[Paper Review] On the period of the Ikeda lift for U(m,m)

Hidenori Katsurada|EPrints - Department of Mathematics, Hokkaido University|Feb 22, 2011

Advanced Algebra and Geometry18 references4 citations

TL;DR

This paper proves a conjecture by Ikeda on the period of the Hermitian Ikeda lift for unitary groups U(m,m), expressing the inner product ⟨Iₘ(f), Iₘ(f)⟩ in terms of special values of the adjoint L-function of a modular form f and its twists by a Dirichlet character χ. The result is derived via Rankin-Selberg L-series and local Siegel series computations, establishing a precise link between automorphic periods and special L-values for Hermitian modular forms.

ABSTRACT

Let K be an imaginary quadratic field, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and and nebentype x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I_m(f) be the lift of f to the space of modular forms of weight 2k+2n for the Hermitian modular group of degree m constructed by Ikeda. We then express the period of I_m(f) in terms of special values of the adjoint L-functions of f. This poves the conjecture concerning the period of the Ikeda lift proposed by Ikeda.

Motivation & Objective

• To resolve Ikeda's conjecture regarding the period of the Hermitian Ikeda lift for unitary groups U(m,m).
• To express the inner product ⟨Iₘ(f), Iₘ(f)⟩ in terms of special values of the adjoint L-function of f and its twists by χ.
• To establish a precise arithmetic relation between automorphic periods and special L-values for Hermitian modular forms.
• To extend the method of Rankin-Selberg series and local Siegel series to the Hermitian setting, particularly for ramified primes in imaginary quadratic fields.

Proposed method

• Construct the Hermitian Ikeda lift Iₘ(f) of a modular form f to U(m,m) of weight 2k+2n and character det⁻ᵏ⁻ⁿ.
• Define a Rankin-Selberg-type Dirichlet series R(s, Iₘ(f)) associated with the lift, whose residue at s = 2k+2n relates to the period ⟨Iₘ(f), Iₘ(f)⟩.
• Reduce the computation of R(s, Iₘ(f)) to local computations of formal power series Ĥₘ,ₚ(d; X, Y, t) via local Siegel series.
• Compute Ĥₘ,ₚ(d; X, Y, t) explicitly for all primes p, including the ramified case in the imaginary quadratic field K = ℚ(√−D).
• Use the explicit local formulas to derive a global expression for R(s, Iₘ(f)) as an Euler product involving L-functions L(s, f, Ad, χⁱ⁻¹) and L(s, χⁱ).
• Compare the residue of R(s, Iₘ(f)) at s = 2k+2n with the period ⟨Iₘ(f), Iₘ(f)⟩ via Proposition 3.1, yielding the main formula.

Experimental results

Research questions

• RQ1 What is the precise expression for the period ⟨Iₘ(f), Iₘ(f)⟩ of the Hermitian Ikeda lift Iₘ(f) in terms of L-functions of f and its twists?
• RQ2 Does the conjecture by Ikeda on the period of the Hermitian Ikeda lift hold for all m = 2n or m = 2n+1?
• RQ3 How do special values of the adjoint L-function L(s, f, Ad) and its twists by χ relate to the inner product of the lift?
• RQ4 Can the Rankin-Selberg method be adapted to compute periods for Hermitian modular forms when the lift is not a theta lift?

Key findings

• The period ⟨Iₘ(f), Iₘ(f)⟩ is expressed as a product of special values: L(1, f, Ad) × ∏ᵢ₌₂ᵐ L(i, f, Ad, χⁱ⁻¹) × ∏ᵢ₌₂ᵐ L(i, χⁱ), up to an elementary factor.
• The residue of the Rankin-Selberg series R(s, Iₘ(f)) at s = 2k+2n is shown to match the inner product ⟨Iₘ(f), Iₘ(f)⟩, confirming the period formula.
• The local series Ĥₘ,ₚ(d; X, Y, t) are computed explicitly for all primes p, including ramified primes, enabling the global computation.
• For m = 2n, the formula involves L(s, f, Ad, χⁱ⁻¹) and L(s, χⁱ) for i = 1 to m, while for m = 2n+1, the character χ is trivial on the central character, and the formula adjusts accordingly.
• The result confirms that the Hermitian Ikeda lift is not generally a theta lift (except for m = 2), justifying the need for a new method beyond theta liftings.
• The method successfully generalizes previous results on the Duke-Imamoglu-Ikeda lift to the Hermitian case, proving the conjecture with a refined treatment of local factors.