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In 2021, the first author and Kalita obtained two general hypergeometric formulas for sums involving certain rising factorials to prove some supercongruence conjectures of Guo related to (B.2) and (C.2). In this paper, we further generalize those formulas by using the WZ-method and the Zeilberger algorithm, respectively., Comment: Submitted for publication on 16th January 2025. Comments are welcome

Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to show that modifications of Ap\'ery's continued fractions can give interesting results including new rapidly convergent continued fractions for certain interesting constants., Comment: 14 pages

Letting $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erd\H{o}s wondered if $\delta_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(\delta_1(n,m))$ has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; $n = 1$. We also solve the question on unimodality of the density of integers whose $k^{th}$ prime is $p$., Comment: 5 pages

We construct a mod $\ell$ congruence between a Klingen Eisenstein series (associated to a classical newform $\phi$ of weight $k$) and a Siegel cusp form $f$ with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch-Kato Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\rho_{\phi}(2-k)\otimes \mathbf{Q}_{\ell}/\mathbf{Z}_{\ell})$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine-Laffaille universal deformation ring of $\overline{\rho}_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\overline{\rho}_{\phi}(k-2))$ is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf{Q}, \textrm{ad}^0\overline{\rho}_{\phi}(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

As a continuation of our earlier paper, we offer a new approach to murmurations and Sato-Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called a-invariant in rank n>2 even for the Sato-Tate law, rather, on a much refined structure, a similar version of which was already observed by Zagier and the senior author when the rank n Riemann hypothesis was established. Namely, instead of the rank n Riemann hypothesis bounds, we use much stronger asymptotic bounds. Accordingly, rank n Sato-Tate law can be established and rank n murmuration can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves., Comment: 14 pages, 4 figures

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on binary systems of linear forms and of Kelley and Meka on Roth's theorem on arithmetic progressions, we show that, for $N \geq \exp((k\log(2/\alpha))^{O(1)})$, any subset $A \subseteq [N]$ of density at least $\alpha$ contains a $k$-configuration. This improves on the previously best known bound $N \geq \exp((2/\alpha)^{O(k^2)})$, due to Shao. As a consequence, it follows that any finite non-empty set $A \subseteq \mathbb{Z}$ contains a subset $B \subseteq A$ of size at least $(\log|A|)^{1+\Omega(1)}$ such that $b_1+b_2 \not\in A$ for any distinct $b_1,b_2 \in B$. This provides a new proof of a lower bound for the Erd\H{o}s--Moser sum-free set problem of the same shape as the best known bound, established by Sanders., Comment: 23 pages

In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the Andr\'e-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.

For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is supported on prime ideals in $F$ over which at least one (rather than each) prime in $K$ has ramification index divisible by $\ell$. This bound holds with a proviso on the Galois group of the normal closure of $K/F$, which is satisfied by towers of Galois extensions, intermediate fields in nilpotent extensions, and intermediate fields in dihedral extensions of degree $8n$, to name a few. We also use our lower bound to prove a new density result on number fields with infinite class field towers., Comment: 23 pages

Let $\mathbb{F}_q$ be the finite field with $q=p^s$ elements, where $p$ is an odd prime and $s$ a positive integer. In this paper, we define the function $f(X)=(cX^q+aX)(X^{q}-X)^{n-1}$, for $a,c\in\mathbb{F}_q$ and $n\geq 1$. We study the dynamics of the function $f(X)$ over the finite field $\mathbb{F}_{q^2}$, determining cycle lengths and number of cycles. We also show that all trees attached to cyclic elements are isomorphic, with the exception of the tree hanging from zero. We also present the general shape of such hanging trees, which concludes the complete description of the functional graph of $f(X)$.

This paper explores the interplay between holonomy, Ihara zeta functions, and cohomological structures within the framework of ratified F-completions of foliated manifolds. We develop a novel formalism for the Gamma-set, a topological multigraph that captures intersection points of foliations with stratified boundaries, and use it to define an Ihara zeta function that encodes the manifold's symmetries. By investigating the holonomy group of spinor fields in relation to the Delta-actions on the Gamma-set, we conjecture a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes. We further analyze how the twist map T impacts cohomology within stratified spaces, highlighting the role of vector bundles and the twisted cochain complex. Applications of our framework to spectral graph theory and tiling in R^n demonstrate the power of holonomy-zeta duality in geometrically rich foliated structures.

