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We derive a formula for the product of two Dirichlet series satisfying Hecke's functional equation. Several examples are given. At the conclusion of our paper, we address prior work on similar formulas., Comment: 11 pages

It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve $x^4+y^4=1$, aka squircle, and the arc length of the lemniscate $(x^2+y^2)^2=x^2-y^2$. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus., Comment: 8 pages, 2 figures

Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott-Halberstam conjecture holds true., Comment: 16 pages

Finding explicit examples of sequences $(x_n)_{n \in \mathbb{N}}$ with Poissonian pair correlations (PPC) turns out to be a difficult problem. Therefore, this property is often studied in a random setting. While it is known that i.i.d. sequences of uniformly distributed random variables generically have Poissonian pair correlations, the case of dependent random variables has not been addressed in the literature yet. The present paper is the first contribution dealing with the dependent setting. More specifically, sequences of jittered samples and random walks on the torus are investigated. We show here that for the former class, the PPC property depends on how the finite sample is extended to an infinite sequence. Moreover, we prove that the random walk on the torus generically has PPC.

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the set of points below the diagonal line. In this paper, we investigate the set of points lying below a parabola. We prove that in some sense, this set of points looks the same from all but two directions, despite having only one non-trivial automorphism. In addition, we study the sizes of these sets, and their intersection numbers with respect to lines., Comment: 9 pages

Let $\mathbb{F}_q$ be the finite field of $q$ elements. We define the Rudin-Shapiro function $R$ on monic polynomials $f=t^n+f_{n-1}t^{n-1}+\dots + f_0\in\mathbb{F}_q[t]$ over $\mathbb{F}_q$ by $$ R(f)=\sum_{i=1}^{n-1}f_if_{i-1}. $$ We investigate the distribution of the Rudin-Shapiro function along the irreducible polynomials. We show, that the number of irreducible polynomials $f$ with $R(f)=\gamma$ for any $\gamma\in\mathbb{F}_q$ is asymptotically $q^{n-1}/n$ as $n\rightarrow\infty$.

We provide a proof of a conjecture by Zhao concerning the structure of certain relations among cyclotomic multiple zeta values in weight two. We formulate this conjecture in a broader algebraic setting in which we give a natural equivalence between two schemes attached to a finite abelian group $G$. In particular, when $G$ is the group of roots of unity, these schemes describe the standard relations among cyclotomic multiple zeta values., Comment: 29 pages. Comments are welcome

Let $(G,+)$ be an Abelian group. Given $h\in \mathbb{Z}^+$, a non-empty subset $A$ of $G$ is called an $S_h$-set if all the sums of $h$ distinct elements of $A$ are different. We extend the concept of $S_h$-set to a more general context in the context of finite vectorial spaces over finite fields. More precisely, a $\emptyset \neq A\subseteq \mathbb{F}_q^r$ is called an $S_h$-linear set if all the linear combinations of $h$ elements of $A$ are different. We establish a correspondence between $q$-ary linear codes and $S_h$-linear sets. This connection allow us to find lower bounds for the maximum size of $S_h$-sets in $\mathbb{F}_q^r$., Comment: 17 pages, 2 figures

Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of \cite{AB24} about counting results related to orthogonality in $\mathcal{K}^n$. For example, we answer an open question from \cite{AB24} by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we use orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, as well as to determine the number of orthogonal bases for a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima.

The question of integer complexity asks about the minimal number of $1$'s that are needed to express a positive integer using only addition and multiplication (and parentheses). In this paper, we propose the notion of $l$-complexity of multiples of $l$, which specializes to integer complexity when $l=1$, prove several elementary results on $2$-complexity of even positive integers, and raise some interesting questions on $2$-complexity and in general $l$-complexity., Comment: To appear in Integers

