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.

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Let N be a fixed integer and f be a holomorphic newform of level q⁠, weight k and trivial nebentypus, where q is a multiple of N⁠. In this article, we prove that the pushforward to the modular curve of level N of the mass measure of f tends weakly to the Haar measure as qk→∞⁠. This generalizes the previous results for modular curve of level 1. The main innovation of this article is to obtain an upper bound for the local integral which cancels the convexity bound of the corresponding L-function in level aspect.

The Colmez conjecture is a formula expressing the Faltings height of an abelian variety with complex multiplication in terms of some linear combination of logarithmic derivatives of Artin L-functions. The aim of this paper to prove an averaged version of the conjecture, which was also proposed by Colmez.

The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity the- ory and number theory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of an isolated singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, in this paper, we prove the Yau Geometric Conjecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six-dimensional simplex. Our new results pave the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also sheds new light on proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.