Ichino–Ikeda Conjecture: Periods & L-values

• Ichino–Ikeda type conjecture is a set of refined period formulas in automorphic representation theory, equating squared global periods to central L-values with explicit local factors.
• It refines the nonvanishing Gross–Prasad principle by providing an exact proportionality constant between automorphic periods and special L-values, integrating Tamagawa measures and component groups.
• The conjecture extends across frameworks including unitary, orthogonal, and Whittaker settings and is supported by techniques such as relative trace formulas and local spherical character comparisons.

The Ichino–Ikeda type conjecture is a family of refined period formulas in automorphic representation theory. In its basic form, it asserts that the square of a global automorphic period—typically a Gross–Prasad, Bessel, Fourier–Jacobi, or Whittaker period—is equal to an explicit central $L$-value multiplied by a product of normalized local period integrals, with an explicit global constant involving Tamagawa measures and component groups. In the orthogonal and unitary settings, it refines the Gross–Prasad nonvanishing principle by replacing the equivalence “period $\neq 0$ if and only if $L(1/2)\neq 0$” with an exact factorization formula (Harris, 2012). Subsequent work has extended this paradigm to endoscopic unitary parameters, Bessel and Fourier–Jacobi periods, Whittaker coefficients, and certain Eisenstein series (Beuzart-Plessis et al., 2020).

1. General form and normalizations

In the refined Gross–Prasad framework, one begins with a pair of classical groups $H \subset G$, an automorphic representation $\pi$ of $G(\mathbb{A}_F)$, and a global period integral $P(\phi)$ defined by integration over $H(F)\backslash H(\mathbb{A}_F)$. The Ichino–Ikeda type formula predicts that

$\frac{|P(\phi)|^2}{\langle \phi,\phi\rangle}$

is proportional to a central special value of an $L$-function and to a product of normalized local factors, almost all equal to $\neq 0$0 at unramified places (Harris, 2012).

For unitary groups, the refined conjecture formulated in low rank has the shape

$\neq 0$1

where $\neq 0$2 denotes quadratic base change to $\neq 0$3, the global measures are Tamagawa, and the local factors $\neq 0$4 are normalized so that $\neq 0$5 for spherical vectors at unramified places (Harris, 2012).

Two structural features recur throughout the literature. First, the global constant is not universal: it depends on the subgroup, the measure normalization, and the component group attached to the relevant global parameter. Second, the local terms are canonical only after division by specific adjoint, Asai, or Rankin–Selberg $\neq 0$6-factors. This is why the same conjectural pattern appears in several different guises rather than as a single invariant formula.

2. Relation to Gross–Prasad and Gan–Gross–Prasad

The unrefined Gross–Prasad philosophy concerns nonvanishing. In the unitary case, the refined statement strengthens the implication

$\neq 0$7

by identifying the exact proportionality constant between the period square and the central $\neq 0$8-value (Harris, 2012).

For $\neq 0$9, the global Gan–Gross–Prasad theorem in endoscopic generality states that for a Hermitian Arthur parameter $L(1/2)\neq 0$0,

$L(1/2)\neq 0$1

such that the diagonal period

$L(1/2)\neq 0$2

induces a nonzero linear form on $L(1/2)\neq 0$3 (Beuzart-Plessis et al., 2020). Parallel equivalences are now known for unitary Bessel periods and for unitary Fourier–Jacobi periods (Beuzart-Plessis et al., 2023, Boisseau et al., 5 Jan 2026).

A common misconception is that the Ichino–Ikeda type conjecture is merely an “explicit version” of nonvanishing. In fact, its refined form encodes substantially more data: the component-group denominator, the exact global $L(1/2)\neq 0$4-ratio, and the local relative characters. It is therefore both a refinement of Gan–Gross–Prasad and a mechanism for global-to-local compatibility.

3. The unitary $L(1/2)\neq 0$5 case

For diagonal periods on $L(1/2)\neq 0$6, the modern unitary formulation uses a Hermitian space $L(1/2)\neq 0$7 of rank $L(1/2)\neq 0$8, the rank-one norm space $L(1/2)\neq 0$9, and the diagonal embedding

$H \subset G$0

If $H \subset G$1 is a cuspidal automorphic representation of $H \subset G$2, tempered everywhere, and $H \subset G$3, then the refined global identity proved in the endoscopic case is

$H \subset G$4

for every nonzero factorizable vector $H \subset G$5 (Beuzart-Plessis et al., 2020). Here

$H \subset G$6

and at almost all places

$H \subset G$7

(Beuzart-Plessis et al., 2020).

