It is conjectured by Professor Zhi-Wei Sun that for each given odd prime $p>100, $ there always exists an solution $(x,y,z)\in[1,p]^3$ to the Pythagoras equation $x^2+y^2=z^2$ such that $x,y,z$ are quadratic residues or non-residues modulo $p$ respectively (eight cases in total). In this paper, we are able to prove the above assertion for all sufficiently large primes $p$, and the method is based on the recent Burgess bound for character sums of forms in many variables due to Lillian B. Pierce and Junyan Xu.

Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry. In this paper, we consider the problem for complete symmetric polynomials. The homogeneous complete symmetric polynomial of degree $m$ in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h_m(x_1,x_2,\ldots$, $x_k):=\sum_{1\leq i_1\leq i_2\leq \cdots \leq i_m\leq k}x_{i_1}x_{i_2}\cdots x_{i_m}.$ A complete symmetric polynomial of degree $m$ over $\mathbb{F}$_q in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h(x_1,\ldots$, $x_k):=\sum_{e=0}^m a_eh_e(x_1,x_2,\ldots$, $x_k),$ where $a_e\in$ $\mathbb{F}$_q and $a_m\not=0$. Let $N_q(h):= \#\{(x_1,\ldots, x_k)\in$ $\mathbb{F}$_q |$ h(x_1,\ldots, x_k)=0\}$ denote the number of $\mathbb{F}$_q-rational points on the affine hypersurface defined by $h(x_1,\ldots, x_k)=0.$ In this paper, we improve the bounds given in [J. Zhang and D. Wan, "Rational points on complete symmetric hypersurfaces over finite fields", Discrete Mathematics, 343(11): 112072, 2020] and [D. Wan and J. Zhang, "Complete symmetric polynomials over finite fields have many rational zeros" Scientia Sinica Mathematica, 51(10): 1677-1684, 2021]. Explicitly, we obtain the following new bounds:
(1) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 3$ variables over $\mathbb{F}$_q of degree $m$ with $1\leq m\leq q-2$. If $q$ is odd, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-m-1)q^{k-2}.$
(2) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 4$ variables over $\mathbb{F}$_q of degree $m$ with $1\leq m\leq q-2$. If $q$ is even, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-\frac{m+1}{2})(q-1)q^{k-3}.$\newline Note that our new bounds roughly improve the bounds mentioned in the above two papers by the factor $\frac{q^2}{6m}$ for small degree $m$.

In this paper we study some determinants and permanents. In particular, we investigate the new-type determinants $$\det [(i^2+cij+dj^2)^{p-2}]_{0≤ i,j≤ p-1}{and}det [(i^2+cij+dj^2)^{p-2}]_{1≤ i,j≤ p-1} $$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.

In this paper, we explicitly compute the Kodaira-Spencer map over a quaternionic Shimura curve over the field of rational numbers, and also compute its effect on the metrics of the Hodge bundle. The former is based on moduli interpretation and deformation theory, and the latter is based on the theory of complex abelian varieties.

In this note, we confirm a conjecture on the existence of test functions for trilinear zeta integrals with regular support, for representations with maximal exponent strictly less than 1/22.

We prove that there is a positive proportion of positive integers which can be uniquely represented as the sum of a Fibonacci number and a prime. We also study the integers of the form $p+a_k$, where $p$ is a prime and $\{ a_k\}$ is an exponential type sequence of integers.

This article is a survey on some recent developement of the prismatic cohomology theory. We will start with some motivation from classical p-adic Hodge theory, and discuss the origine of the prismatic cohomolgy theory and its basic results. We will then put emphasis on the notion of prismatic crystals, their cohomological properties, and the relationship with the cohomology of classical crystalline crystals.

Manin's conjecture predicts the quantitative behaviour of rational points on algebraic varieties. For a primitive positive definite quadratic form $Q$ with integer coefficients, the equation $x^3=Q(\boldsymbol{y})z$ represents a class of singular cubic hypersurfaces. In this paper, we introduce Manin's conjecture for these hypersurfaces, and describe the ideas, methods, and related results. Generalizations are treated in the last section.

This paper is a survey on mod $p$ Langlands program, with a focus on the history of development and some recent progress in the case of $GL_2$.

We give an introduction to the Vandiver conjecture and some related research in the literature. We show that $A_0=A_2=\cdots=A_{32}=0$, where $A$ is the $p$-Sylow subgroup of the ideal class group of $\mathbb{Q}(\zeta_{p})$. Finally, we propose a new conjecture on the distribution of irregular primes with numerical verifications.

Faltings proposed a $p$-adic analogue of Simpson's correspondence between Higgs bundles on projective complex manifolds and finite dimensional $\mathbb{C}$-representation of the fundamental group. In this paper, we will give an overview of this work and recent progress on finite dimensional $p$-adic representations of the fundamental group of a $p$-adic curve. In the last section, we will briefly discuss some related works.

In this paper, aggregate zero density bounds for a family of automorphic $\mathrm{L}$-functions are deduced from bounds for a sum of integral power moments of such $\mathrm{L}$-functions. More precisely, let $I$ be a set of certain automorphic representations $\pi$, and let $c(\pi)$ be a non-negative coefficient for each $\pi\in I$ such that $\sum_{\pi\in I}c(\pi)$ converges. Assume that \begin{equation*} \sum_{\pi\in I} c(\pi) \int_T^{T+T^\alpha} \bigg| \mathrm{L}\bigg(\frac12+{\rm i}t,\pi\bigg) \bigg|^{2\ell} dt \ll_\varepsilon T^{\theta+\varepsilon} \sum_{\pi\in I} c(\pi) \end{equation*} for certain $\ell\geq1$, $0<\alpha\leq1$ and $\theta\geq\alpha$. Upper bounds for the following aggregate zero density \begin{equation*} \sum_{\pi\in I} c(\pi) N_\pi(\sigma,T,T+T^\alpha) \end{equation*} will be proved, where $N_\pi(\sigma,T_1,T_2)$ is the number of zeros $\rho=\beta+{\rm i}\gamma$ of $\mathrm{L}(s,\pi)$ in $\sigma<\beta<1$ and $T_1\leq\gamma\leq T_2$.

We extend the definition of central strong approximation with Brauer- Manin obstruction which is valid for all singular varieties. We show that a variety defined by a polynomial represented by an isotropic binary quadratic form satisfies central strong approximation with Brauer-Manin obstruction by explicit blowing-up. This is the last case of the whole generalization of Watson’s results about Diophantine equations reducible to quadratics.

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell, and obtain an explicit formula on the LusztigVogan map from the coweight lattice to the set of dominant coweights.

We survey some recent developments of the restriction problems for real untiary group. In particular we briefly explain a proof of the local Gan-Gross-Prasad conjecture for real unitary groups.

We demonstrate the importance of duality and traces in symmetric monoidal categories through a series of concrete examples. In particular, we give an introduction to a new application in ′etale cohomology: the characterization of universal local acyclicity and the relative Lefschetz-Verdier trace formula.