Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 112 JENS FUNKE In Proposition we will give an extension of Theorem to F having logarithmic singularities inside D. By the usual unfolding argument see BF06 section 4 we have Lemma . Let N 0 or N 0 such that N ị Qx 2. Then aN v E i F z 0 VvX z . xep Ln rx d If F is smooth on X then by Theorem we obtain aN v tF N E - L i ddcF z 0 ựVX z N 0 XE1 Ln lrx D aN v E ddcF z e VVX z N 0 N ị Qx 2 xep Ln x d For N m2 unfolding is typically not valid since in that case rx is trivial. In the proof of Theorem in BF06 we outline Lemma . Let N m2. Then aN v E i dlF z Ede0 VvX 7z x L 2niJv . .d dF z Eeũ VVX Yz I -- M Yep Ồ í SdỖF z E - X . -- M Y p Note that with our choice of the particular lattice L in we actually have r L-m2 m and as representatives we can take m _2m k 0 . m 1 . Finally we have a0 v y F z E T vX z . We split this integral into two pieces a 0 for X 0 and a v a0 v a 0 for X 0. However unless F is at most mildly increasing the two individual integrals will not converge and have to be regularized in a certain manner following Bor98 BF06 . For a 0 v we have only one T-equivalence class of isotropic lines in L since r has only one cusp. We denote by 0 QX0 the isotropic line spanned by the primitive vector in L X0 02 . Note that the pointwise stabilizer of 0 is rTO the usual parabolic subgroup of r. We obtain Lemma . a0 -L r 9 F zH 2n J M CM POINTS AND WEIGHT 3 2 MODULAR FORMS 113 a0 v -L ỉreB 4 F z E Ể dỆ VVnX0 yz I n Jm yeĩ rn -x J Ồ re3 4dF z E Ể Ệ VVnX0 yz I n Jm yer rn -x J - 21 rt ddF z E ỉ Ệ0 VVnX0 yz . n Jm Here 2 indicates that the sum only extends over n 0. 3. The lift of modular functions . The lift of the constant function. The modular trace of the constant function F 1 is already very interesting. In that case the modular trace of index N is the geometric degree of the 0-cycle Z N t1 N deg Z N E 1. X L N I X I For p 1 this is twice the famous Kronecker-Hurwitz class number H N of positive definite binary .