The conjectural form of this identity was already visible in the unitary refinement proved in the first cases $H \subset G$8 and, under a theta-lift hypothesis, $H \subset G$9 (Harris, 2012). The stable tempered case for $\pi$0 was completed by isolating the cuspidal spectrum on the Jacquet–Rallis relative trace formula, yielding the expected formula with trivial global component group in the stable setting (Beuzart-Plessis et al., 2019). The endoscopic extension then handled general Hermitian Arthur parameters, not only simple cuspidal ones (Beuzart-Plessis et al., 2020).

The same formalism has been pushed beyond the cuspidal corank-one setting. For $\pi$1-regular Hermitian Arthur parameters, Eisenstein periods admit a regularized Ichino–Yamana period, and the resulting identity keeps the same shape: $\pi$2 with $\pi$3 incorporating completed Rankin–Selberg and Asai factors (Beuzart-Plessis et al., 2023).

4. Bessel, Fourier–Jacobi, and Whittaker variants

The Ichino–Ikeda pattern is not confined to diagonal Gross–Prasad periods. For unitary Bessel periods, one fixes $\pi$4, writes $\pi$5, defines a Bessel subgroup $\pi$6, and considers

$\pi$7

The corresponding refined formula is

$\pi$8

with almost all local factors equal to $\pi$9 (Beuzart-Plessis et al., 2023).

For unitary Fourier–Jacobi periods, the subgroup is $G(\mathbb{A}_F)$0, equipped with a Heisenberg–Weil representation $G(\mathbb{A}_F)$1, and the global period is

$G(\mathbb{A}_F)$2

The proved refined identity is

$G(\mathbb{A}_F)$3

where

$G(\mathbb{A}_F)$4

and $G(\mathbb{A}_F)$5 at almost all places (Boisseau et al., 5 Jan 2026). The global Fourier–Jacobi Gan–Gross–Prasad conjecture and its refined form are now established via comparison of Liu’s relative trace formulae, including positive corank (Boisseau et al., 2024).

Orthogonal Bessel settings furnish further archetypes. For $G(\mathbb{A}_F)$6, the refined formula for tempered cuspidal representations gives

$G(\mathbb{A}_F)$7

with $G(\mathbb{A}_F)$8 for almost all $G(\mathbb{A}_F)$9 (Furusawa et al., 2022). A related refined formula for non-endoscopic Yoshida lifts on $P(\phi)$0 has global constant $P(\phi)$1, matching the expected component-group factor (Corbett, 2015).

Finally, there are Whittaker analogues. Lapid–Mao formulated an Ichino–Ikeda type conjecture for Whittaker–Fourier coefficients on quasi-split reductive groups (Lapid et al., 2013). For quasi-split unitary groups, the proved formula reads

$P(\phi)$2

for any irreducible cuspidal globally generic automorphic representation of $P(\phi)$3 (Morimoto, 2024).

5. Local factors, relative characters, and proof technology

The conjecture is local-global in a strong sense. Its local input is not a bare matrix coefficient integral, but a normalized relative character. In the unitary diagonal case, the normalized local factor is an integral over $P(\phi)$4 divided by $P(\phi)$5 (Beuzart-Plessis et al., 2020). In Bessel and Fourier–Jacobi settings, the local factor is defined by stable or regularized integration against a local Bessel or Heisenberg–Weil model (Beuzart-Plessis et al., 2023, Boisseau et al., 5 Jan 2026). In Whittaker settings, one integrates matrix coefficients against a generic character on the maximal unipotent and normalizes by Asai factors (Morimoto, 2024).

A second indispensable ingredient is local comparison of spherical characters between a classical-group side and a linear-group side. For unitary Gan–Gross–Prasad, the comparison takes the form

$P(\phi)$6

for matching local test functions, completing the local step in Zhang’s program (Beuzart-Plessis, 2018). In the Fourier–Jacobi comparison, the local identity is

$P(\phi)$7

with explicit $P(\phi)$8 depending on $P(\phi)$9, $H(F)\backslash H(\mathbb{A}_F)$0, $H(F)\backslash H(\mathbb{A}_F)$1, and the discriminant of the Hermitian space (Boisseau et al., 5 Jan 2026).

