets (Delorme et al., 2015).

This structure shows that the geometric side is not merely a sum over “regular semisimple classes.” Depending on the model, it may also involve irregular orbital integrals, germ expansions, weighted orbital integrals, and truncation polynomials (Lu, 2024, Delorme et al., 2015).

3. Spectral side: periods, relative characters, and full-spectrum expansions

The spectral side arises by inserting the spectral decomposition of the kernel. In the most schematic form, if $F$7 is an orthonormal basis and $F$8, then one obtains

$F$9

with $H\subset G\times G$0 determined by the test function and the representation parameter (Blomer, 2019).

Different period problems produce different spectral distributions. In the toric and Kuznetsov formulas for $H\subset G\times G$1, one defines toric relative characters $H\subset G\times G$2 from toric periods and Whittaker relative characters $H\subset G\times G$3 from Whittaker periods. The global toric RTF and the Kuznetsov trace formula with non-standard test functions both extend to measures on the Satake spectrum, together with point-evaluations at automorphic characters (Sakellaridis, 2014).

For $H\subset G\times G$4, the spectral side is explicitly a weighted average of Rankin–Selberg $H\subset G\times G$5-functions over the full spectrum. With $H\subset G\times G$6 and $H\subset G\times G$7 fixed on $H\subset G\times G$8, one has

$H\subset G\times G$9

where the local zeta integrals represent $\chi:H(\mathbb A_F)\to\mathbb C^\times$0 (Yang, 2023).

Local relative trace formulas may exhibit a different analytic profile. For $\chi:H(\mathbb A_F)\to\mathbb C^\times$1 relative to $\chi:H(\mathbb A_F)\to\mathbb C^\times$2, the spectral side is given in terms of discrete series, regularized normalized periods $\chi:H(\mathbb A_F)\to\mathbb C^\times$3, and normalized $\chi:H(\mathbb A_F)\to\mathbb C^\times$4-functions of Harish-Chandra, integrated over unitary characters $\chi:H(\mathbb A_F)\to\mathbb C^\times$5 and the unit circle $\chi:H(\mathbb A_F)\to\mathbb C^\times$6 (Delorme et al., 2016).

A common misconception is that the spectral side is always purely cuspidal. Several constructions explicitly include continuous or Eisenstein contributions, and some formulas are designed to keep them visible rather than eliminate them by truncation (Yang, 2023, Delorme et al., 2016).

4. Comparison, transfer, and nonstandard matching

A central use of relative trace formulas is comparison between two different period problems. The local input is a matching of test functions and orbital integrals; the global output is an identity between relative trace distributions and, ultimately, between periods or $\chi:H(\mathbb A_F)\to\mathbb C^\times$7-values.

In the $\chi:H(\mathbb A_F)\to\mathbb C^\times$8 “beyond endoscopy” setting, the matching is explicitly nonstandard. For the toric quotient $\chi:H(\mathbb A_F)\to\mathbb C^\times$9 and the Whittaker quotient $H$0, one defines an involutive integral transform

$H$1

and proves that $H$2 induces a topological isomorphism

$H$3

together with a fundamental lemma for Hecke translates of basic vectors (Sakellaridis, 2012). Globally, the transfer operator $H$4 yields

$H$5

which gives a new proof of Waldspurger’s formula for toric periods (Sakellaridis, 2014).

The twisted linear-period formula on $H$6 introduces local transfer factors $H$7 so that for matching regular semisimple orbits $H$8,

$H$9

At unramified odd places where the quaternion algebra splits and $G$0 is unramified, the unit elements match: $G$1 This relative fundamental lemma is combined with spectral isolation to compare elliptic parts of the two trace formulas (Xue et al., 2022).

Twisted relative trace formulas generalize both Langlands–Shelstad’s twisted trace formula and Jacquet–Lai’s relative trace formula. In the unitary-group setting, matching stable twisted orbital integrals $G$2 with stable relative orbital integrals $G$3 provides the local bridge between a twisted formula and an untwisted one (Getz et al., 2017).

