We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb {Q}(\sqrt {d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and ${\lvert d \rvert < 100}$.

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

Let F be a finite extension of ${\mathbb Q}_p$. Let $\Omega$ be the Drinfeld upper half plane, and $\Sigma^1$ the first Drinfeld covering of $\Omega$. We study the affinoid open subset $\Sigma^1_v$ of $\Sigma^1$ above a vertex of the Bruhat–Tits tree for $\text{GL}_2(F)$. Our main result is that $\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that $\text{Pic}({\mathbf Y})[p] = 0$ for ${\mathbf Y}$ the Deligne–Lusztig variety of $\text{SL}_2\!\left({\mathbb F}_q\right)$. One formal consequence is a description of the representation $H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$ of $\text{GL}_2(\mathcal{O}_F)$ as the p-adic completion of $\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.

Romyar Sharifi has constructed a map $\varpi _M$ from the first homology of the modular curve $X_1(M)$ to the K-group $K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[1/2]$, where $\zeta _M$ is a primitive Mth root of unity. Sharifi conjectured that $\varpi _M$ is annihilated by a certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with $\operatorname {\mathrm {\mathbf {Z}}}_p$ for a prime $p\geq 5$ dividing M. More recently, Sharifi and Venkatesh proved the conjecture for Hecke operators away from M. In this note, we prove two main results. First, we give a relation between $\varpi _M$ and $\varpi _{M'}$ when $M' \mid M$. Our method relies on the techniques developed by Sharifi and Venkatesh. We then use this result in combination with results of Fukaya and Kato in order to get the Eisenstein property of $\varpi _M$ for Hecke operators of index dividing M.

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$.

In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$. In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$, thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$, where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

We establish analogues for trees of results relating the density of a set ${E \subset \mathbb {N}}$, the density of its set of popular differences and the structure of E. To obtain our results, we formalize a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting, we use a recent result of the first author with Björklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.

When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations

\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}

$d=o\!\left(k^{1/4}\right)$

Let ${\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\overline{p}}(n-1){\overline{p}}(n+1)/{\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

In this paper, we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well-established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit, such as the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM-points.

We establish a Harder–Narasimhan formalism for modifications of $G$-bundles on the Fargues–Fontaine curve. The semi-stable stratum of the associated stratification of the ${B^+_{{\rm dR}}}$-Grassmannian coincides with the variant of the weakly admissible locus defined by Viehmann, and its classical points agree with those of the basic Newton stratum. When restricted to minuscule affine Schubert cells, the stratification corresponds to the Harder–Narasimhan stratification of Dat, Orlik and Rapoport. We also study basic geometric properties of the strata, and the relation to the Hodge–Newton decomposition.

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa 30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$. We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$. As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$. Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$.

We study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages—it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set

\begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*}

$ \alpha\in[0,1] $

$ \alpha $

Lin introduced the partition function $\text {PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\text {PDO}_t(n)$, and conjectured certain congruences modulo $3^{k+2}$ for $k\geq 0$. He proved the conjecture for $k=0$ and $k=1$ [‘The number of tagged parts over the partitions with designated summands’, J. Number Theory 184 (2018), 216–234]. We prove the conjecture for $k=2$. We also study the lacunarity of $\text {PDO}_t(n)$ modulo arbitrary powers of 2 and 3. Using nilpotency of Hecke operators, we prove that there exists an infinite family of congruences modulo any power of 2 satisfied by $\text {PDO}_t(n)$.

We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$, which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.