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arxiv: 2605.18294 · v2 · pith:4ACG4LGAnew · submitted 2026-05-18 · 🧮 math.NT

On the Periods of Ikeda-Yamana Lift for the Unitary Group I

Jin Higashitani This is my paper

Pith reviewed 2026-05-25 06:15 UTC · model grok-4.3

classification 🧮 math.NT
keywords Ikeda-Yamana liftunitary groupHermitian modular formsperiodsL-functionsHecke eigenformsautomorphic formsCM extensions
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The pith

The period of the Yamana lift I_n(f) equals a product of special L-values attached to the Hermitian form f.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an explicit formula expressing the period of the automorphic form obtained by applying the Yamana lift to a Hermitian modular form. For Hecke eigenforms f of weight (κ_v) and level one, with n odd, this period is written directly in terms of special values of L-functions attached to f. The setting is a totally real field F with quadratic CM extension E. The result extends earlier work of Katsurada on Ikeda's conjecture to the unitary group case. A reader would care because the formula links the arithmetic of automorphic forms on unitary groups to the special values of L-functions without needing to compute the period directly on the target group.

Core claim

Yamana constructed a lift from Hermitian modular forms to automorphic forms on the unitary group. For an odd positive integer n and Hecke eigenform f, the period perd{I_n(f), I_n(f)} is expressed in terms of special values of certain L-functions attached to f. This is an extension of Katsurada's result concerning Ikeda's conjecture.

What carries the argument

The Yamana lift I_n(f), which produces an automorphic form on the unitary group from a Hermitian modular form f of the stated weight and level one.

If this is right

  • The period of I_n(f) becomes computable from known L-values rather than from the geometry of the unitary group.
  • The formula supplies an explicit arithmetic expression for the squared norm of the lifted form.
  • The result holds uniformly for all Hecke eigenforms f once the lift is defined.
  • Special values of the L-functions attached to f control the size of the period on the unitary side.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formula may allow one to deduce algebraicity or integrality statements for the periods from known properties of the L-values.
  • Similar period formulas might exist for the even-n case or for forms of higher level once the lift is constructed there.
  • The expression could be used to compare periods across different lift constructions such as those from symplectic to orthogonal groups.

Load-bearing premise

The Yamana lift I_n(f) exists and yields an automorphic form on the unitary group for every Hermitian modular form f of the given weight and level one.

What would settle it

Compute the period directly for a concrete low-weight Hecke eigenform f with small n and check whether it matches the product of L-values given by the formula.

read the original abstract

Let $F$ be a totally real field and $E$ be a quadratic CM extension field of $F$. Let $n$ be an odd positive integer. Yamana constructed a lift from Hermitian modular forms to automorphic forms on the unitary group. We denote by $\mathrm{I}_n(f)$ the form obtained by applying this lift to the Hermitian modular form $f$ of weight $(\kappa_v)_{v|\infty}$ and level 1. We then express the period $\perd{\mathrm{I}_n(f), \mathrm{I}_n(f)}$ of $\mathrm{I}_n(f)$ for Hecke eigenforms $f$ in terms of special values of certain $L$-functions attached to $f$. This is an extension of Katsurada's result concerning Ikeda's conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper extends Katsurada's result on Ikeda's conjecture. For a totally real field F, quadratic CM extension E of F, and odd positive integer n, Yamana's lift produces an automorphic form I_n(f) on the unitary group from a Hermitian modular form f of weight (κ_v)_{v|∞} and level 1. For Hecke eigenforms f the manuscript expresses the period perd{I_n(f), I_n(f)} in terms of special values of L-functions attached to f.

Significance. If the claimed period formula holds, the result supplies an explicit relation between periods of lifts to unitary groups and L-values of the underlying Hermitian forms. This continues a line of work connecting periods, special values, and conjectures of Ikeda type, and would be of interest to researchers studying automorphic forms on unitary groups and their arithmetic invariants.

minor comments (2)
  1. The abstract uses the notation perd{I_n(f), I_n(f)} without defining the period symbol or the normalization of the inner product; a brief definition or reference in the introduction would improve readability.
  2. The statement that the lift exists for every such f is taken from Yamana's prior work; a short sentence recalling the precise hypotheses under which the lift is known to be nonzero would help readers assess the scope of the new period formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript arXiv:2605.18294. The referee summary correctly describes the main result: an extension of Katsurada's work expressing the period of the Ikeda-Yamana lift I_n(f) for Hecke eigenforms f in terms of special values of L-functions attached to f. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity; derivation extends cited prior results without self-reduction

full rationale

The abstract states that the period of the Yamana lift I_n(f) is expressed in terms of L-values attached to f, as an extension of Katsurada's result on Ikeda's conjecture. The lift itself is attributed to Yamana's construction (external citation), and the period comparison is presented as a new expression rather than a re-derivation or fit. No equations are given, no self-citations are invoked as load-bearing, and no parameter is fitted then renamed as prediction. The central claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described or can be extracted.

pith-pipeline@v0.9.0 · 5659 in / 1064 out tokens · 37236 ms · 2026-05-25T06:15:16.427273+00:00 · methodology

