Abstract
Let \(K=\mathbf{Q}(\sqrt{-D})\) be an imaginary quadratic field with discriminant \(-D,\) and \(\chi \) the Dirichlet character corresponding to the extension \(K/\mathbf{Q}\). Let \(m=2n\) or \(2n+1\) with n a positive integer. Let f be a primitive form of weight \(2k+1\) and character \(\chi \) for \(\varGamma _0(D),\) or a primitive form of weight 2k for \(SL_2(\mathbf{Z})\) 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 \(2k+2n\) and character \(\det ^{-k-n}\) for the Hermitian modular group \(\varGamma _K^{(m)}\) constructed by Ikeda. We then express the period \(\langle I_m(f), I_m(f) \rangle \) of \(I_m(f)\) in terms of special values of the adjoint L-function of f and its twist by the character \(\chi \). This proves the conjecture concerning the period of the Hermitian Ikeda lift proposed by Ikeda.
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References
Andrianov, A.N.: Quadratic Forms and Hecke Operators, vol. 286. Springer, Berlin (1987)
Böcherer, S., Dummigan, N., Schulze-Pillot, R.: Yoshida lifts and Selmer groups. J. Math. Soc. Jpn. 64, 1353–1405 (2012)
Böcherer, S.: Eine Rationalitätsatz für formale Heckereihen zur Siegelschen Modulgruppe, vol. 56. Abh. Math. Sem. Univ., Hamburg (1986)
Brown, J.: Saito–Kurokawa lifts and applications to the Bloch–Kato conjecture. Compos. Math. 143(2), 290–322 (2007)
Brown, J., Keaton, R.: Congruence primes for Ikeda lifts and the Ikeda ideal. Pac. J. 274, 27–52 (2015)
Gritsenko, V.A.: The Maass space for \({\rm SU}(2,2)\). The Hecke ring, and zeta functions. (Russian) Translated in Proc. Steklov Inst. Math. 1991, no. 4, 75–86. Galois theory, rings, algebraic groups and their applications (Russian). Trudy Mat. Inst. Steklov. 183 (1990), 68–78, 223–225
Ibukiyama, T., Katsurada, H.: An explicit formula for Koecher-Maaß Dirichlet series for the Ikeda lifting, vol. 74. Abh. Math. Sem., Hamburg (2004)
Ibukiyama, T., Katsurada, H.: Koecher–Maaß Series for Real Analytic Siegel Eisenstein Series, Automorphic Forms and Zeta Functions. World Scientific Publications, Hackensack (2006)
Ikeda, T.: Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture. Duke Math. J. 131(3), 469–497 (2006)
Ikeda, T.: On the lifting of hermitian modular forms. Compos. Math. 144, 1107–1154 (2008)
Jacobowitz, R.: Hermitian forms over local fields. Am. J. Math. 84, 441–465 (1962)
Katsurada, H.: Congruence of Siegel modular forms and special values of their standard zeta functions. Math. Z. 259, 97–111 (2008)
Katsurada, H.: Congruence Between Duke-Imamoḡlu-Ikeda Lifts and Non-Duke-Imamoḡlu-Ikeda Lifts, vol. 64, pp. 109–129. Comment. Math. Univ, St. Pauli (2015)
Katsurada, H.: Koehcer–Maass series of the Ikeda lift for \(U(m, m)\). Kyoto J. Math. 55, 321–364 (2015)
Katsurada, H., Kawamura, H.: On Andrianov type identity for a power series attached to Jacobi forms and its applications. Acta Arith. 145, 233–265 (2010)
Katsurada, H., Kawamura, H.: On Ikeda’s conjecture on the period of the Duke–Imamoglu–Ikeda lift. Proc. Lond. Math. Soc. 111, 445–483 (2015)
Klosin, K.: The Maass space for \(U(2,2)\) and the Bloch–Kato conjecture for the symmetric square motive of a modular form. J. Math. Soc. Jpn. 67, 797–859 (2015)
Kojima, H.: An arithmetic of Hermitian modular forms of degree two. Invent. Math. 69, 217–227 (1982)
Kohnen, W., Skoruppa, N.-P.: A certain Dirichlet series attached to Siegel modular forms of degree 2. Invent. Math. 95, 541–558 (1989)
Krieg, A.: The Maass spaces on the Hermitian half-space of degree 2. Math. Ann. 289, 663–681 (1991)
Oda, T.: On modular forms associated with indefinite quadratic forms of signature \((2, n-2),\). Math. Ann. 231, 97–144 (1977)
Rallis, S.: \(L\)-functions and Oscillator representation. Lecture Notes in Math, vol. 1245. Springer, Berlin (1987)
Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976)
Shimura, G., Euler products and Eisenstein series. In: CBMS Regional Conference Series in Mathematics, vol. 93, American Mathematical Society (1997)
Shimura, G.: Arithmeticity in the theory of automorphic forms. In: Mathematical Surveys and Monographs, vol. 82. American Mathematical Society (2000)
Shimura, G.: Collected Papers, vol. II. Springer, New York (2002)
Sugano, T.: Jacobi Forms and the Theta Lifting, vol. 44. Comment Math. Univ., St. Pauli (1995)
Acknowledgments
The author was partly supported by the JSPS KAKENHI Grant Numbers 24540005, 25247001 and 23224001. The author thanks T. Ikeda for useful comments. The author also thanks the referee for pointing out many errors in the original version of our paper.
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Dedicated to Professor Yasuo Morita on the occasion of his 70th birthday.
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Katsurada, H. On the period of the Ikeda lift for U(m, m). Math. Z. 286, 141–178 (2017). https://doi.org/10.1007/s00209-016-1758-y
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DOI: https://doi.org/10.1007/s00209-016-1758-y