In this paper, we consider the family of monic integer polynomials of degree $n > 1$ with prime coefficients. We determine the asymptotic density of polynomials in the family with squarefree discriminant as $n$ goes to infinity. We also determine the number of irreducible polynomials $f(x)$ in the family such that $\mathbb{Z}[x]/(f(x))$ is the ring of integers of $\mathbb{Q}[x]/(f(x))$.

In 1997, Van Hamme proposed 13 supercongruences on truncated hypergeometric series. Van Hamme's (B.2) supercongruence was first confirmed by Mortenson and received a WZ proof by Zudilin later. In 2012, using the WZ method again, Sun extended Van Hamme's (B.2) supercongruence to the modulus $p^4$ case, where $p$ is an odd prime. In this paper, by using a more general WZ pair, we generalize Hamme's (E.2) and (F.2) supercongruences, as well as two supercongruences by Swisher, to the modulus $p^4$ case. Our generalizations of these supercongruences are related to Euler polynomials. We also put forward a relevant conjecture on $q$-congruences for further study., Comment: 15 pages

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. As an application of our results, we prove the following Diophantine statements: 1. Using a result of Pollington and Velani, we show that if $\mu$ is a self-conformal measure in $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $\alpha>0$ such that for all $\beta>\alpha$ we have $$\mu\left(\left\{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^{\beta}}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right\}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $\mu$ is an affinely irreducible self-similar measure, then $\mu$ almost every $\mathbf{x}$ is not a singular vector.

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions.

We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character $\chi$, we give an explicit conjecture relating the ideal factorization of $L(E,\chi,1)$ to the Galois module structure of the Tate-Shafarevich group of $E/K$, where $\chi$ factors through the Galois group of $K/\mathbb{Q}$. We provide numerical evidence for this conjecture using the methods of visualization and $p$-descent. For the latter, we present a procedure that makes performing an $11$-descent over a $C_5$ number field practical for an elliptic curve $E/\mathbb{Q}$ with complex multiplication. We also expect that our method can be pushed to perform higher descents (e.g. $31$-descent) over a $C_5$ number field given more computational power., Comment: 18 pages

We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growth of convex sets., Comment: 28 pages

In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive integer points of the Markov mutation invariant and its variant. As an application, several classes of Diophantine equations with cluster algebraic structures are exhibited., Comment: 43 pages, any comments are welcome!

We construct an arithmetic analogue of the quantum local systems on the moduli of curves, and study its basic structure. Such an arithmetic local system gives rise to a uniform way of assigning a Galois cohomology class of the first geometric \'etale cohomology of a smooth proper curve over a number field.

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an irreducible automorphic representation of $\mathrm{GSp}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification., Comment: 37 pages; To appear in Research Directions in Number Theory-Women in Numbers 6

Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2., Comment: 27 pages, comments are very welcome!

In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject., Comment: Dedicated to Athanase Papadopoulos for the occasion of his 65th birthday

Wesley Ivan Hurt and Ali Sada both independently proposed studying an infinite array where the $0$'th row consists of the non-negative integers $0,1,2,\ldots$ in increasing order. Thereafter the $n$'th row is formed from the $(n-1)$'th row by "jumping" the single entry $n$ by $n$ places to the right. Sada also defined a sequence $s(n)$ defined to be the first number that $n$ jumps over. In this note I show how the Hurt-Sada array and Sada's sequence are intimately connected with the golden ratio $\varphi$ and Zeckendorf representation. I also consider a number of related sequences.