Given a finite abelian group $\Gamma$, we study the distribution of the $p$-part of the class group $\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$ with Galois group isomorphic to $\Gamma$. We first construct a discrete valuation ring $e\mathbb{Z}_p[\Gamma]$ for each primitive idempotent $e$ of $\mathbb{Q}_p[\Gamma]$, such that 1) $e\mathbb{Z}_p[\Gamma]$ is a lattice of the irreducible $\mathbb{Q}_p[\Gamma]$-module $e\mathbb{Q}_p[\Gamma]$, and 2) $e\mathbb{Z}_p[\Gamma]$ is naturally a quotient of $\mathbb{Z}_p[\Gamma]$. For every $e$, we study the distribution of $e\operatorname{Cl}(K):=e\mathbb{Z}_p[\Gamma] \otimes_{\mathbb{Z}_p[\Gamma]} \operatorname{Cl}(K)[p^{\infty}]$, and prove that there is an ideal $I_e$ of $e\mathbb{Z}_p[\Gamma]$ such that $e\operatorname{Cl}(K) \otimes (e\mathbb{Z}_p[\Gamma]/I_e)$ is too large to have finite moments, while $I_e \cdot e\operatorname{Cl}(K)$ should be equidistributed with respect to a Cohen--Lenstra type of probability measure. We give conjectures for the probability and moment of the distribution of $I_e\cdot e\operatorname{Cl}(k)$, and prove a weighted version of the moment conjecture in the function field case. Our weighted-moment technique is designed to deal with the situation when the function field moment, obtained by counting points of Hurwitz spaces, is infinite; and we expect that this technique can also be applied to study other bad prime cases. Our conjecture agrees with the Cohen--Lenstra--Martinet conjecture when $p\nmid |\Gamma|$, and agrees with the Gerth conjecture when $\Gamma=\mathbb{Z}/p\mathbb{Z}$. We also study the kernel of $\operatorname{Cl}(K) \to \bigoplus_e e\operatorname{Cl}(K)$, and show that the average size of this kernel is infinite when $p^2\mid |\Gamma|$., Comment: with an appendix by Peter Koymans

For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \sigma(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } \sigma(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\} \end{align*} and \begin{align*} \cal{D}_{r,u,m}(n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x

Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of divisors function. In this paper we relate zero-free regions for the Riemann zeta-function to maximal order for the sum of powers of divisors function. In particular, we give equivalent conditions for the Riemann hypothesis in terms of the sum of the $1/2$-th powers of divisors function. As a by-product, we also give equivalent conditions for the Riemann hypothesis in terms of the partial Euler product for the Riemann zeta-function., Comment: 26 pages

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let $A$ be an abelian variety with RM over a number field whose attached Galois representation has large image. Then the number of Elkies primes (in a suitable range) for reductions of $A$ modulo primes converges weakly to a Gaussian distribution around its expected value. This refines and generalizes results obtained by Shparlinski and Sutherland in the case of non-CM elliptic curves, and has implications for the complexity of the SEA point counting algorithm for abelian surfaces over finite fields.

For a fixed irrational $\theta>0$ with a prescribed irrationality measure function, we study the correlation $\int\limits_{1}^{X}\Delta\left(x\right)\Delta\left(\theta x\right)dx$, where $\Delta$ is the Dirichlet's Delta Function from the Divisor Problem. It is known that when $\theta$ have finite irrationality measure there's a decorrelation given in terms of its measure and a strong decorrelation is obtained at every positive irrational number, except, maybe, at the Liouville numbers. We prove that for the irrationals with a prescribed irrationality measure function $\psi$, a decorrelation can be obtained in terms of $\psi^{-1}$.

In this paper, we consider a generalization of McKay correspondence in positive characteristic regarding Euler characteristic of crepant resolutions given by finite subgroups of the special linear group. As the main result, we prove that the generalization holds for groups with a specific semidirect product structure. Apart from such groups, two certain examples for groups with more difficult structure are also given., Comment: 17 pages

For $i\in \{1,2,3\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for each $i\in \{1,2\}$, the Riemann hypothesis is equivalent to the condition $\int_2^X E_i(x)\,\mathrm{d}x>0$ for all $X>2$, and we examine assumptions under which the equivalence also holds for $i=3$. In addition, we generalize this to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.

Kantor's Thesis was the first step in unifying the Chabauty-Kim and Lawrence-Venkatesh methods via relative completion. In this work, we refine Kantor's approach by addressing its limitations, achieving the first unification where a dimension inequality between local and global Galois cohomology implies Diophantine finiteness for curves. This results in a new conditional proof of Faltings' and Siegel's theorems and introduces a novel p-adic analytic method for computing rational points on hyperbolic curves, offering advantages over the Chabauty-Kim and Lawrence-Venkatesh approaches. Our technical contributions are threefold. First, we establish the density of Kantor's p-adic period map. Second, our method applies to all curves of genus g >= 2, extending beyond the specific modular curves considered in Kantor's thesis. Third, we resolve the representability problem for Kantor's global Selmer stack. Previously, Kantor's method required additional conjectures in p-adic Hodge theory to represent his Selmer stack--a priori a rigid analytic stack--in a category with a suitable dimension theory. We overcome this by replacing the unipotent completions used in Kim's framework with the unipotent radicals of Kantor's relative completions, derived from monodromy representations associated with the relative cohomology of a Kodaira-Parshin family. Kantor's method is a step toward the Effective Faltings Problem, which seeks not only to establish finiteness of rational points on hyperbolic curves but to compute them explicitly. While the Lawrence-Venkatesh method is unconditional, it has not yet been made effective for any curve. In contrast, the Chabauty-Kim method, though conditional, has been made effective in various settings. Our Unipotent Chabauty-Kim-Kantor method addresses key challenges in Kantor's program and highlights its potential for both theoretical and computational advances., Comment: 45 pages, comments are welcome

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture., Comment: 20 pages, 7 figures

We systematically study relative and absolute ${\Delta}_{\mathrm{dR}}^+$-crystals on the (log-) prismatic site of a smooth (resp.~ semi-stable) formal scheme. Using explicit computation of stratifications, we classify (local) relative crystals by certain nilpotent connections, and classify (local) absolute crystals by certain enhanced connections. By using a $p$-adic Riemann--Hilbert functor and an infinite dimensional Sen theory over the Kummer tower, we globalize the results on absolute crystals and further classify them by certain small (global) $\mathbb{B}_{\mathrm{dR}}^+$-local systems., Comment: 73 pages. comments welcome!

In this article, we study the relation between the universal deformation rings and big Hecke algebras in the residually reducible case. Following the strategy of Skinner-Wiles and Pan's proof of the Fontaine-Mazur conjecture, we prove a pro-modularity result. Based on this result, we also give a conditional big $R=\mathbb{T}$ theorem over some totally real fields, which is a generalization of Deo's result., Comment: 30 pages

It is proved that the Epstein zeta-function corresponding to a binary positive definite quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line., Comment: 75 pages

We fill a gap in the study of the Hausdorff dimension of the set of exact approximation order considered by Fregoli [Proc. Amer. Math. Soc. 152 (2024), no. 8, 3177--3182].

Let $T_m(N,2k)$ denote the $m$-th Hecke operator on the space $S_{2k}(\Gamma_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$. In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$. We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms., Comment: 22 pages

A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two., Comment: 30 pages

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we use the information about prime factorization of the values taken by such polynomials at sufficiently large integer arguments along with the information about their root location in the complex plane. Further, these techniques are extended to bivariate polynomials over arbitrary fields using non-Archimedean absolute values, yielding extensions of the irreducibility results of M. Ram Murty and S. Weintraub to bivariate polynomials., Comment: 11 pages

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating functions, distribution of zeros for individual entries of the matrices and new type of differential-difference structure. We further speculate about other potentials of the connection formulas found. Part of our proofs makes use of creative telescoping in a matrix setting$-$the strategy which is not yet developed algorithmically., Comment: 21 pages, 3 figures

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where $p$ is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient., Comment: 22 pages

In this article we introduced algebraic sieves, i.e. selection procedures on a given finite set to extract a particular subset. Such procedures are performed by finite groups acting on the set. They are called sieves because there are certain sets of numbers which, with appropriate groups, can select, for example, a set of primes, think of the famous Eratosthenes sieve. In this article we have given a general definition of algebraic sieves. And we also introduced the notion of invariant and multi-invariant functions, certain permutations on the sieve set which, in the invariant case, commute with the action of a given sieve-selecting group and the automorphism of that group, and multi-invariants which commute with all groups and their respective automorphisms. By means of such functions we have given symmetries on such sieves. In particular, we studied certain groups of symmetries of invariant functions. Then, using such notions, we studied a particular example, the Goldbach sieve, where the selector groups are dihedral groups and the selected set consists of the primes and, in some cases, the numbers $1$ and $N-1=p$, with $N$ being even and p prime, which satisfies the Goldbach conjecture for $N$. We have shown that one of these symmetry groups is isomorphic to a subgroup or affinity group of the ring of integers modulo N with N an integer even $\mathbb{Z}_{N}$.

