2023/06/13 by Hiraku Atobe, Masataka Chida, Atobe, Hiraku +7
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2306.07582
Let f be a primitive form of weight 2k+j-2 for SL2(Z), and let P be a prime ideal of the Hecke field of f. We denote by Spm(Z) the Siegel modular group of degree m. Suppose that k is congruent to 0 modulo 4, j is congruent to 0 modulo 4, and that P divides the algebraic part of L(k+j,f). Put \bf k=(k+j/2,k+j/2,j/2+4,j/2+4). Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform F in the space of modular forms of weight (k+j,k) for Sp2(Z) such that [I2(f)]\bf k is congruent to A(I)4(F) modulo P. Here, [I2(f)]\bf k is the Klingen-Eisenstein lift of the Saito-Kurokawa lift I2(f) of f to the space of modular forms of weight \bf k for Sp4(Z), and A(I)4(F) is a certain lift of F to the space of cusp forms of weight \bf k for Sp4(Z). As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of F and some quantities related to the Hecke eigenvalues of f. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.