A cusp of a Hecke group $G_q$ has two natural lifts to the ring of integers of a cyclotomic field. These lifts are called here odd vanishing cycles. All lifts of all cusps together form a discrete subset of ${\mathbb C}$ of some exquisite beauty. They form one or two or four orbits of a certain subgroup of the matrix Hecke group. The subgroup can be considered as a monodromy group and is an analog of a rank 2 Coxeter group, so of a dihedral group. The paper has a research part and a larger survey part., Comment: 48 pages, 12 figures

In this paper, we give some finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities, where the semi-decomposable form is factorized into a family of homogeneous polynomials with the distributive constant exceeding a certain number., Comment: 8 pages

For $ k \geq 2 $, let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) = 1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s) = \mathtt{g}(a_1, a_2, \ldots, a_k;s)$, represents the largest integer that has at most $s$ representations in terms of $a_1, a_2, \ldots, a_k$ with non-negative integer coefficients. In this article, we provide a formula for the generalized Frobenius number of three consecutive triangular numbers, $\mathtt{g}(t_{n}, t_{n+1}, t_{n+2};s) $, valid for all $s \geq 0$ where $t_n$ is given by $\binom{n+1}{2}$. Furthermore, we present the proof of Komatsu's conjecture

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of $\lambda_k-\lambda_{k-1}$ over any thick part of the moduli space of closed Riemann surfaces approaches the limit $\frac{1}{4}$., Comment: 56 pages, 11 figures, comments welcome

Let ${\rm \mathbf{H}}(\nu^{u}_{\rm CL})$ be the entropy of the Cohen-Lenstra measure on finite abelian $p$-groups associated to an integral unit-rank $0 \le u \in \mathbb{N}$. In this note, we show that $0 < {\rm \mathbf{H}}(\nu^{u}_{\rm CL}) < \infty$ for all $u$, ${\rm \mathbf{H}}(\nu^{u}_{\rm CL})$ is a strictly decreasing function of $u \ge 0$, and ${\rm \mathbf{H}}(\nu^{u}_{\rm CL}) \xrightarrow{u \to \infty} 0$. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian $p$-groups., Comment: 7 pages

Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from $\ell$-adic sheaves to motivic sheaves. In particular, we prove independence of $\ell$ of the $L$-parameters constructed there., Comment: 19 pages

We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight., Comment: 23 pages

We study representations of integers as sums of $h$ elements from subsets of $k$-th powers $\mathbb{N}^k$ and $k$-th powers of primes $\mathbb{P}^k$ for $k \geq 1$. Extending work by Vu, Wooley, and others, we show that for $h \geq h_k = O(8^k k^2)$, if $F$ is a regularly varying function satisfying $\lim_{n\to\infty} F(n)/\log n = \infty$ and certain growth conditions, then there exists $A \subseteq \mathbb{N}^k$ such that $r_{A,h}(n) \sim \mathfrak{S}_{k,h}(n)F(n)$, where $\mathfrak{S}_{k,h}(n)$ is the singular series from Waring's problem. For $h \geq 2^k + 1$, we show the existence of $A \subseteq \mathbb{P}^k$ with $r_{A,h}(n) \sim \mathfrak{S}^*_{k,h}(n) c n^\kappa$ for $0 < \kappa < h/k - 1$ and $c > 0$, where $\mathfrak{S}^*_{k,h}(n)$ is the singular series from Waring-Goldbach's problem. Additionally, for $0 \leq \kappa \leq h/k - 1$ and functions $\psi$ satisfying $\psi(x) \asymp_{\lambda} \psi(x^{\lambda})$ for every $\lambda>0$, if $\log x \ll x^{\kappa} \psi(x) \ll x^{h/k-1}/(\log x)^h$, then there exists $A \subseteq \mathbb{P}^k$ with $r_{A,h}(n) \asymp n^\kappa \psi(n)$ for $n$ satisfying certain congruence conditions, producing thin subbases of prime powers when $\kappa = 0$, $\psi = \log$., Comment: 26 pages, comments welcome

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This manuscript shows that this is indeed the case at least under broad conditions on ramification and deduce from them new properties of the Iwasawa Main Conjecture for modular forms. This manuscript is dedicated to the memory of Jo\"el Bella\"iche.

