International Journal of Number Theory

In this paper, we prove a duality formula for the sum of the finite/symmetric multiple zeta-star values of fixed weight, depth, and height, which was conjectured by Kaneko. This result is a generalization of “height-one duality” proved by Hoffman and an analogue of the duality formula for the sum of the multiple zeta-star values proved by Li.

Let $A$ and $B$ be nonempty finite subsets of $\mathbb{Z}$. Freiman’s $3k-4$ Theorem states that if $|A+A|\le 3\left|A\right|-4$, then $A$ is contained in a short arithmetic progression. Freiman generalized his theorem establishing that if $|A+B|\le \left|A\right|+\left|B\right|+min\{|A|,|B|\}-4$, then $A$ and $B$ are contained in short arithmetic progressions with common difference. Take $S\subseteq A\times B$ and write $A\stackrel{S}{+}B=\{a+b:(a,b)\in S\}$. There have been several attempts to generalize Freiman’s statements for restricted sumsets $A\stackrel{S}{+}B$. In the last few years, there have been some results establishing that (under reasonable technical conditions) if $\left|A\stackrel{S}{+}B\right|<\left|A\right|+\left|B\right|+(c-d)min\{|A|,|B|\}$ for an absolute constant $c$ and $d$ such that $d\to 0$ whenever $\frac{|(A\times B)\setminus S|}{|A\times B|}\to 0$, then there are arithmetic progressions $C$ and $D$ with common difference such that $\frac{|A\setminus C|}{\left|A\right|}$ and $\frac{|B\setminus D|}{\left|B\right|}$ are small (in terms of $\frac{|(A\times B)\setminus S|}{|A\times B|}$), $\left|C\right|\le \left|A\stackrel{S}{+}B\right|-\left|B\right|(1-e)$ and $\left|D\right|\le \left|A\stackrel{S}{+}B\right|-\left|A\right|(1-e)$ where $e\to 0$ whenever $\frac{|(A\times B)\setminus S|}{|A\times B|}\to 0$. Furthermore, in some of the papers where these results appear, it was conjectured that the best possible value of $c$ such that the same conclusion is reached is $c=1$. In this paper we confirm this conjecture.

We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of Margenstern.

Let $X$ be a curve defined over $\mathbb{Q}$ and let $t\in \mathbb{Q}(X)$ be a non-constant rational function on $X$ of degree $v\ge 2$. For every rational number $a/b$ pick a point $P_{a/b}\in X(\overline{\mathbb{Q}})$ such that $t(P_{a/b})=a/b$. In this paper, we obtain lower bounds on the number of distinct fields among $\mathbb{Q}(P_{a/b})$ with $1\le a,b\le N$ under some assumptions on $t$. We show that if $t$ has a pole of order at least 2 or if there is a rational number $\alpha$ such that $t-\alpha$ has a zero of order at least 2, then the set $\{\mathbb{Q}(P_{a/b})|1\le a,b\le N\}$ contains $\gg \frac{N^2}{{(logN)}^2}$ elements. We also obtain partial results when $t$ does not have a pole of order at least two.

In their 2019 paper, Lee and Park presented a formula for the arithmetic Chern–Simons invariant. This formula gives a relation between this invariant and the local Hasse invariants at certain finite primes. Given a number field having real embeddings, we present alternative formulas to give relations between the arithmetic Chern–Simons invariant and the local Hasse invariants at certain primes including all real ones. As an application, we propose a new problem which concerns the existence of a certain $3$-cocycle. If the answer to this problem is positive, the obtained statement is an analogue of the Albert–Brauer–Hasse–Noether theorem.

We describe the construction and properties of a singular theta lift for the orthogonal group $SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann, Kane and Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura–Shintani correspondence. This work provides extensions of the theta lifts considered by Hövel and Bringmann, Kane and Viazovska.

With the results on $\mathrm{\text{GL}}(3)$ and the adjoint lift, a restricted version of analytic strong multiplicity one is proved which holds for $\mathrm{\text{SL}}(2)$, the restriction being that the representations are required to be generic with respect to the same additive character. In this case there exists a positive constant $c$ depending on $𝜀>0$ and the number field $F$ only, such that $\pi$ can be decided completely by its local components ${\pi }_v$ with norm $N(v)<c\cdot C{(\pi )}^{6+𝜀}$, where $C(\pi )$ is the analytic conductor of $\pi$ with respect to the adjoint lift. A similar argument also allows us to simplify the proof of the restricted version of strong multiplicity one for $\mathrm{\text{SL}}(2)$ originally given in [J. Chai and Q. Zhang, A strong multiplicity one theorem for $\mathrm{\text{S}}{L}_2$, Pacific J. Math. 285(2) (2016) 345–374].

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in $S$-units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584].

We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many $n$. This complements earlier work of Filaseta and Moy, who studied Galois groups of $\left(\genfrac{}{}{0.0pt}{}{n}{0}\right)+\left(\genfrac{}{}{0.0pt}{}{n}{1}\right)X+\cdots +\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)X^k$ for more general pairs $(n,k)$, but had to admit a possibly infinite exceptional set specifically for $k=6$ of at most logarithmic growth in $n$. The proof rests upon invoking Faltings’ theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.

Following recent studies of vanishing coefficients in arithmetic progressions in infinite $q$-series products, we further prove that such phenomenon also exists in a family of quotients of theta functions with odd moduli. Moreover, we find that some arithmetic progressions with vanishing coefficients could be shared by an infinite family of quotients of theta functions. Finally, we present several related conjectures that merit further investigation.

Let ${ℰ}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to ${ℰ}$ is a graph which has a vertex for each elliptic curve over $\mathbb{Q}$ of ${ℰ}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in ${ℰ}$ to another elliptic curve in ${ℰ}$, with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to ${ℰ}$ is the isogeny graph associated to ${ℰ}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The main result of the article is a classification of the $2$-adic Galois image at each vertex of the isogeny-torsion graphs whose associated $\mathbb{Q}$-isogeny class consists of elliptic curves over $\mathbb{Q}$ with complex multiplication.

We prove two unconditional results on Fourier coefficients of modular forms: Let $f$ be a Hecke eigen cusp form of weight $k$ and level $N$ ($N$ square free), and ${\lambda }_f(n)$ be the normalized Hecke eigenvalues. Let $n_f$ be the least prime $p$ such that ${\lambda }_f(p)<0$. Then we show that in a family of modular forms of weight $k$ and level $N$, except for a density zero set, $n_f\ll {(log{k}^{2}N)}^{\alpha }$ for some $\alpha >0$. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to $O({(log{k}^{2}N)}^{\alpha })$ for some $\alpha >0$.

Let $f$ be a Drinfeld modular form of level ${\mathrm{\Gamma }}_0(𝔪)$ which is an eigenform for the Hecke operator $T_{𝔭}$ ($𝔭$ a prime of ${𝔽}_q[T]$). We study the relations between the Fourier coefficients of $f$ and the $𝔭$-adic valuation of its eigenvalue (slope). We use formulas for some of the Fourier coefficients of $T_{𝔭}f$ to provide bounds and estimates on the slopes and, in particular, to find necessary conditions for “large” slopes, whose existence is closely connected with conjectures on oldforms and newforms.

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett’s double coset decomposition which works in our more general situation.