Lifting conjectures from vector valued Siegel modular forms of degree two (Q2839765)

When considering automorphic form on groups other than subgroups of \(\mathrm{SL}(2,\mathbb{Z})\) (e.g., subgroups of \(\mathrm{Sp}(4,\mathbb{Z})\), subgroups of \(\mathrm{SL}(2,\mathbb{Z}_F)\) for \(\mathbb{Z}_F\) the ring of integers of some number field \(F\)) and their \(L\)-functions, a good place to start is to consider the space of lifts, i.e., those automorphic forms whose Hecke data and \(L\)-function data are determined by another automorphic form defined on a ``smaller'' group. Famous examples of such lifts are the Saito-Kurokawa lift from modular forms on \(\mathrm{SL}(2,\mathbb{Z})\) to modular forms on \(\mathrm{Sp}(4,\mathbb{Z})\) (and the Ikeda lift, its generalization to \(\mathrm{Sp}(2n,\mathbb{Z})\)) and the Doi-Naganuma lift from modular forms on \(\mathrm{SL}(2,\mathbb{Z})\) to Hilbert modular forms on \(\mathrm{SL}(2,\mathbb{Z}_F)\) for some totally real quadratic field extension \(F/\mathbb{Q}\) (and base change, its generalization to other extensions \(E/F\)).NEWLINENEWLINESuch lifts can be discovered in a number of ways, but many times they are discovered as a result of the following kind of computation: one computes a modular form \(F\) on some arithmetic group \(G\) via some method. For example, maybe it can be computed as a sum of theta series. Suppose further, that enough coefficients of \(F\) are known so that one can compute enough Hecke eigenvalues to determine the first few Euler factors of the \(L\)-function \(L(F,s)=\prod_{p} L_p(F,s)\) associated to \(F\). It is predicted by the Generalized Ramanujan Petersson Conjecture that most Euler factors, if the \(L\)-function is normalized so that its functional equation relates \(s\) and \(1-s\), will have all their roots on the unit circle. So, if the Euler factor \(L_p(F,s)\) that you have in your hand does not have all its roots on the unit circle, you probably have a lift.NEWLINENEWLINEIn a paper by the reviewer, \textit{C. Poor} and \textit{D. S. Yuen} [Bull. Aust. Math. Soc. 80, No. 1, 65--82 (2009; Zbl 1234.11062)], such an experiment was carried for modular forms on \(\mathrm{Sp}(8,\mathbb{Z})\) and, in particular, the seven Hecke eigenforms in weight 16 were considered carefully (note: in the paper we use \(\mathrm{Sp}(4,\mathbb{Z})\) to denote the group of \(8\times 8\) symplectic matrices but here we use \(\mathrm{Sp}(8,\mathbb{Z})\)). Of those seven forms, four were shown to be known lifts and the other three were conjectured to be some kind of unknown lifts. The conjecture was based solely on the fact that the Euler factors at the prime 2 had some nonunimodular roots.NEWLINENEWLINEIn the paper under review, two conjectured lifts to modular forms on \(\mathrm{Sp}(8,\mathbb{Z})\) are described, lifts that would explain the existence of the forms computed in the paper referenced above.NEWLINENEWLINEConjecture 2.1. For any even natural number \(m\geq 4\), let \(f\) be a vector-valued Siegel eigenform of degree 2 and weight \((4,2m-8)\). Then there should exist a Siegel eigenform \(F\) of degree 4 and weight \(m\) such that the following relations hold: NEWLINE\[NEWLINE\begin{aligned} L(s,F,\mathrm{St})&=\zeta(s)L(s-\tfrac12,f,\mathrm{Sp})L(s+\tfrac12,f,\mathrm{Sp})\\ L(s,F,\mathrm{Sp})&=\zeta(s-1)\zeta(s)\zeta(s+1)L(s,f,\mathrm{St})L(s-12,f,\mathrm{Sp})L(s+12,f,\mathrm{Sp}). \end{aligned}NEWLINE\]NEWLINE Conjecture 2.2. For any even natural number \(m\geq 4\), let \(g\) be a vector-valued Siegel eigenform of degree 2 and weight \((m-2,2)\) and let \(f\) an elliptic eigenform of weight \(2m-4\). Then there should exist a Siegel eigenform \(F\) of degree 4 and weight \(m\) such that NEWLINE\[NEWLINE\begin{aligned} L(s,F,\mathrm{St})&=L(s+m-2,f)L(s+m-3,f)L(s,g,\mathrm{St})\\ L(s,F,\mathrm{Sp})&=L(s-\tfrac12,g,\mathrm{Sp})L(s+\tfrac12,g,\mathrm{Sp})L(s,f\otimes g). \end{aligned}NEWLINE\]NEWLINE In these conjectures the \(\mathrm{Sp}\) means the spin \(L\)-function of \(F\), the \(\mathrm{St}\) means the standard \(L\)-function of \(F\) and \(L(s,f\otimes g)\) is the convolution \(L\)-function of \(f\) and \(g\). In the paper, the author also states generalizations of these conjectures to higher degree.NEWLINENEWLINEThe evidence for the conjectures as given in the paper is as follows: {\parindent=6mm \begin{itemize}\item[1.] once the form on the smaller group has been identified, its \(L\)-function is computed and it is checked that the Euler factors as outlined in the conjectures agree. \item[2.] the conjectural functional equations and Gamma factors of the completed the \(L\)-functions on each side of the equations in the conjectures are compared. \item[3.] as predicted by the Langlands philosophy this lifting corresponds to a mapping between the \(L\)-groups of the form being lifted and its image. \item[4.] it is shown in the case when the vector-valued Siegel modular forms are actually scalar-valued (i.e., are of weight \((m,0)\) for some \(m\)) these liftings correspond to known lifts. NEWLINENEWLINE\end{itemize}} This gives a good recipe for how someone in the future might want to give evidence for a conjectured lift and also how one might be able to predict the existence of lifts.