We calculate a G_2-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group E_6, and relate this period to the Ginzburg-Rallis period of cusp forms on GL_6. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of GL_6.

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we prove a local trace formula for the Ginzburg-Rallis model. By applying this trace formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel work of [Wan15] in which we proved the geometric side of the trace formula.

We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and are related to local exterior square L-functions.

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For each positive characteristic multiple zeta value defined by Thakur, the first and third authors constructed a t-module such that a certain coordinate of a logarithmic vector of a specified algebraic point is a rational multiple of that multiple zeta value. The main result in this paper gives explicit formulae for all of the coordinates of this logarithmic vector in terms of Taylor coefficients of t-motivic multiple zeta values and t-motivic Carlitz multiple star polylogarithms.

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg L-functions and the Godement-Jacquet L-functions associated to automorphic cuspidal representations over global function fields.

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In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer n to be a congruent number in terms of the Legendre symbols for the prime factors of n. Our criterion generalizes previous results of Heegner, Birch–Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate–Shafarevich number $\mathcal{L}(n)$ in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross–Zagier formula of Yuan–Zhang–Zhang.

In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.