Let $F$ be an Ikeda lift, $λ_F(m)$ be the eigenvalue corresponding to the Hecke operator $T(m)$. We show that $λ_F(p)$ is positive for all large enough primes $p$. This is proved for Ikeda lifts of all genus. The second result is specific for genus 4 Ikeda lifts. If $F$ is genus 4 Ikeda lift, we show that, given $r$ there exists a constant $c_r$ such that $λ_F(p^r)$ is positive for all $p>c_r$.
Let k and n be positive even integers. For a cuspidal Hecke eigenform g in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(g) be the Duke-Imamoglu-Ikeda lift of g in the space of cusp forms of weight k for Sp(n,Z), and f the primitive form of weight 2k-n for SL(2,Z) corresponding to g under the Shimura correspondence. We then characterize prime ideals giving congruence between I(g) and another cuspidal Hecke eigenform not coming from the Duke-Imamoglu-Ikeda lift in terms of the specilal values of the Hecke L-function and the adjoint L-function of f.
We consider lifts from two elliptic modular forms to Siegel modular forms of odd degrees which are special cases of Miyawaki-Ikeda lifts. Assuming non-vanishing of these Miyawaki-Ikeda lifts, we show that the spinor L-functions of these Miyawaki-Ikeda lifts are products of some kind of symmetric power L-functions determined by original two elliptic modular forms.
We compute the twisted Koecher-Maass series of the first and second kind of the Ikeda type lift for the exceptional group of type $E_{7,3}$. As an application, we obtain their rationality result.
Some generalizations of the Maass relation for Siegel modular forms of higher degrees have been obtained by several authors. In the present article we first give a new generalization of the Maass relation for Siegel-Eisenstein series of arbitrary degrees. Furthermore, we show that the Duke-Imamoglu-Ibukiyama-Ikeda lifts satisfy this generalized Maass relation with some modifications. As an application of the generalized Maass relation in the present article we give a new proof of the Miyawaki-Ikeda lifts of two elliptic modular forms. Namely, we compute the standard L-function of the Miyawaki-…
Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Sp(n,Z), and f the primitive form of weight 2k-n for SL(2,Z) corresponding to h under the Shimura correspondence. We then express the ratio <I(h),I(h)>/< h, h > of the period of I(h) to that of h in terms of special values of certain L-functions of f. This proves the conjecture proposed by Ikeda concerning the period of the Duke-Imamoglu-Ikeda lift.
We give an explicit formula for the twisted Koecher-Maass series of the Duke-Imamoglu-Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin-Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura.
Let $k$ and $n$ be positive even integers. For a Hecke eigenform $h$ in the Kohnen plus subspace of weight $k-n/2+1/2$ for $\varGamma_0(4)$, let $I_n(h)$ be the Duke-Imamoglu-Ikeda lift of $h$ to the space of cusp forms of weight $k$ for $Sp_n(\mathbb Z)$. We then give an estimate of the Fourier coefficients of $I_n(h)$. It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form.
In our previous work, he second and the third named authors constructed the Ikeda type lift for the exceptional group $E_{7,3}$ from an elliptic modular cusp form. In this paper, we prove an explicit formula for the period or the Petersson norm of the Ikeda type lift in terms of the product of the special values of the symmetric square $L$-function of the elliptic modular form. There are similar works done by the first author with his collaborator, but new technical inputs are required and developed to overcome some difficulties coming from the hugeness of $E_{7,3}$.
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 <I_m(f), I_m(f) > of I_m(f) in terms of special values of the adjoint L-functions of f. This poves the conjecture concerning the peri…
For two Hecke eigenforms $h_1$ and $h_2$ in the Kohnen plus space of half-integral weight, let $I_n(h_1)$ and $I_n(h_2)$ be the Duke-Imamoglu-Ikeda lift of $h_1$ and $h_2$, respectively, which are Siegel cusp forms with respect to $Sp_n(\ZZ)$. Moreover, let $E_{n/2+1/2}$ be the Cohen Eisenstein series of weight $n/2+1/2$. We then express the Rankin-Selberg convolution $R(s,I_n(h_1),I_n(h_2))$ of $I_n(h_1)$ and $I_n(h_2)$ in terms of a certain Dirichlet series $D(s,h_1,h_2,E_{n/2+1/2})$, which is similar to the triple convolution product of $h_1, h_2$ and $E_{n/2+1/2}$. We apply our formula to …
In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants w…
Let K be an imaginary quadratic field with discriminant -D, 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 level D with Neben character 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 Hermitian modular forms of degree m constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s,I(m,f)) of I(m,f). This is a generalization of [Y. Mizuno, Manuscripta Math. 119(2006), 175-216].
