The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant function. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type L-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special derivative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.

In this paper we prove the \pm-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at p such that a_p = 0. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0.

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We study a partially linearized version of the relative trace formula for the arithmetic Gan--Gross--Prasad conjecture for the unitary group U(V). The linear factor in this relative trace formula provides an SL_2-symmetry which allows us to prove by induction the arithmetic fundamental lemma over Q_p when p is odd and p ≥ dim V.

In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors.