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preprint2020arXiv

Quantitative non-divergence and Diophantine approximation on manifolds

The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.

preprint2020arXiv

Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$.

preprint2020arXiv

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times θ_χ)$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $θ_χ$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $χ$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to \emph{any} family whose associated matrix coefficients have such a decay property.

preprint2020arXiv

A nonlinear version of the Newhouse thickness theorem

Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $τ(C_1)\cdot τ(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $τ(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset \mathbb{R}, i=1,\cdots, d$, be some Cantor sets (perfect and nowhere dense) with convex hull $[0,1]$. Suppose $f(x_1,\cdots, x_{d-1},z)\in \mathcal{C}^1$ is a continuous function defined on $\mathbb{R}^d$. Denote the continuous image of $f$ by $$f(K_1,\cdots, K_d)=\{f(x_1, \cdots x_{d-1},z):x_i\in K_i,z\in K_d, 1\leq i\leq d-1\}.$$ If for any $(x_1, \cdots, x_{d-1},z)\in [0,1]^d$, we have $$(τ(K_i))^{-1}\leq \left|\dfrac{\partial_{x_i} f}{\partial_z f}\right|\leq τ(K_d),1\leq i\leq d-1$$ then $f(K_1,\cdots, K_d)$ is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.

preprint2020arXiv

Thue inequalities with few coefficients

Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll sm^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.

preprint2020arXiv

The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory I

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of $\mathrm{Spec}\,\mathbb{Z}$. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim's conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim's conjecture.

preprint2020arXiv

Projective spaces as orthogonal modular varieties

We construct $16$ reflection groups $Γ$ acting on symmetric domains $\mathcal{D}$ of Cartan type IV, for which the graded algebras of modular forms are freely generated by forms of the same weight, and in particular the Satake--Baily--Borel compactification of $\mathcal{D} / Γ$ is isomorphic to a projective space. Four of these are previously known results of Freitag--Salvati Manni, Matsumoto, Perna and Runge. In addition we find several new modular groups of orthogonal type whose algebras of modular forms are freely generated.

preprint2020arXiv

Some implications of the $2$-fold Bailey lemma

The $2$-fold Bailey lemma is a special case of the $s$-fold Bailey lemma introduced by Andrews in 2000. We examine this special case and its applications to partitions and recently discovered $q$-series identities. Our work provides a general comparison of the utility of the $2$-fold Bailey lemma and the more widely applied $1$-fold Bailey lemma. We also offer a discussion of the $spt_M(n)$ function and related identities.

preprint2020arXiv

Uniqueness and two shared set problems of L-Function and certain class of meromorphic function

Starting with a question of Yuan-Li-Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58(2)(2018), 249-262] we have studied the uniqueness of a meromorphic function f and an L-function L sharing two finite sets. At the time of execution of our work, we have pointed out a serious lacuna in the proof of a recent result of a of Sahoo-Halder [ Some results on L-functions related to sharing two finite sets, Comput. Methods Funct. Theo., 19(2019), 601-612] which makes most of the part of the Sahoo-Halder's paper under question. In context of our choice of sets, we have rectified Sahoo-Halder's result in a convenient manner.

preprint2020arXiv

The Chabauty--Coleman method and p-adic linear forms in logarithms

Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mordell-Weil generators of its Jacobian. The proof uses the Chabauty-Coleman method, together with effective methods in transcendence theory. We also discuss generalisations to the Chabauty-Kim method.

preprint2020arXiv

A Bessel $δ$-method and hybrid bounds for $\mathrm{GL}_2$

Let $g$ be a primitive holomorphic or Maass newform for $Γ_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $δ$-identity associated to $g$. Among other applications, we prove the following hybrid subconvexity bound $$ L\left(1/2+it,g\otimes χ\right)\ll_{g,\varepsilon} (q(1+|t|))^{\varepsilon}q^{3/8}(1+|t|)^{1/3} $$ for any $\varepsilon>0$, where $χ\bmod q$ is a primitive Dirichlet character with $(q, D)=1$. This improves the previous known result.

preprint2020arXiv

$k$--Fibonacci numbers with two blocks of repdigits

A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we find all $k$--Fibonacci numbers that are concatenations of two repdigits. This generalizes prior results which dealt with the above problem for the particular cases of Fibonacci and Tribonacci numbers.

preprint2020arXiv

The density of rational lines on hypersurfaces: A bihomogeneous perspective

Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$ variables. We give an asymptotic formula of the pairs of integer points $(\mathbf x, \mathbf y)$ with $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ which generate a line lying in the hypersurface defined by $F$, provided that $n > 2^{d-1}d^4(d+1)(d+2)$. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of $X$ and $Y$.