Miyawaki type lifts are kinds of Langlands functorial lifts and a special case was first conjectured by Miyawaki and proved by Ikeda for Siegel cusp forms. Since then, such a lift for Hermitian modular forms was constructed by Atobe and Kojima , and for half-integral weight Siegel cusp forms by Hayashida, and we constructed Miyawaki type lift for ${\rm GSpin}(2,10)$. Recently Ikeda and Yamana generalized Ikeda type construction and accordingly did Miyawaki type lift for Hilbert cusp forms in a remarkable way. In all these works, a construction of Miyawaki type lift takes two steps as follows: …
Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k-n/2+1/2, let f be the corresponding primitive form of weight 2k-n for SL(2,Z) under the Shimura correspondence, and I(h) the Duke-Imamoglu-Ikeda lift of h to the space of cusp forms of weight k of genus n. Moreover, let FJ(I(h),1) be the first Fourier-Jacobi coefficient of I(h) and s(FJ(I(h),1)) be the cusp form in the generalized Kohnen plus space of weight k-1/2 corresponding to FJ(I(h),1) under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series o…
In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $U(n,n)(A_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_ϕ$ (of an elliptic modular form $ϕ$) on $U(n,n)(A_{\mathbf{Q}})$ proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence …
We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebra and in a tensor product of such algebras. In particular, we formulate and prove Rankin's lemma in genus two. An application to a lifting from (GSp2 \times GSp2) to GSp4 is given using Ikeda-Miyawaki constructions.
We prove a result on the distribution of Hecke eigenvalues, $μ_F(p^r)$ (for $r=1,2$ or $3$) of a non Saito--Kurokawa lift $F$ of degree $2$. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an $F$, which is the best possible in terms of orders of magnitude.
Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.
By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on $E_{7,3}$, called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes the previous results on level one forms. But there are new phenomena in higher levels; first, we can handle non-trivial central characters. Second, the lift depends only on the restriction of the Hecke cusp form to $SL_2$. Hence any twist of the cusp form gives rise to the same lift. However for square free le…
We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory an…
Given a congruence of Hecke eigenvalues between newforms of weight $2$, we prove, under certain conditions, a congruence between corresponding weight-$3/2$ forms.
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
Let $\frak T_2$ (resp. $\mathfrak{T}$) be the Hermitian symmetric domain of $Spin(2,10)$ (resp. $E_{7,3}$). In the previous work, we constructed holomorphic cusp forms on $\mathfrak{T}$ from elliptic cusp forms with respect to $SL_2(\mathbb{Z})$. By using such cusp forms we construct holomorphic cusp forms on $\mathfrak{T}_2$ which are similar to Miyawaki lift in symplectic groups established by T. Ikeda.
Let $f$ be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\mathrm{Sym}^n f$ for every $n \geq 1$. We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over $\mathbb{Q}$.
We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the…
In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy-Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semi-simple descend in the twisted space according to Waldspurger's fundamental work "L'endoscopie tordue n'est pas si tordue".
Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…
Kyoto University（京都大学）；Tokyo Denki University（东京电机大学）；Osaka University（大阪大学）；Hokkaido University（北海道大学）；Muroran Institute of Technology（室兰工业大学）；Tohoku University（东北大学）
Let $f$ and $g$, of weights $k'>k\geq 2$, be normalised newforms for $Γ_0(N)$, for square-free $N>1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $λ\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,\frac{k+k'-2}{2})$. Under certain hypotheses, we prove that $λ$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we a…
Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $χ:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the …