We produce canonical sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding fundamental domains are connected. We present some pictures and examples at the end., Comment: 4 figures

The higher Euler-Kronecker constants of a number field $K$ are the coefficients in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function about $s=1$. These coefficients are mysterious and seem to contain a lot of arithmetic information. In this article, we study these coefficients. We prove arithmetic formulas satisfied by them and prove bounds. We generalize certain results of Ihara.

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on the construction of degree-3 \'etale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes., Comment: Comments are very welcome!

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums., Comment: 25 pages, 3 figures

Generating functions $h_r(\tau)$ of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants, are known to be higher depth mock modular forms satisfying a specific modular anomaly equation, with depth determined by the D4-brane charge $r$. We develop a method to solve the anomaly equation for arbitrary charges, in terms of indefinite theta series. This allows us to find the generating functions up to modular forms that can be fixed by computing just a finite number of Fourier coefficients of $h_r$., Comment: 45+38 pages

We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $\eta$-function, analogous to the sequence $\delta_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density $D(r)$ for a normalized eta-power $\eta^r$ measuring the proportion of primes $\ell$ for which the order at infinity of $U_\ell (\eta^r)$ modulo 2 is maximal. We relate $D(r)$ to a notion of density measuring nonzero prime Fourier coefficients introduced by Bella\"iche, and use this to completely classify the vanishing of and establish upper bounds for $D(r)$. Furthermore, for several infinite families of $\eta$ powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bella\"iche, we explicitly compute the densities $D$. We rely on Galois-theoretic techniques developed by Bella\"iche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bella\"iche's unpublished results on densities of mod-$2$ modular forms., Comment: 40 pages, 3 figures, 1 appendix

We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\bigl\{\lfloor m^c \rfloor \colon m \in \mathbb{N} \bigr\}$ for any non-integer $c > 2$., Comment: 22 pages, 1 figure

Suppose we partition the integers into finitely many cells. Can we always find a solution of the equation $x^2+y^2=z^2$ with $x,y,z$ on the same cell? What about more general homogeneous quadratic equations in three variables? These are basic questions in arithmetic Ramsey theory, which have recently been partially answered using ideas inspired by ergodic theory and tools such as Gowers-uniformity properties and concentration estimates of bounded multiplicative functions. The aim of this article is to provide an introduction to this exciting research area, explaining the main ideas behind the recent progress and some of the important challenges that lie ahead., Comment: 36 pages, contribution to the Proceedings of the 9th European Congress of Mathematics

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of characteristic $l$ odd dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$.

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-(\log N)^c}$ for any $c < \frac{1}{4}$., Comment: 30 pages

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all } n\in\mathbb{N} \}, $$ then $$ S(2)+S(2) = [0,1]. $$ He also conjectured that this result is unique in the sense that if you exclude partial quotients from $1$ to $k-1$ with $k\geq3$, then the Lebesgue measure $\lambda$ of the set of numbers which can be expressed as a sum of two continued fractions with no partial quotients from $\{1,\ldots,k-1\}$ is equal to $0$, that is $$\lambda\Bigl( S(k)+S(k) \Bigl)= 0 \text{ for }k\geq 3.$$ In this paper, we disprove the conjecture of Cusick by showing that $$ S(k)+S(k) \supseteq \left[0,\frac{1}{k-1}\right]. $$ The proof is constructive and does not rely on ideas from previous works on the topic. We also show the existence of countably many 'gaps' in $S(k)+S(k)$, that is intervals, for which the endpoints lie in $S(k)+S(k)$, while none of the elements in the interior do so. Finally, we prove several results on the sums $$ S(m)+S(n) $$ for $m\neq n$., Comment: 22 pages, 1 figure, comments are appreciated

