For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $η(mτ+k/h)$, $mh^2|N$ and $k\in\mathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.
For a positive integer $N$, let $\mathscr{C}_N(\mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $\mathscr{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion subgroup of $J_0(N)$. We prove that two groups $\mathscr{C}_N(\mathbb{Q})$ and $\mathscr{C}(N)$ are equal when $N=p^2M$ for any prime $p$ and any squarefree integer $M$. To achieve this we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we deter…
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,χ}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.
In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function valid for any integer and fully explicit.
In [6], Kohnen proves that if $Γ=Γ_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $Γ$ having a divisor supported at the cusps is an $η$-product. Under the condition of having rational Fourier coefficients, we are able to extend Kohnen's result to the case where $N$ is the square of a prime. If the rationality condition does not hold, we show that the statement is no longer true by providing a family of counter-examples that generalizes naturally the Dedekind $η$-function. This paper fits within the framework of generalized modular forms in the sense of Kn…
For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$. More specifically, for a given prime $\ell$, we construct a rational cuspidal divisor $Z_\ell(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class of the divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the lin…
We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
Let $Γ$ be the Fuchsian group of the first kind. For an even integer $m\ge 4$, we study $m/2$-holomorphic differentials in terms of space of (holomorphic) cuspidal modular forms $S_m(Γ)$. We also give in depth study of Wronskians of cuspidal modular forms and their divisors.
Rational points on $X_0(N)^*$ when $N$ is non-squarefree
Brown University（布朗大学）；Universität Würzburg（维尔茨堡大学）；Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik（汉诺威莱布尼茨大学代数、数论与离散数学研究所）；Laboratoire de mathématiques Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学数学实验室）
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq 0,4,7 \,(\operatorname{mod}8)$ is represented as $n= x_1^2 + x_2^2 + x_3^3$ for integers $x_1,x_2,x_3$ such that the product $x_1x_2x_3$ has at most 72 prime divisors.
Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ and $d$ be a fundamental discriminant. Hoffstein and Kontorovich ask for a bound on the least $|d|$ (if it exists) such that the central value $L(1/2, π\otimes χ_d) \neq 0$. The bound should be given in terms of the weight, Laplace eigenvalue and/or level of $π$. Let $f$ be a holomorphic twist-minimal newform of even weight $\ell$, odd cubefree level $N$, and trivial nebentypus. When $π\cong π_f$ and the squarefree part of $N$ is of appropriate size, we conditionally improve upon level aspect res…
Let $F_\infty=\mathbb{F}_q(\!(1/T)\!)$ be the completion of $\mathbb{F}_q(T)$ at $1/T$. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat-Tits building of $\mathrm{PGL}_r(F_\infty)$, $r\geq 2$, generalizing an earlier construction of Gekeler for $r=2$. We then apply this theory to study modular units on the Drinfeld symmetric space $Ω^r$ over $F_\infty$, and the cuspidal divisor groups of Satake compactifications of certain Drinfeld modular varieties. In particular, we obtain a higher dimensional analogue of a result of Ogg for classical modular curves …
Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case , where we can consider convenient generalized …
We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence $\{U_n\}_{n\ge 0}$ of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}.\] A $k$-Wall-Sun-Sun prime is a prime $p$ such that \[U_{π_k(p)}\equiv 0 \pmod{p^2},\] where $π_k(p)$ is the length of the period of $\{U_n\}_{n\ge…
Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$. We say $f(x)$ is \emph{monogenic} if $Θ=\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers ${\mathbb Z}_K$ of $K={\mathbb Q}(θ)$, where $f(θ)=0$. If $Θ$ is not a basis for ${\mathbb Z}_K$, we say that $f(x)$ is \emph{non-monogenic}. Let $k\ge 1$ be an integer, and let $(U_n)$ be the sequence defined by \[U_0=U_1=0,\quad U_2=1 \quad \mbox{and}\quad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} \quad \mbox{for $n\ge 3$}.\] It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2…
