Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture. (English) Zbl 1112.11022

Ikeda, Tamotsu

Duke Math. J. 131, No. 3, 469-497 (2006).

The existence of liftings of modular forms from one group to another is a very interesting phenomenon with ramifications in the theory of \(L\)-functions and Galois representations, as well as in generalizations of the Ramanujan conjecture. The paper under review continues the author’s significant contributions to the subject, by constructing a lifting from Siegel eigenforms of degree \(r\) to Siegel eigenforms of degree \(r+2n\). All the forms considered are of level \(1\).
Let \(f\) be a normalized Hecke eigenform of degree \(1\) and weight \(2k\). In a previous article [Ann. Math. (2) 154, No. 3, 641–681 (2001; Zbl 0998.11023)], the author constructed a lifting of \(f\) to a Siegel cusp form \(F\) of even degree \(2n+2r\) and weight \(k+n+r\), whose standard \(L\)-function is \[ \zeta(s)\prod_{i=1}^{2n+2r}L(s+k+n+r-i,f). \] Here \(n\) and \(r\) can be any positive integers such that \(n+r\equiv k\pmod{2}\). The construction of \(F\) uses an auxiliary half-integral form \(h\) in the Kohnen plus-subspace.
Suppose now that we are given a Hecke eigenform \(g\) of degree \(r\) and weight \(k+n\), with standard \(L\)-function \(L(s,g,\mathrm{st})\). By a procedure called “pullback to a block diagonal subset”, the author constructs a cusp form \(\mathcal{F}_{h,g}\) of degree \(2n+r\); if nonzero, this cusp form is a Hecke eigenform and its standard \(L\)-function is \[ L(s,\mathcal{F}_{g,h},\mathrm{st})=L(s,g,\mathrm{st}) \prod_{i=1}^{2n} L(s+k+n-i,f). \] An application of this result is given in Sections 2 and 7, by proving a conjecture of I. Miyawaki [Mem. Fac. Sci., Kyushu Univ., Ser. A 46, No. 2, 307–339 (1992; Zbl 0780.11022)].
It would be good to figure out when the lifting \(\mathcal{F}_{g,h}\) is nonzero. The author proposes a conjecture in this direction, in the form of a formula relating the Petersson inner product \(\langle \mathcal{F}_{g,h},\mathcal{F}_{g,h}\rangle\) to a certain \(L\)-value constructed from \(f\) and \(g\). The paper concludes with some numerical evidence for this conjecture.

Reviewer: Alexandru Ghitza (Waterville)

MSC:

11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Keywords:

Siegel modular forms; lifting

Citations:

Zbl 0998.11023; Zbl 0780.11022

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