This article examines Gaussian processes generated by monotonically modulating stationary kernels. An explicit isometry between the original and the modulated reproducing kernel Hilbert spaces is established, preserving eigenvalues and normalization. The expected number of zeros over the interval $[0,T]$ is shown to be exactly $\sqrt{-\ddot{K}(0)}(\theta(T)-\theta(0))$, where $\ddot{K}(0)$ is the second derivative of the kernel at zero and $\theta$ is the modulating function., Comment: submitted to Mathematical Communications

In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and lie within the interval $(2n,4n]$. This led us to derive a new representation for the third order mock theta function $f_3(q)$ and an analogue of the fundamental identity for the smallest part partition function Spt$(n)$, both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function $\phi_3(q)$., Comment: Submitted for publication

Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve. Our proof proceeds via computing the $\mathbf{Q}(\zeta_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(\zeta_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(\zeta_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.

The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a generalization of the Euler numbers depending on an integer parameter d where one takes the coefficients of the expansion of 1/(1+x^d/d!+x^{2d}/(2d)!+...). These numbers have been shown to have many interesting properties despite being much less studied. And the techniques used have been mainly algebraic. We propose a combinatorial model for them as signed sums over ordered partitions. We show that this approach can be used to prove a number of old and new results including a recursion, integrality, and various congruences. Our methods include sign-reversing involutions and M\"obius inversion over partially ordered sets., Comment: 14 pages, 1 figure

We prove a general irreducibility result for geometrically induced coadmissible equivariant $\mathcal{D}$-modules on rigid analytic spaces. As an application, we geometrically reprove the irreducibility of certain locally analytic representations previously constructed by Orlik-Strauch., Comment: 56 pages

The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants under a quadratic field extension. This is applied to obtain bounds in terms of the index and the u-invariant of the base field., Comment: 17 pages

In this work, we relate two recent constructions that generalize classical (genus-zero) polylogarithms to higher-genus Riemann surfaces. A flat connection valued in a freely generated Lie algebra on a punctured Riemann surface of arbitrary genus produces an infinite family of homotopy-invariant iterated integrals associated to all possible words in the alphabet of the Lie algebra generators. Each iterated integral associated to a word is a higher-genus polylogarithm. Different flat connections taking values in the same Lie algebra on a given Riemann surface may be related to one another by the composition of a gauge transformation and an automorphism of the Lie algebra, thus producing closely related families of polylogarithms. In this paper we provide two methods to explicitly construct this correspondence between the meromorphic multiple-valued connection introduced by Enriquez in e-Print 1112.0864 and the non-meromorphic single-valued and modular-invariant connection introduced by D'Hoker, Hidding and Schlotterer, in e-Print 2306.08644., Comment: 55 + 26 pages

Let $K$ be a field and $\phi(z)\in K[z]$ be a polynomial. Define $\Phi(z) := \frac{1}{\phi(z)} \in K(z).$ For $n \in\mathbb{N}^* $, let the $n$-th iterate of $\Phi(z)$ be defined as $\Phi^{(n)}(z) = \underbrace{\Phi \circ \Phi \circ \cdots \circ \Phi}_{n \text{ times}}(z).$ We express the \(\Phi^{(n)}(z)\) in its reduced form as \( \Phi^{(n)}(z) = \frac{f_{n,\phi}(z)}{g_{n,\phi}(z)}, \) where \(f_{n,\phi}(z)\) and \(g_{n,\phi}(z)\) are coprime polynomials in \(K[z]\). A polynomial $\phi(z) \in K[z]$ is called inversely stable over $K$ if every $g_{n,\phi}(z)$ in the sequence $\{g_{n,\phi}(z)\}_{n=1}^\infty$ is irreducible in $K[z]$. This paper investigates the inverse stability of the binomials $\phi(z) = z^d + c$ over $K$.

We show that the integer group determinants for the general affine group of degree one, $GA(1,q)$ with $q=p^k$ a prime power, take the form $D=AB^{q-1},$ where $A$ is a $\mathbb Z_{q-1}$ integer group determinant and $B\equiv A \bmod q$. This generalizes the result for $k=1$. When $2^k-1$ is a Mersenne prime we show that this condition is both necessary and sufficient for $GA(1,2^k).$ The same is true for $GA(1,9)$ and $GA(1,27)$.