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Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    D. Blasius. Hilbert modular forms and the Ramanujan conjecture. InNoncommutative geometry and number theory, volume E37 ofAspects Math., pages 35–56. Friedr. Vieweg, Wiesbaden, 2006

    work page 2006

  2. [2]

    Casselman

    B. Casselman. Introduction to admissible representations of p-adic groups.unpublished notes, 1995

    work page 1995

  3. [3]

    P. Feit. Explicit formulas for local factors in the Euler products for Eisenstein series.Nagoya Math. J., 113:37–87, 1989

    work page 1989

  4. [4]

    Gelbart, I

    S. Gelbart, I. Piatetski-Shapiro, and S. Rallis.Explicit constructions of automorphic L-functions, volume 1254 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1987

    work page 1987

  5. [5]

    J. R. Getz and H. Hahn.An introduction to automorphic representations—with a view towardtrace formulae, volume 300 ofGraduate Textsin Mathematics. Springer, Cham, [2024]©2024

    work page 2024

  6. [6]

    Ibukiyama and H

    T. Ibukiyama and H. Katsurada. Koecher-Maaß series for real analytic Siegel Eisenstein series. In Automorphic forms and zeta functions, pages 170–197. World Sci. Publ., Hackensack, NJ, 2006

    work page 2006

  7. [7]

    Ibukiyama, H

    T. Ibukiyama, H. Katsurada, and H. Kojima. Period of the Ikeda-Miyawaki lift.J. Number Theory, 269:341–369, 2025

    work page 2025

  8. [8]

    Ichino and T

    A. Ichino and T. Ikeda. On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture.Geom. Funct.Anal., 19(5):1378–1425, 2010

    work page 2010

  9. [9]

    T. Ikeda. On the lifting of elliptic cusp forms to Siegel cusp forms of degree2n.Ann. of Math. (2), 154(3):641–681, 2001

    work page 2001

  10. [10]

    T. Ikeda. On the lifting of Hermitian modular forms.Compos. Math., 144(5):1107–1154, 2008

    work page 2008

  11. [11]

    Katsurada

    H. Katsurada. Koecher-Maass series of the Ikeda lift forU(m, m).Kyoto J. Math., 55(2):321–364, 2015

    work page 2015

  12. [12]

    Katsurada

    H. Katsurada. On the period of the Ikeda lift forU(m, m).Math. Z., 286(1-2):141–178, 2017

    work page 2017

  13. [13]

    Abh.Math.Semin.Univ.Hambg., 88(1):67– 86, 2018

    H.Katsurada.PeriodoftheadelicIkedaliftforU(m, m). Abh.Math.Semin.Univ.Hambg., 88(1):67– 86, 2018

    work page 2018

  14. [14]

    Katsurada and H.-a

    H. Katsurada and H.-a. Kawamura. Ikeda’s conjecture on the period of the Duke-Imamo¯ glu-Ikeda lift. Proc. Lond. Math. Soc. (3), 111(2):445–483, 2015

    work page 2015

  15. [15]

    OnthefunctionalequationssatisfiedbyEisensteinseries, volumeVol.544of Lecture Notes in Mathematics

    R.P.Langlands. OnthefunctionalequationssatisfiedbyEisensteinseries, volumeVol.544of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1976

    work page 1976

  16. [16]

    I. G. Macdonald. The volume of a compact Lie group.Invent.Math., 56(2):93–95, 1980

    work page 1980

  17. [17]

    Murase and T

    A. Murase and T. Sugano. Inner product formula for Kudla lift. InAutomorphic forms and zeta functions, pages 280–313. World Sci. Publ., Hackensack, NJ, 2006

    work page 2006

  18. [18]

    H. Saito. Explicit formula of orbitalp-adic zeta functions associated to symmetric and Hermitian matrices. Comment. Math. Univ. St. Paul., 46(2):175–216, 1997

    work page 1997

  19. [19]

    J.-J. Sansuc. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math., 327:12–80, 1981. 53

    work page 1981

  20. [20]

    Scharlau

    W. Scharlau. Quadratic and Hermitian forms, volume 270 of Grundlehren der mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences]. Springer-Verlag, Berlin, 1985

    work page 1985

  21. [21]

    Shimura.Euler products and Eisenstein series, volume 93 ofCBMS Regional Conference Series in Mathematics

    G. Shimura.Euler products and Eisenstein series, volume 93 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997

    work page 1997

  22. [22]

    Shimura.Arithmeticity in the theory of automorphic forms, volume 82 ofMathematical Surveys and Monographs

    G. Shimura.Arithmeticity in the theory of automorphic forms, volume 82 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000

    work page 2000

  23. [23]

    S. Yamana. On the lifting of Hilbert cusp forms to Hilbert-Hermitian cusp forms.Trans. Amer. Math. Soc., 373(8):5395–5438, 2020

    work page 2020

  24. [24]

    D. Zagier. Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), volume Vol. 627 ofLecture Notes in Math., pages 105–169. Springer, Berlin-New York, 1977

    work page 1976