International Journal of Number Theory

Let $E/\mathbb{Q}$ be an elliptic curve and $p\in \{5,7,11\}$ be a prime. We determine the possibilities for $E(\mathbb{Q})({\zeta }_p){)}_{\mathrm{\text{tors}}}$. Additionally, we determine all the possibilities for $E{(\mathbb{Q}({\zeta }_{16}))}_{\mathrm{\text{tors}}}$ and $E{(\mathbb{Q}({\zeta }_{27}))}_{\mathrm{\text{tors}}}$. Using these results, we can determine the possibilities for $E{(\mathbb{Q}({\mu }_{p^{\infty }}))}_{\mathrm{\text{tors}}}$ for $p\in \{2,3,5,7,11\}$.

For a modular elliptic curve $A=X_0(49)$ and its quadratic twists $A^{(M)}$, we give equivalent conditions such that the $2$-Selmer group $S_2(A^{(M)}/\mathbb{Q})$ is minimal, namely, it is of order $2$. One of these conditions is described by the $L$-value $L(A^{(M)}/\mathbb{Q},1)$. The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of $M$). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai.

Let $\{a_1,\dots ,a_n\}\subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erdős asked whether this implies $a_n\ge c\cdot 2^n$ for some universal $c>0$. We prove, slightly extending a result of Elkies, that for any $a_1,\dots ,a_n\in {ℝ}_{>0}$,

In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature $(n-2,2)$ with $n\ge 3$. We prove a conjecture of Gross that the superspecial loci and their lift in the integral models can be parametrized by certain homogeneous spaces. As applications, we provide mass formulas for the superspecial loci. We also indicate how Gross’s conjectures can be proved in the case of Coxeter type Shimura varieties using the group theoretic methods of Görtz and He.

Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews’ results involving parity in Rogers–Ramanujan–Gordon identities. Their result partially answered an open question of Andrews’. The open question was to involve parity in overpartition identities. We extend Sang, Shi and Yee’s work to arbitrary moduli, and also provide a missing case in their identities. We also unify proofs of Rogers–Ramanujan–Gordon identities for overpartitions due to Lovejoy and Chen et al.; Sang, Shi and Yee’s results; and ours. Although verification type proofs are given for brevity, a construction of series as solutions of functional equations between partition generating functions is sketched.

Let $F$ be a finite extension of ${\mathbb{Q}}_p$. Every supersingular mod $p$ representation of ${\mathrm{\text{GL}}}_2(F)$ is a quotient of a certain universal module, say ${\pi }_r.$ The space of pro-$p$-Iwahori invariants of ${\pi }_r$ is instrumental in classifying the supersingular mod $p$ representations of ${\mathrm{\text{GL}}}_2({\mathbb{Q}}_p)$. For $F\ne {\mathbb{Q}}_p,$ the space of pro-$p$-Iwahori invariants of ${\pi }_r$ has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod $p$ representations of ${\mathrm{\text{GL}}}_2(F)$ is introduced by Anandavardhanan and Borisagar using the Iwahori–Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for ${\mathrm{\text{GL}}}_2(F)$ obtained from the spherical approach and the Iwahori–Hecke approach. We also compute an explicit basis of the pro-$p$-Iwahori invariant space of ${\pi }_r$ for ${\mathrm{\text{GL}}}_2(F)$ using the Iwahori–Hecke approach. This method allows us to extend Hendel’s computation slightly in the range of $p$-adic digits from $2<r_j<p-3$ to $1<r_j<p-1.$

Classical multiple zeta values can be viewed as iterated integrals of the differentials $\frac{dt}{t},\frac{dt}{1-t}$ from $0$ to $1$. In this paper, we reprove Brown’s theorem: For $a_i,b_i,c_{ij}\in \mathbb{Z}$, the iterated integral of the form

What is more, we show that if $p_i(t),1\le i\le N,$ are in a $\mathbb{Q}[t,1/t,1/(1-t)]$-algebra generated by multiple polylogarithms and their dual, and if $q_{ij}(t),1\le i<j\le N$, are in a $\mathbb{Q}[t,1/t]$-algebra generated by logarithm, then the iterated integral

As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals

Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level.

In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map $n^s$ can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.

Let f be a Hecke–Maass cusp form of weight zero for ${SL}_2(\mathbb{Z})$ and ${\lambda }_f(n)$ be the nth Fourier coefficient. For almost all $𝜗,$ we have

For all $\alpha >1$ of type $\tau <\infty ,$ we also show that

We obtain two parametric solutions of the diophantine equation $\varphi (x_1,x_2,x_3)=\varphi (y_1,y_2,y_3)$ where $\varphi (x_1,x_2,x_3)$ is the octic form defined by $\varphi (x_1,x_2,x_3)=x_1^8+x_2^8+x_3^8-2x_1^4{x}_2^4-2x_1^4{x}_3^4-2x_2^4{x}_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank $5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.

We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of ${\mathrm{\text{SL}}}_2(ℝ)$, Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for $G={\mathrm{\text{SL}}}_2(ℂ)\times \cdots \times {\mathrm{\text{SL}}}_2(ℂ)$ and $\mathrm{\Gamma }<G$ a lattice subgroup, the points of $G/\mathrm{\Gamma }$ with bounded orbits under a one-parameter Ad-semisimple subgroup of $G$ form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of [R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers, Glasgow Math. J. 52 (2010) 349–355] about the set of badly approximable complex numbers.

In this paper, we investigate the average