Academic literature on the topic 'Saito–Kurokawa lifts'

AbstractIn this paper we obtain special value results for L-functions associated to classical and paramodular Saito–Kurokawa lifts. In particular, we consider standard L-functions associated to Saito–Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on GL(2). The results are obtained by combining classical and representation theoretic arguments.

In this paper, we show how one can use an inner product formula of Heim giving the inner product of the pullback of an Eisenstein series from Sp10 to Sp 2 × Sp 4 × Sp 4 with a new-form on GL2 and a Saito–Kurokawa lift to produce congruences between Saito–Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp 4 × GL 2. The congruence is then used to produce nontrivial torsion elements in an appropriate Selmer group, providing evidence for the Bloch–Kato conjecture.

博士
國立臺灣大學
數學研究所
106
We give explicit pullback formulas for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulas provide non-trivial examples for the refined Gross-Prasad conjecture for (SO(5), SO(4)) in the non-tempered cases. As an application, we obtain new cases for Deligne’s conjecture for central critical values of certain automorphic L-functions for GL(3) × GL(2).

In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic newforms of integral weight, which says that if two such forms f1, f2 have the same eigenvalues of the p-th Hecke operator Tp for almost all primes p, then f1 = f2. The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree 2. These objects have a Fourier expansion indexed by certain matrices of size 2 over an imaginary quadratic field. We show that Hermitian cusp forms of weight k for the Hermitian modular group of degree 2 are determined by their Fourier coe cients indexed by matrices whose determinants are essentially square–free. Moreover, we give a quantitative version of the above result. is is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. is result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from GSp(4) to GL(4). We expect similar applications. We also discuss few results on the square–free Fourier coefficients of elliptic cusp forms. In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms li ed from classical elliptic modular forms on the upper half plane H. If g is such an elliptic modular form of integral weight k on SL(2, Z) then we consider its Saito–Kurokawa li Fg and certain ‘restricted’ L2-norm N(Fg ) (which we refer to as the mass) associated with it. Pullback of a Siegel modular form F (( z z ¨ )) to H × H is its restriction to z = 0, which we denote by F |z=0. Conjectures of Ikeda relate such pullbacks to central values of L-functions. In fact, when a Siegel modular form arises as a Saito–Kurokawa li (say F = Fg ), results of Ichino relate the pullbacks to the central values of certain GL(3)×GL(2) L-functions. Moreover, it has been observed that comparison of the (normalized) norm of Fg with the norm of its pullback provides a measure of concentration of Fg along z = 0. We use the amplification method to improve the currently known bound for N(Fg ).