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arXiv1999

Lattices and codes with long shadows

In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".

math.NT↗
arXiv1999

The canonical arithmetic height of subvarieties of an abelian variety over a finitely generated field

This paper is the sequel of our paper "Arithmetic height functions over finitely generated fields" (cf. math.NT/9809016). In this paper, we define the canonical height of subvarieties of an abelian variety over a finitely generated field over Q. We also prove that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point.

math.NT↗
arXiv2002

On isomorphisms between deformation rings and Hecke rings

We give a different approach to the well-known modularity lifting results of Wiles and Taylor. Instead of Taylor-Wiles systems we use a Galois cohomological discovery of R. Ramakrishna. This paper is 2 years older than math.NT/0210296 where another approach to modularity lifting results was presented.

math.NT↗
arXiv2004

Tame class field theory for arithmetic schemes

We extend the unramified class field theory for arithmetic schemes of K. Kato and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme and let $D$ be a divisor on $X$ whose vertical irreducible components are normal schemes. Theorem: There exists a natural reciprocity isomorphism \[ \rec_{X,D}: \CH_0(X,D) \liso \tilde π_1^t(X,D)^\ab\. \] Both groups are finite. This paper corrects and generalizes my paper "Relative K-theory and class field theory for arithmetic surfaces" (math.NT/0204330)

math.NT↗
arXiv2006

Diophantine approximation on planar curves: the convergence theory

The convergence theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.

math.NT↗
arXiv2007

Renormalization of Multiple $q$-Zeta Values

In this paper we shall define the renormalization of the multiple $q$-zeta values (M$q$ZV) which are special values of multiple $q$-zeta functions $ζ_q(s_1,...,s_d)$ when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang (math.NT/0606076v3) on the renormalization of Euler-Zagier multiple zeta values. We show that our renormalization process produces the same values if the M$q$ZVs are well-defined originally and that these renormalizations of M$q$ZV satisfy the $q$-stuffle relations if we use shifted-renormalizations for all divergent $ζ_q(s_1,...,s_d)$ (i.e., $s_1\le 1$). Moreover, when $\qup$ our renormalizations agree with those of Guo and Zhang.

math.NT↗
arXiv2024

Vanishing properties of Kloosterman sums and Dyson's conjectures

In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$.

math.NT↗
arXiv2011

Criterion for the integrality of the Taylor coefficients of mirror maps in several variables

We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series $q_i({\mathbf z})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$, with ${\mathbf z}=(z_1,...,z_d)$ and where $F({\mathbf z})$ and $G_i({\mathbf z})+\log(z_i)F({\mathbf z})$, $i=1,...,d$ are particular solutions of certain A-systems of differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated with the sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in "Critère pour l'intégralité des coefficients de Taylor des applications miroir" [J. Reine Angew. Math.]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on the formal congruences between formal series, proved by Krattenthaler and Rivoal in "Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps" [arXiv:0804.3049v3, math.NT]. This criterion involves the integrality of the Taylor coefficients of new univariate mirror maps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin.

math.NT↗
arXiv2006

Summation of Series Defined by Counting Blocks of Digits

We discuss the summation of certain series defined by counting blocks of digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (γ+ \log \frac{4}π)/2$. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.

math.NT↗
arXiv2013

Sur la théorie spectrale des métriques intégrables sur une surface de Riemann compacte

We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-integrable metrics. As an application, we recover the following identity: [ζ'_{Δ_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,ω_\infty); \bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in arXiv:1301.1792 [math.NT].

math.DG↗
arXiv2025

Classification of Finite Groups With Equal Left and Right Quotient Sets

In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A \subseteq G$, we have $|AA^{-1}| = |A^{-1}A|$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].

math.GR↗
arXiv2005

A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity

Let K be a field and F denote the prime field in K. Let \tilde{K} denote the set of all r \in K for which there exists a finite set A(r) with {r} \subseteq A(r) \subseteq K such that each mapping f:A(r) \to K that satisfies: if 1 \in A(r) then f(1)=1, if a,b \in A(r) and a+b \in A(r) then f(a+b)=f(a)+f(b), if a,b \in A(r) and a \cdot b \in A(r) then f(a \cdot b)=f(a) \cdot f(b), satisfies also f(r)=r. Obviously, each field endomorphism of K is the identity on \tilde{K}. We prove: \tilde{K} is a countable subfield of K, if char(K) \neq 0 then \tilde{K}=F, \tilde{C}=Q, if each element of K is algebraic over F=Q then \tilde{K}={x \in K: x is fixed for all automorphisms of K}, \tilde{R} is equal to the field of real algebraic numbers, \tilde{Q_p}={x \in Q_p: x is algebraic over Q}.

math.NT↗
arXiv1998

Mordell-Lang plus Bogomolov

We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A is isogenous to the product of an abelian variety and a torus. The proof makes use of the Mordell-Lang conjecture, the Bogomolov conjecture, and an equidistribution theorem.

math.NT↗
arXiv2016

Explicit estimates for the zeros of Hecke $L$-functions

Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates.

math.NT↗
arXiv2021

Fields of dimension one algebraic over a global or local field need not be of type $C_{1}$

Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ satisfying the following: (i) $E$ has dimension dim$(E) \le 1$, i.e. the Brauer group Br$(E ^{\prime })$ is trivial, for every algebraic extension $E ^{\prime }/E$; (ii) finite extensions of $E$ are not $C _{1}$-fields. This, applied to the maximal algebraic extension $K$ of the field $\mathbb{Q}$ of rational numbers in the field $\mathbb{Q} _{p}$ of $p$-adic numbers, for a given prime $p$, proves the existence of an algebraic extension $E _{p}/\mathbb{Q}$, such that dim$(E _{p}) \le 1$, $E _{p}$ is not a $C _{1}$-field, and $E _{p}$ has a Henselian valuation of residual characteristic $p$.

math.NT↗
arXiv2021

Sidon sets for linear forms

Let $φ(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h $ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $φ$-Sidon set if, for all $h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $ (a'_1,\ldots, a'_h) \in A^h$, the relation $φ(a_1,\ldots, a_h) = φ(a'_1,\ldots, a'_h)$ implies $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for the linear form $φ$ if and only if the set of coefficients of $φ$ has distinct subset sums. In a normed vector space with $φ$-Sidon sets, every infinite sequence of vectors is asymptotic to a $φ$-Sidon set of vectors. Results on $p$-adic perturbations of $φ$-Sidon sets of integers and bounds on the growth of $φ$-Sidon sets of integers are also obtained.

math.NT↗
arXiv2025

Inverse problems for sumset sizes of finite sets of integers

Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $( |hA| )_{h=1}^{\infty}$ and $( |hA| )_{h=1}^{\infty}$.

math.NT↗