数论

We study families of abelian varieties over smooth proper curves with small $l$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an…

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the…

Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^\omega$ is defined by \[ T (x_1, x_2, \ldots, x_\nu, \ldots) = (x_1 + \alpha, x_2 + h(x_1), \ldots, x_\nu + h(x_1 + (\nu-2)\beta), \ldots) \] with $\alpha\in \mathbb{R}$…

Extending our work on the $k$-tuple conjecture, we apply those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. We track the evolution of these $58$ counterexamples from inadmissible…

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This…

In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup $\mathbb{Z}/N\mathbb{Z}$. We show that, via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group…

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can…

The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant…

The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\rho=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx…

Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<\alpha<\beta<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant…

In 1878, Sylvester proved Cayley's Conjecture that the coefficients of the Gaussian $q$-binomial coefficients are unimodal. In 1990, O'Hara famously discovered a constructive combinatorial proof, and in 2013, Pak and Panova proved the…

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324 \ldots$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. This…

We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an $\operatorname{SL}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions…

The asymptotic behavior of the mean values of multiple zeta functions is of significant interest due to its close connection with the Riemann zeta function. In this paper, we establish asymptotic behavior of the mean square values of Barnes…

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$…

We describe the local-global compatibility of local Plancherel measures and the Tamagawa measure under the Jacquet-Langlands correspondence. We apply the notion of densities of modules over a discrete group, which generalizes the dimensions…

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro…

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote…

For any $\sigma$ with $0\leq \sigma\leq 1$ and any $T>10$ sufficiently large, let $N_{\zeta}(\sigma,K,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\zeta_{K}(s)$ with $|\gamma|\leq T$ and $\beta\geq \sigma$ and the zero being counted…