These examples show that orbit-by-orbit equality is not the only comparison mechanism. In several relative settings, the correct transfer is an integral transform on orbital integrals, and the “fundamental lemma” must be formulated accordingly (Sakellaridis, 2012, Sakellaridis, 2014).

5. Analytic and arithmetic applications

Relative trace formulas are used to isolate periods, prove moment identities, and derive bounds for $G$4-functions. In analytic number theory, the Kuznetsov-type formula is the prototype: integrating the kernel against unipotent characters produces a spectral sum of Fourier–Whittaker coefficients and a geometric sum of Kloosterman sums. This framework underlies applications to subconvexity, equidistribution, and bounds on Fourier coefficients (Blomer, 2019).

A particularly explicit application is Yang’s regularized torus-period formula on $G$5, which proves the Burgess subconvex bound

$G$6

Its geometric side is organized into degenerate, dual, and regular orbits, while the spectral side isolates the desired central $G$7-value by amplification (Yang, 2023).

On $G$8, a relative trace formula with $G$9-periods yields a Kuznetsov-type identity whose geometric side has four contributing double-coset representatives: the identity, the long Weyl element $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$0, and two intermediate Weyl elements $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$1. The resulting formula contains symplectic Kloosterman sums and archimedean Bessel kernels (Comtat, 2021).

For $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$2, the relative trace formula on $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$3 weighted by cusp forms on $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$4 gives a second moment evaluation and simultaneous nonvanishing in the level aspect. The geometric side contains Rankin–Selberg $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$5-functions for $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$6 together with explicit holomorphic functions, and both sides admit meromorphic continuation (Yang, 2023).

Geometric applications are also prominent. On a compact hyperbolic Riemann surface $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$7 with a closed geodesic $(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$8, the relative trace formula equates the period spectrum

$(h_\ell,h_r)\colon g\mapsto h_\ell g h_r^{-1},$9

with the ortholength spectrum of geodesic segments meeting $H_1$00 orthogonally. For the period Weyl law,

$H_1$01

and twisted variants yield simultaneous nonvanishing results (Martin et al., 2015).

In a different direction, the “general simple relative trace formula” leads to a relative Weyl law: $H_1$02 as $H_1$03 (Getz et al., 2014).

6. Operator-theoretic and scattering variants

The term “relative trace formula” also appears in spectral and scattering theory. For two smooth obstacles $H_1$04, with Dirichlet Laplacians $H_1$05 and free Laplacian $H_1$06, Hanisch–Strohmaier–Waters prove that for $H_1$07,

$H_1$08

extends to a trace-class operator. Its trace is given by a modified Birman–Krein formula

$H_1$09

where $H_1$10. In the Casimir case $H_1$11,

$H_1$12

which rigorously justifies determinant formulas used in the physics literature (Hanisch et al., 2020).

The electromagnetic version replaces the scalar Dirichlet Laplacian by relative and absolute Laplacians on divergence-free vector fields with Lipschitz boundaries. On $H_1$13, with relative and absolute boundary conditions, one defines operators $H_1$14 and $H_1$15 via functional calculus and proves that they are trace class for a class of admissible functions that may be polynomially bounded at infinity or have a $H_1$16-type singularity at zero. The relative trace formula is

$H_1$17

with

$H_1$18

For $H_1$19, this yields a determinant formula for the electromagnetic Casimir energy and rigorously justifies the “TGTG-formula” for metallic boundary conditions (Strohmaier et al., 2021).

A one-dimensional analogue appears for differential operators of arbitrary order. If $H_1$20 with $H_1$21, then under short-range assumptions one has

$H_1$22

and the perturbation determinant satisfies $H_1$23, relating the resolvent trace directly to a normalized Wronskian (Ostensson et al., 2011).

These operator-theoretic formulas show that “relative trace” need not be restricted to automorphic kernels. A plausible implication is that the unifying principle is cancellation relative to a simpler comparison problem: separated obstacles, single-body operators, free dynamics, or simpler boundary data (Hanisch et al., 2020, Strohmaier et al., 2021).