CyberSec Research

In this note, using Borger's theory of periodic Witt vectors, we construct integral refinements of the arithmetic subalgebras associated with Bost-Connes systems for general number fields.

We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any natural notion of length.

We present a special class of examples of automorphic lifts of multiple tensor products of automorphic representations in the sense of matching $L$-functions, motivated by combinatorial identities for Schur polynomials and a celebrated result of Newton and Thorne.

Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds $h!n \; \lesssim \; F_h(n)^h \; \lesssim \; h^h n$ are well-known, comparatively little has been established for $h>2$. In this paper, we make significant improvements to both of the best-known bounds on $F_h(n)$ for sufficiently large $h$. For the lower bound, we use a probabilistic approach along with the Berry-Esseen Theorem to improve upon the best-known asymptotic result due to Yu. We also establish the first nontrivial asymptotic upper bound on $F_h(n)$ by leveraging a construction for additive bases of finite cyclic groups due to Jia and Shen. In particular, we show that given any $\epsilon>0$, for sufficiently large $h$, we have \[ \left(\frac{1}{2}-\epsilon\right)h!\sqrt{2\pi e} n\; \leq \; F_h(n)^h \; \leq \; \left(\left(\frac{\sqrt{3}}{2}+\epsilon\right)h\right)^h n. \]

Let ${\pmb b}=\{b_0,\,b_1,\,\ldots\}$ be the known sequence of numbers such that $b_0\neq0$. In this work, we develop methods to find another sequence ${\pmb a}=\{a_0,\,a_1,\,\ldots\}$ that is related to ${\pmb b}$ as follows: $a_n=a_0\,b_{n+m}+a_1\,b_{n+m-1}+\ldots+a_{n+m}\,b_0$, $n\in\mathbb{N}\cup\{0\}$, $m\in\mathbb{N}$. We show the connection of $\lim_{n\to\infty}a_n$ with $a_0,\,a_1,\,\ldots,\,a_{m-1}$ and provide varied examples of finding the sequence ${\pmb a}$ when ${\pmb b}$ is given. We demonstrate that the sequences ${\pmb a}$ may exhibit pretty patterns in the plane or space. Also, we show that the properly chosen sequence ${\pmb b}$ may define ${\pmb a}$ as some famous sequences, such as the partial sums of the Riemann zeta function, etc.

We introduce a special class of multiple Dirichlet series whose terms are supported on a variety and which admit an Euler product structure. We show that these series arise naturally from twisted moments of automorphic \( L \)-functions associated with Dirichlet twists. We proposed several conjectures on the analytic properties of these series.

There exist numerous results in the literature proving that within certain families of totally real number fields, the minimal rank of a universal quadratic lattice over such a field can be arbitrarily large. Kala introduced a technique of extending such results to larger fields -- e.g. from quadratic fields to fields of arbitrary even degree -- under some conditions. We present improvements to this technique by investigating the structure of subfields within composita of number fields, using basic Galois theory to translate this into a group-theoretic problem. In particular, we show that if totally real number fields with minimal rank of a universal lattice $\geq r$ exist in degree $d$, then they also exist in degree $kd$ for all $k\geq3$.

Arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are obtained by using the cyclotomic classes of order 2 with respect to $n=p^sq^t$, where $p\equiv3 \mathrm{mod} 4$, $\gcd(\phi(p^s),\phi(q^t))=2$, $l$ is a primitive root modulo $q^t$ and $\mathrm{ord}_{p^s}(l)=\phi(p^s)/2$. The form of these cyclotomic classes enables us to further generalize the results obtained in \cite{ref1}. The explicit expressions of primitive idempotents of minimal ideals in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are also obtained.

We show that a totally degenerate limit of discrete series representation admits a choice of n cohomology group that is nonvanishing at a canonically defined degree. We then show that the combinatorial complexes used by Soergel to compute these cohomology groups satisfies Serre duality. We conclude that this produces two n cohomology groups, each for a totally degenerate limit of discrete series of U(n+1) and U(n), which are nonvanishing at the same degree. This suggests Gan Gross Prasad type branching laws for the TDLDS of unitary groups of any rank.

We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering numbers possesses a natural density $d(\mathcal{C})$ and prove that $0.103230 < d(\mathcal{C}) < 0.103398.$ Our approach adapts methods developed by Behrend and Del\'eglise for bounding the density of abundant numbers, by introducing a function $c(n)$ that measures how close an integer $n$ is to being a covering number with the property that $c(n) \leq h(n) = \sigma(n)/n$. However, computing $d(\mathcal{C})$ to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for $d(\mathcal{A})$, the density of abundant numbers, namely $0.247619608 < d(\mathcal{A}) < 0.247619658$. We also show the count of primitive covering numbers up to $x$ is $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + \epsilon\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers.

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new results towards the Bloch--Kato conjecture in analytic rank one, assuming that the fixed prime $p$ is inert in the relevant imaginary quadratic field.

We study statistical properties of Fourier coefficients of automorphic forms on GL(n). For most Hecke-Maass cusp forms, we give the asymptotic number of nonvanishing coefficients, show that there is a positive proportion of sign changes among them, when these are real, and describe the asymptotic density of these signs. We generalize the results by J\"a\"asaari obtained in the case of self-dual forms of GL(3) and our method moreover circumvents the assumption of the Generalized Ramanujan Conjecture.

Let \( \pi \) be a polarized, regular algebraic, cuspidal automorphic representation of \( \GL_n(\bb{A}_F) \) where \( F \) is totally real or imaginary CM, and let \( (\rho_\lambda)_\lambda \) be its associated compatible system of Galois representations. We prove that if \( 7\nmid n \) and \( 4 \nmid n \) then there is a Dirichlet density \( 1 \) set of rational primes \( \mc{L} \) such that whenever \( \lambda\mid \ell \) for some \( \ell\in \mc{L} \), then \( \rho_\lambda \) is irreducible.

Hypergeometric sequences obey first-order linear recurrence relations with polynomial coefficients and are commonplace throughout the mathematical and computational sciences. For certain classes of hypergeometric sequences, we prove linear growth estimates on their Weil heights. We give an application of our effective results towards the Membership Problem from Computer Science. Recall that Membership asks to procedurally determine whether a specified target is an element of a given recurrence sequence.

We investigate the structure of the higher Chow groups $CH^2(E,1)$ for an elliptic curve $E$ over a global function field $F$. Focusing on the kernel $V(E)$ of the push-forward map $CH^2(E,1)\to F^{\times}$ associated to the structure map $E\to \operatorname{Spec}(F)$, we analyze the torsion part $V(E)$ based on the mod $l$ Galois representations associated to the $l$-torsion points $E[l]$.

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for prime discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\operatorname{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.

We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of $\mathbb F_{p^n}\setminus \mathbb F_p$ into disjoint cycles of even length. This allows a natural interpretation of the cycles as permutation cycles. Finally, we explore chaining under more than two polynomial bases.

In this short note, we prove the equivalence of Grothendieck-Katz $p$-curvature Conjecture with Conjecture F in Ekedahl-Shepherd-Barron-Taylor. More precisely, we show that Conjecture F implies the $p$-curvature conjecture, and that the $p$-curvature Conjecture implies Conjecture F for the foliation attached to a vector bundle with integrable connection.