We determine the characteristic polynomials of the matrices $[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and $[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number $q\not =0,1$. As an application, for complex numbers $a,b,c$ with $b\not =0$ and $a^2\not =4b$, and the sequence $(w_m)_{m\in \mathbb Z}$ with $w_{m+1}=aw_m-bw_{m-1}$ for all $m\in \mathbb Z$, we determine the exact value of $\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.

Let $\pi $ be a cuspidal, cohomological automorphic representation of an inner form G of $\operatorname {{PGL}}_2$ over a number field F of arbitrary signature. Further, let $\mathfrak {p}$ be a prime of F such that G is split at $\mathfrak {p}$ and the local component $\pi _{\mathfrak {p}}$ of $\pi $ at $\mathfrak {p}$ is the Steinberg representation. Assuming that the representation is noncritical at $\mathfrak {p}$, we construct automorphic $\mathcal {L}$-invariants for the representation $\pi $. If the number field F is totally real, we show that these automorphic $\mathcal {L}$-invariants agree with the Fontaine–Mazur $\mathcal {L}$-invariant of the associated p-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

Let f be an $L^2$-normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that

$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

We establish analogous estimates in the setting of definite quaternion algebras.

We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field $F$. Our definition relies on a canonical measure $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

We obtain asymptotic formulas for the number of matrices in the congruence subgroup

$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$

For an elliptic curve E defined over a number field K and $L/K$ a Galois extension, we study the possibilities of the Galois group Gal$(L/K)$, when the Mordell–Weil rank of $E(L)$ increases from that of $E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at $s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

For N integer $\ge 1$, K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth Fermat curve. We show that the Manin–Drinfeld principle holds for $N=3$, but not for $N=5$. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$, for n even integer, both dominate $X(2)$, which produces a morphism between Jacobians $J_N\rightarrow J(n)$. We prove that the latter has image $0$ or an elliptic curve of j-invariant $0$. In passing, we give a description of the homology of $X^{\prime }_{N}$.

In this paper, we prove that the bound

\begin{equation*}\max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}}\end{equation*}

$A \subset \mathbb R$

\begin{equation*}\max \{ |16A|, |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c},\end{equation*}

$c\gt 0$

$\mathbb R$

$3/2$

\begin{equation*}|AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{\frac{2}{3}-c^{\prime}},\end{equation*}

$c,C \gt 0$

A set $S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leqslant s_2$) are distinct. A set $S\subset {\mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb {F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.

For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, Tr. Mat. Inst. Steklova 132 (1973), 247–250].

We prove the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups.

In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2) 172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$-horospheres in the space of affine lattices.