Globally, proofs are built from relative trace formulas. The diagonal unitary case uses the Jacquet–Rallis relative trace formula, together with isolation of the desired cuspidal packet on the spectral side (Beuzart-Plessis et al., 2019). The endoscopic extension computes the $H(F)\backslash H(\mathbb{A}_F)$2-generic spectral contributions in two ways—via truncation and intertwining periods, and via Whittaker zeta integrals—and proves their equality (Beuzart-Plessis et al., 2020). Fourier–Jacobi proofs compare two relative trace formulae due to Liu, one on the unitary side and one on the linear side, and identify matching test functions spectrally as well as geometrically (Boisseau et al., 2024, Boisseau et al., 5 Jan 2026). Lower-rank and special-lift cases often proceed instead through Waldspurger’s formula, theta correspondence, Ichino’s triple product formula, and Rallis inner product formulas [(Harris, 2012); (Corbett, 2015); (Furusawa et al., 2022)].

6. Proven cases, scope, and related directions

A substantial portion of the conjectural landscape has now been proved. The first unitary cases $H(F)\backslash H(\mathbb{A}_F)$3 and $H(F)\backslash H(\mathbb{A}_F)$4 were established using Waldspurger and theta methods (Harris, 2012). The stable case for $H(F)\backslash H(\mathbb{A}_F)$5 was completed by isolation of the cuspidal spectrum (Beuzart-Plessis et al., 2019). All endoscopic cases for $H(F)\backslash H(\mathbb{A}_F)$6 were then proved, including the refined formula with explicit $H(F)\backslash H(\mathbb{A}_F)$7 (Beuzart-Plessis et al., 2020). Bessel periods on unitary groups and some Eisenstein cases are also covered (Beuzart-Plessis et al., 2023). Fourier–Jacobi periods on unitary groups are now proved in full, including the Ichino–Ikeda type refinement (Boisseau et al., 2024, Boisseau et al., 5 Jan 2026). For $H(F)\backslash H(\mathbb{A}_F)$8, the refined formula is known for tempered irreducible cuspidal automorphic representations (Furusawa et al., 2022), and for non-endoscopic Yoshida lifts one obtains an exact factor $H(F)\backslash H(\mathbb{A}_F)$9 in the refined formula (Corbett, 2015). Whittaker-period analogues for quasi-split unitary groups are likewise established (Morimoto, 2024).

The scope of each theorem remains sensitive to hypotheses. Temperedness is often imposed to guarantee absolute convergence and holomorphy of local zeta integrals. Several arguments assume totally real or CM global fields, discrete series at archimedean places, or regularity conditions on Hermitian Arthur parameters (Furusawa et al., 2022, Beuzart-Plessis et al., 2023). At the same time, the extension to endoscopic parameters and to Eisenstein series shows that the Ichino–Ikeda type paradigm is not restricted to simple cuspidal packets (Beuzart-Plessis et al., 2020, Beuzart-Plessis et al., 2023).

A related but distinct local direction is the Hiraga–Ichino–Ikeda formal degree conjecture, which identifies formal degrees or Plancherel densities with adjoint $\frac{|P(\phi)|^2}{\langle \phi,\phi\rangle}$0-factors and component groups. For unitary groups, explicit Plancherel formulas on $\frac{|P(\phi)|^2}{\langle \phi,\phi\rangle}$1 were used to deduce the formal degree conjecture in the unitary case (Beuzart-Plessis, 2018). Substantial further cases are known for unipotent representations and for discrete series in principal series of split $\frac{|P(\phi)|^2}{\langle \phi,\phi\rangle}$2-adic groups (Opdam, 2018, Ricci, 24 Jun 2025). This suggests that the “Ichino–Ikeda type” viewpoint extends beyond global periods to a broader local representation-theoretic framework in which adjoint $\frac{|P(\phi)|^2}{\langle \phi,\phi\rangle}$3-factors, component groups, and canonical models are governed by the same normalization principles.