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.

8(1):187-187

The study of the present paper is inspired by Gangl, Kaneko and Zagier's result of the connection with double zeta values and modular forms. We introduce double Eisenstein series $E_{r,s}$ in positive characteristic with double zeta values $\zeta_A(r,s)$ as their constant term and compute the t-expansions of the double Eisenstein series. Moreover, we derive the shuffle relations of double Eisenstein series which match the shuffle relations of double zeta values in \cite{Huei2015}.

The estimate of integral points in an n-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third named author formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n  3) real right-angled simplices. The previous results on the conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. However, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [22]. In this paper, on the one hand, the similar method has been applied to prove the conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a signicant progress since it sheds light on proving the Yau Number Theoretic Conjecture in full generality. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.

In the classical theory of multiple zeta values (MZV's), Furusho proposed a conjecture asserting that the p-adic MZV's satisfy the same Q-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho's conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations.

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In 1962 Sprind\v{z}uk proved Mahler's conjecture in both the real and complex cases. Baker gave a generalized result by using a modified version of Sprind\v{z}uk's method. Later, Baker and Schmidt derived the Hausdorff dimensions of sets which are defiend in terms of approximation by algebraic numbers of bounded degrees by using Baker's theorem. In this article we will prove two analogue theorems in the fields of formal power series over finite finite fields.

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.

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2$s$, with $${1\leqslant s\leqslant k^{2}-1}$$ . In this way, we prove the main conjecture for such even moments when $${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}$$ , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of$s$, these finding application to the asymptotic formula in Waring’s problem.

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