tailieunhanh - Analytic Number Theory A Tribute to Gauss and Dirichlet Part 11

Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 192 KEN ONO where A p f z e l ma mn m n 1 Ẹ xEZ x2 m2p mod 2 x2 m2p q 5 xEZ x m mod 2 x2 m2 p q 4 V B ĩ . z 2e Ơ n Ơ n hi -n w x2. J- f z z t 2_ U1 n I Ơ1 n a n v q n 1 xEZ and where e 1 2 for 1 and is 1 otherwise. As usual Ơ1 x denotes the sum of the positive divisors of x if x is an integer and is zero if x is not an integer. Bringmann Rouse and the author have shown BOR05 that these generating functions are also modular forms of weight 2. In particular we obtain a linear map d. . Ao r0 A I P where the map is defined for the subspace of those functions with constant term 0 . Theorem . Bringmann Ono and Rouse Theorem of BOR05 Suppose that p 1 mod 4 is prime and that 1 or is an odd prime with p -1- If f z En -TO a n qn e A0 rg with a 0 0 then the npnprfitinn fq i nP f qci n J p Í 0 id qTi X. Í r I T y tí 2 Ì I yenei a vi ny J un io ei f z s on J V 12 I J- 0 P p J In Section 3 we combine the geometry of these surfaces with recent work of Bruinier and Funke BF06 to sketch the proof of Theorem . In this section we characterize these modular forms T p z when f z J1 z j z 744. In T s J X z -L z X z terms of the classical Weber functions f1 z and f2 z V2 n z n z we have the following exact description. Theorem . Bringmann Ono and Rouse Theorem of BOR05 If p 1 mod 4 is prime then adp rz- n 2z n 2Pz E4 Pz f2 2z 2f2 2Pz 2 t . t .x J z ----------------4 6-----------------f1 4z f2z f1 4Pz f2 Pz d . Although Theorem gives a precise description of the forms ij z it is interesting to note that they are intimately related to Hilbert class polynomials the polynomials given by Hd x n x j T e Z x T ECd where CD denotes the equivalence classes of CM points with discriminant D. Each Hd x is an irreducible polynomial in Z x which generates a class field extension of Q ự D . Define Np z as the multiplicative norm of T1jJ1 z Np z n 1 IM. M eF0 p SL2 Z SINGULAR MODULI FOR MODULAR CURVES AND SURFACES 193 If N z is the normalization of Np z with leading .