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm., Comment: 32 pages, comments are welcome

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with respect to a prime $ p $ can be explicitly described in terms of the Newton polygon of the polynomial $ g(x) $. Our results provide deeper insights into how the Newton polygon of a polynomial evolves under iteration and composition, with applications to the study of dynamical irreducibility, eventual stability, non-monogenity of tower of number fields, etc., Comment: Any comments or suggestions are welcome. Any additional applications are appreciated. 26 pages

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on $ \mathcal D $ that are given by Poincar\'e series of polynomial type and correspond via the classical lift to the Poincar\'e series of certain $ K $-finite matrix coefficients of integrable discrete series representations of $ G $. We provide an example application of our results in the case when $ G=\mathrm{SU}(p,q) $ and $ K=\mathrm S(\mathrm U(p)\times\mathrm U(q)) $ with $ p\geq q\geq1 $., Comment: 24 pages, 2 tables

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Dr\`ezet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers., Comment: 11 pages, 4 figures

This article generalizes Venkatesh's structure theorem for the derived Hecke action on the Hecke trivial cohomology of a division algebra over an imaginary quadratic field to division algebras over all number fields. In particular, we show that the Hecke trivial cohomology of a division algebra is a free module for the action of the strict derived Hecke algebra. Moreover, the strict derived Hecke algebra possesses a rational form that preserves the canonical rational structure on the cohomology during the derived Hecke action. The main ingredients in our improvement are a careful study of the congruence classes in the torsion cohomology of the arithmetic manifold and the author's new result on the reduction map in the $K$-theory of the ring of integers in number fields., Comment: Comments welcome!

We provide a theory of the Fourier-Jacobi expansion for automorphic forms on simple adjoint groups of some general class. This theory respects the Heisenberg parabolic subgroups, whose unipotent radicals are the Heisenberg groups uniformly explained in terms of the notion of cubic norm structures. Based on this theory of the Fourier expansion, we prove that automorphic forms generating quaternionic discrete series representations automatically satisfy the moderate growth condition except for the cases of the group of $G_2$-type and special orthogonal groups of signature $(4,N)$. This should be called ``K\"ocher principle'' verified already for the case of the quaternion unitary group $Sp(1,q)$ for $q>1$ by the author. We also prove that every term of the Fourier expansion with a non-trivial central character for cusp forms generating quaternionic discrete series has no contribution by the discrete spectrum of the Jacobi group, which is a non-reductive subgroup of the Heisenberg parabolic subgroup. This is obtained by showing that generalized Whittaker functions of moderate growth for the Schr\"odinger representations are zero under some assumption of the separation of variables, which suffices for our purpose to establish such consequence.

We prove that there are infinitely many integers, which can represent as sum of a square-free integer and a prime $p$ with $||\alpha p+\beta||

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number theory, naturally extend the classical theory. This generalization transitions from the idele class group, which acts as the adelic analog of Galois groups, to a geometric framework associated with schemes and the ring of integers of global fields. This perspective provides a conceptual explanation for the role of the adele class space in the spectral realization of L-function zeros and identifies the idele class group as a generic point in this context. The sector $X_{\mathbb{Q}}$ of the adele class space corresponding to the Riemann zeta function gives the class field counterpart of the scaling topos. The main result is the construction of a functor mapping finite abelian extensions of $\mathbb{Q}$ to finite covers of $X_{\mathbb{Q}}$, with the monodromy of periodic orbits of length $\log p$ under the scaling action corresponding to the Galois action of the Frobenius at the prime p., Comment: 30 pages

Our main objective in the present article is to study the factorization problem for triple-product $p$-adic $L$-functions, particularly in the scenarios when the defining properties of the $p$-adic $L$-functions involved have no bearing on this problem, although Artin formalism would suggest such a factorization. Our analysis, which is guided by the ETNC philosophy, recasts this problem as a comparison of diagonal cycles, Beilinson--Kato elements, and Heegner cycles., Comment: This is the first paper in a two-part series replacing an earlier submission (arXiv:2301.08383), which is no longer intended to be published as a stand-alone paper. 45 pages, comments are very welcome

This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet formula for continuants. Using this result, we provide a continuant-based formulation for well-known identities associated with Lucas sequences., Comment: Journal of Integer Sequences (2023)