preprint2020arXiv

On Cycles of Generalized Collatz Sequences

We explore the cycles and convergence of Generalized Collatz Sequence, where $3n+1$ in original collatz function is replaced with $3n+k$. We present a generating function for cycles of GCS and show a particular inheritance structure of cycles across such sequences. The cycle structure is invariant across such inheritance and appears more fundamental than cycle elements. A consequence is that there can be arbitrarily large number of cycles in some sequences. GCS can also be seen as an integer space partition function and such partitions along with collatz graphs are inherited across sequences. An interesting connection between cycles of GCS and certain exponential Diophantine equations is also presented.

preprint2020arXiv

Möbius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields

We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the Pólya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $\mathbb{F}_q[u]$-analog of Chowla's conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $\mathbb{F}_q[u]$.

preprint2020arXiv

Optimal mean value estimates beyond Vinogradov's mean value theorem

We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjectured conclusions. This is the first occasion on which bounds of this quality have been attained for Diophantine systems not of Vinogradov type. As a consequence of this progress, whenever $u \ge 3v$ we obtain the Hasse principle for systems consisting of $v$ cubic and $u$ quadratic diagonal equations in $6v+4u+1$ variables, thus attaining the convexity barrier for this problem.

preprint2020arXiv

Elliptic curves with non-abelian entanglements

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1]) \cap K(E[m_2])$ is a non-abelian extension of $K$. There is an infinite set $\mathcal{S}$ of modular curves whose $K$-rational points capture all elliptic curves over $K$ without complex multiplication that have this property. Our main theorem explicitly describes the (finite) subset of $\mathcal{S}$ consisting of those modular curves having genus zero. In the case $K = \mathbb{Q}$, this has applications to the problem of determining when the Galois representation on the torsion of $E$ is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example.

preprint2020arXiv

On the model theory of higher rank arithmetic groups

Let $Γ$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $Γ$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $Γ$ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first order theory of $Γ$ is undecidable, that all finitely generated subgroups of $Γ$ are definable, and that $Γ$ is characterized by a single first order sentence among all finitely generated groups.

preprint2020arXiv

On an inequality of Bushnell--Henniart for Rankin--Selberg conductors

We prove a division algebra analogue of an ultrametric inequality of Bushnell--Henniart for Rankin--Selberg conductors. Under the Jacquet--Langlands correspondence, the two versions are equivalent.

preprint2020arXiv

On the $k$-generalized Fibonacci numbers with negative indices

In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the absolute value of the second real root of $T_k(x)$ is minimal among the roots. Combining this with a deep result of Bugeaud and Kaneko \cite{BK} we prove that there are only finitely many perfect powers in $(F_n^{(k)})_{n\in \Z}$, provided $k$ is even. Another consequence is that, if $k$ and $l$ denote even integers then the equation $F_m^{(k)} = \pm F_n^{(l)}$ has only finitely many effectively computable solutions in $(n,m)\in \Z^2$. In the case $k=l=4$ we establish all solutions of this equation.

preprint2020arXiv

Basic and bibasic identities related to divisor functions

Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.

preprint2020arXiv

Classical Iwasawa theory and infinite descent on a family of abelian varieties

For primes $q \equiv 7 \mod 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field $K = \mathbb{Q}(\sqrt{-q})$, which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.

preprint2020arXiv

Legendre Symbol of $\prod f(i,j) $ over $ 0<i<j<p/2, \ p\nmid f(i,j) $

Let $p>3$ be a prime. We investigate Legendre symbol of $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ , where $4c(a+b+c)k \equiv (4ac-b^2)\pmod{p}.$ Parallel results are given for $\displaystyle \prod_{i,j=1 \atop p\nmid f(i,j) }^{(p-1)/2} \bigg(\frac{ f(i,j) }{p}\bigg).$ Then we show that $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ can be evaluated explicitly when k=2,4,5,9,10 or k is a square. And for several classes of f(i,j) these two kinds of products can be evaluated explicitly. Finally when f is a linear form we give unified identities for these products. Thus we prove these kind of problems raised in Sun's paper.

preprint2020arXiv

The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes

The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although we believe that the automorphism group of an elliptic curve over a finite field is well known in the community, we could not find this in the literature. Nevertheless, in this paper we show the group structure of the automorphism group of an elliptic curve over a finite field. More importantly, we characterize subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field. Despite of theoretical interest on this topic, our research is largely motivated by constructions of optimal locally repairable codes. The first research to make use of automorphism group of function fields to construct optimal locally repairable codes was given in a paper \cite{JMX20} where automorphism group of a projective line was employed. The idea was further generated to an elliptic curve in \cite{MX19} where only automorphisms fixing the point at infinity were used. Because there are at most $24$ automorphisms of an elliptic curve fix