Guo-Jacquet Conjecture Framework

• Guo-Jacquet Conjecture is a framework in automorphic representation theory that connects periods of automorphic forms with central L-values.
• It employs relative and infinitesimal trace formulae alongside weighted orbital integrals to compare spectral and geometric distributions.
• The conjecture influences the study of distinguished representations, inner-form transfers, and local harmonic analysis using methods like Poisson summation.

The Guo–Jacquet conjecture is a conjectural framework in automorphic representation theory that relates periods of automorphic forms on forms of general linear groups to special values of standard $L$-functions at the center, and is formulated through the comparison of relative trace formulae for symmetric pairs and their inner forms. In the cited literature it is described as a higher-rank generalization of Waldspurger’s results on toric periods and central $L$-values for $\mathrm{GL}(2)$, with both global and local incarnations and with a technical apparatus built from infinitesimal trace formulae, weighted orbital integrals, and transfer identities (Li, 2021, Li, 2020, Chaudouard et al., 6 Feb 2026).

1. Conjectural setting and basic objects

A recurrent model for the Guo–Jacquet program is the symmetric pair

$$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$

with $D$ a central division algebra, together with corresponding inner forms and base-change analogues (Li, 2019, Li, 2021). In a more general formulation, one starts with a central simple algebra $\fg$ over a number field $F$ containing a quadratic field extension $E/F$, sets $G=\fg^\times$, lets $\fh=\mathrm{Cent}_{\fg}(E)$, and takes $L$0; the symmetric space is then $L$1 (Li, 2020).

The infinitesimal, or Lie-algebra, version replaces the symmetric space by its tangent space at the identity. In the central-simple-algebra setting this tangent space is

$L$2

and $L$3 acts on $L$4 by conjugation (Li, 2020). This infinitesimal passage is not merely auxiliary: comparison of local terms at the infinitesimal level often implies comparison at the group level by descent methods, while the spectral side is replaced by Fourier transforms of geometric distributions (Li, 2021).

The central analytic difficulty is the comparison of orbital integrals and their weighted versions on the two sides of the relative trace formula. The cited papers treat this difficulty by adapting Arthur-style truncation, proving local and global Poisson-type identities, and making regular semi-simple terms explicit as weighted orbital integrals (Li, 2019, Li, 2020).

2. Periods, distinguished representations, and the spectral interpretation

In the global formulation developed for symmetric spaces attached to involutions, one considers periods over fixed-point subgroups of involutions. For an automorphic form $L$5, the period is written

$L$6

and a representation is $L$7-distinguished if this period is non-zero (Chaudouard et al., 6 Feb 2026).

The resulting trace formula is designed to connect spectral distributions built from such periods with geometric distributions built from relative orbital integrals. In the formulation of the general symmetric-space trace formula, the main identity is

$L$8

where $L$9 is spectral and $\mathrm{GL}(2)$0 is geometric (Chaudouard et al., 6 Feb 2026). If $\mathrm{GL}(2)$1 corresponds to an irreducible cuspidal automorphic representation $\mathrm{GL}(2)$2, then

$\mathrm{GL}(2)$3

This makes precise the principle that the spectral side is organized by relative characters generalizing linear periods (Chaudouard et al., 6 Feb 2026).

The papers describe the Guo–Jacquet conjecture as asserting deep links between automorphic distinguishedness, transfers between inner forms, and special $\mathrm{GL}(2)$4-values. A plausible implication is that the conjecture should be understood not as a single isolated identity, but as a program in which period integrals, inner-form transfer, and comparison of relative trace formulae are mutually constraining structures (Chaudouard et al., 6 Feb 2026, Li, 2021).

3. Infinitesimal trace formulae and weighted orbital integrals

A major advance in the subject is the establishment of infinitesimal variants of the Guo–Jacquet trace formula. For the case $\mathrm{GL}(2)$5, the infinitesimal trace formula is presented as a Poisson summation formula obtained by an analogue of Arthur’s truncation process, equating sums of distributions attached to rational points in the categorical quotient with the corresponding sums for Fourier transforms (Li, 2019). In the more general central-simple-algebra setting, one proves

$\mathrm{GL}(2)$6

with $\mathrm{GL}(2)$7 defined by truncation and extraction of the constant term in the truncation parameter (Li, 2020).

The truncation is essential because the naïve sum over orbits is not absolutely convergent. The papers define truncated kernels $\mathrm{GL}(2)$8, prove absolute convergence of the truncated sums, and show that the resulting distributions are polynomial or exponential-polynomial in the truncation parameter $\mathrm{GL}(2)$9 (Li, 2019, Li, 2020). This is the relative analogue of Arthur’s method, but implemented on the infinitesimal symmetric space.

For regular semi-simple terms, the geometric side becomes explicit. In the global infinitesimal formulation, the corresponding distribution is a weighted orbital integral: $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$0 or, in the split-block setting, the analogous formula with $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$1 and the character $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$2 (Li, 2020, Li, 2019). These expressions supply the precise local-global objects that must be matched under transfer.

