The Second Term of an Eisenstein Series

The leading term of Eisenstein series and L-functions are well-known to be extremely important and to have very important arithmetic implications. For example, one may consider the analytic class number formula, The Siegel-Weil formula, the Birch and Swinnerton-Dyer conjecture, Stark’s conjectures, and Beilinson’s and Bloch-Kato conjectures. These are well-documented and are central in current number theory research. In this note, we would like to indicate, via examples, that the second term of an Eisenstein series or an L-function at a critical point is also very interesting and encodes important arithmetic. We start with Kronecker’s first limit formula in section one. Then we turn to some recent work of Kudla, Rapoport, and the author in sections 2, 3, and 5. The basic theme is that both the leading and the second term of certain Eisenstein series at some (critical) point are generating functions of some interesting arithmetic data. In section 2, we explain that the classical Eisenstein series of weight 1 associated to an imaginary quadratic field has a symmetric center s = 0. Its central value is the generating function of the counting function of integral ideals of given norm. Its central derivative is related to the CM elliptic curves with extra endomorphism which anti-commutes with the CMs ([KRY1], [Ya1]). In section 3, we explain that Zagier’s famous Eisenstein series of weight 32 can be viewed as the generating function of the degree of Heegner cycles on the modular curve and that its ‘derivative’ is the generating function of Faltings height of these Heegner cycles ([KRY3], [Ya2]). Here s = 1 2 is not the symmetric center of the Eisenstein series. In section 5, we extend this result to the case of Shimura curves, and give a brief summary of the main theorem in [KRY2]. Section 4 gives an introduction to the construction of Eisenstein series in adelic language, which is needed in section 5 and is used in all our work. In the last section, we briefly summarize a recent work of Kudla on the integral of a Borcherds’ modular form, where one again sees the importance of the derivative of Eisenstein series at a critical point where the value does not vanish. We are content with examples in this note, and refer to [Ku1-2] for the general framework and conjectures on this subject.