We demonstrate the general outlines of a method for obtaining analytic expressions for certain types of general arithmetical sums. In particular, analytical expressions for a general arithmetical sum whose terms are summed over either the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2+da^2 = N$ or the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2-da^2 = N$ are derived. As one of the consequences, we propose a possible improvement of the Robin-Lagarias criteria for the Riemann hypothesis., Comment: 25 pages, feedback are welcomed

We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level are independent of the actions of the $p$-adic Lie groups $G$ and $G_b$ of the two towers; this generalizes a result of Pan for the Lubin-Tate and Drinfeld spaces for $GL_2$. We apply this theory to show that the $p$-adic Jacquet-Langlands functor of Scholze commutes with the passage to locally analytic vectors, and is compatible with central characters of Lie algebras. We also prove that the compactly supported de Rham cohomology of the two towers are isomorphic as smooth representations of $G\times G_b$., Comment: 41 pages, comments are welcome!

We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$, where $\mathcal{O}_K$ is the ring of integers of $K$, is isomorphic to the Paley-Wiener shuffle algebra associated to a Hecke $L$-function corresponding to $K$., Comment: 25 pages

In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all flat group subschemes of relative dimension at most $t$, gives a set of bounded total height. Our main tools include the Ax--Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan--Zhang. As two applications, we generalize Silverman's specialization theorem to a higher dimensional base, and establish a bounded height result towards Zhang's ICM Conjecture., Comment: 42 pages; comments are welcome!

We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-\omega(k)), $$ where $\omega(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $\nu=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $\nu\geq 0,$ we prove for positive $n$ that $$ p(n)=\frac{1}{g_{\nu}(n,0)}\left(\alpha_{\nu}\cdot \sigma_{2\nu-1}(n)+ \mathrm{Tr}_{2\nu}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_{\nu}(n,k)\cdot p(n-\omega(k))\right), $$ where $\sigma_{2\nu-1}(n)$ is a divisor function, $\mathrm{Tr}_{2\nu}(n)$ is the $n$th weight $2\nu$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_{\nu}(n,k)$ is a polynomial in $n$ and $k.$ The $\nu=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot \tau(n). $$

By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which come from elliptic newforms of weight $2k$ without supercuspidal local components. In particular, our results give a partial answer to Gross' conjecture., Comment: 55 pages

Let $\varphi$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda$ be a mixing horseshoe of $\varphi$. Given a smooth real function $f$ defined on $S$, we define for points $\eta$ in the unstable Cantor set of the pair $(\varphi,\Lambda)$, a generalization, $k_{\varphi,\Lambda,f}(\eta)$, of the best constant of Diophantine approximation for irrational numbers. We study the set of points $\eta$ for which the sets $k_{\varphi,\Lambda,f}^{-1}((-\infty,\eta])$ and $k_{\varphi,\Lambda,f}^{-1}(\eta)$ have the same Hausdorff dimension and when the Hausdorff dimension of $\Lambda$ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, $\mathcal{L}_{\varphi,\Lambda,f}$, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra.

We obtain Fuchs decomposition theorem for regular singular differential modules over a large class of differential rings. We provide a definition of regularity inspired by differential Galois theory and we deduce the classical equivalence with vector spaces endowed with an automorphism., Comment: 34 Pages

We consider sums of Hurwitz class number $H_{m,M}(n)=\sum_{t\equiv m (\text{mod} M)}{H(4n-t^2)}$, where $H(N)$ denotes the Hurwitz class number. In this article, we consider the case of $M=7$. By completing the mixed mock modular form generated by $H_{m,7}(n)$, We obtain the formula of modular forms consist of a computable part and a part from newform 49.2.a.a whose prime terms of Fourier expansion has a connection with with $p=x^2+7y^2$ $(p\equiv 1,2,4 \mod 7)$.

These are notes based on four lectures given at the Heidelberg spring school on non-archimedean geometry and eigenvarieties. None of the contents are original work. Our goal is to explain the construction of eigenvarieties in various different contexts, including the prototypical example of the Coleman--Mazur eigencurve. We will also discuss some of the common geometric properties of eigenvarieties., Comment: 30 pages, to appear in a volume of M\"unster Lectures in Mathematics