A further result at the infinitesimal level is the weighted fundamental lemma for unit elements at almost all unramified places, proved by reducing the comparison of weighted orbital integrals to Labesse’s work on base change for $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$3 (Li, 2020). This is a structural step toward any full trace-formula comparison.

4. Local harmonic analysis and comparison identities

The local theory isolates weighted orbital integrals on infinitesimal symmetric spaces as the main comparison objects. For $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$4, regular semi-simple $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$5, and auxiliary Levi-parabolic data, the weighted orbital integral is

$$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$6

and the cited work proves local constancy, support properties, descent formulas, and $$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$7equivariance relations (Li, 2021).

At the trace-formula level one obtains both non-invariant and invariant local identities. The non-invariant formula satisfies

$$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$8

while the invariant formula satisfies

$$(G,H)=\bigl(\mathrm{GL}_{2n,D},\ \mathrm{GL}_{n,D}\times \mathrm{GL}_{n,D}\bigr),$$9

These are local analogues of Arthur-style symmetry under Fourier transform, but formulated for symmetric spaces rather than groups (Li, 2021).

The local comparison problem is sharpened in the study of matched infinitesimal symmetric spaces associated with $D$0 and an inner form $D$1. For matching Levi subgroups $D$2, matching regular semi-simple orbits $D$3, and matching points $D$4, the principal identity proved is

$D$5

where the $D$6-terms are Fourier transforms of invariant weighted orbital integrals and $D$7 denotes explicit transfer factors (Li, 2024). The same work proves a vanishing statement: $D$8 unless $D$9 when the relevant sign data do not match (Li, 2024).

These results are explicitly presented as preparatory to the non-invariant comparison of infinitesimal Guo–Jacquet trace formulae and as an infinitesimal, weighted “fundamental lemma” for Guo–Jacquet (Li, 2024).

5. Singular terms, nilpotent contributions, and descent

The regular semi-simple terms do not exhaust the geometric side. For the pair $\fg$0, the unipotent contribution of the Guo–Jacquet relative trace formula admits a fine expansion in terms of global nilpotent integrals: $\fg$1 with each $\fg$2 absolutely convergent for very cuspidal $\fg$3 (Chaudouard, 2019).

These nilpotent terms are further expressed via zeta integrals. For scaled variants,

$\fg$4

for $\fg$5 sufficiently small, and they satisfy the homogeneity relation

$\fg$6

under the stated size conditions on $\fg$7 and $\fg$8 (Chaudouard, 2019). This gives a precise description of how singular geometric terms behave under scaling and shows that the unipotent contribution can be analyzed by Eulerian zeta-type methods.

In the general trace formula for symmetric spaces, non-regular geometric data are handled by a descent procedure to the centralizer. The paper states that this descent expresses any geometric distribution in terms of the nilpotent contribution of infinitesimal trace formulas studied in previous papers (Chaudouard et al., 6 Feb 2026). This places the nilpotent analysis not at the periphery but at the center of the general geometric expansion.

A plausible implication is that the singular part of the Guo–Jacquet program is structurally recursive: regular semi-simple distributions are treated by weighted orbital integrals, whereas non-regular distributions are reduced to nilpotent contributions on smaller symmetric spaces.

6. Local variants, related conjectures, and common confusions

Some of the cited literature uses the phrase “local (Guo-)Jacquet conjecture” for distinction problems over a quadratic extension $\fg$9. In that setting, with $F$0 and $F$1, an irreducible smooth representation $F$2 is distinguished if it admits a non-zero $F$3-invariant linear functional, and $F$4-distinguished if it admits a non-zero linear functional that is $F$5-equivariant with respect to the quadratic character $F$6; a necessary condition is

$F$7

The cited paper proves that every smooth irreducible ladder representation of $F$8 satisfying $F$9 possesses the expected distinction properties relative to $E/F$0, namely that it is either distinguished or $E/F$1-distinguished, with precise combinatorial criteria, and that a proper ladder representation cannot be both distinguished and $E/F$2-distinguished (Gurevich, 2014).

This local distinction problem should not be conflated with Jacquet’s local converse conjecture for $E/F$3, which is characterized by twisted $E/F$4-factors and proved in the cited papers by means of Bessel functions, Whittaker models, Howe vectors, and kernel formulas (Chai, 2016, Chai, 2014, Adrian et al., 2014). Those results concern characterization of generic representations from $E/F$5-factors rather than periods and relative trace formulae.

A separate, entirely unrelated source of confusion is the arithmetic paper “On a conjectural congruence of Guo,” which proves the supercongruence

$E/F$6

for $E/F$7 and $E/F$8, and explicitly states that the Guo conjecture treated there has no direct relation to the Guo–Jacquet conjecture from the theory of automorphic representations and global periods (Wang et al., 2020).

The contemporary picture in the cited literature is therefore highly stratified: the Guo–Jacquet conjecture in its main sense concerns periods, relative trace formulae, and central $E/F$9-values; infinitesimal trace formulae and weighted orbital integrals provide the analytic infrastructure for comparison; local distinction results furnish representation-theoretic evidence in special families; and unrelated “Guo” or “Jacquet” conjectures must be separated carefully by context.