Explicit unobstructed primes for modular deformation problems of squarefree level

Modular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree

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…

无平方因子水平下的模p新形式及其在Monsky Hecke稳定滤过上的应用

Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

本文提出无平方因子水平下模p新形式的代数定义，解释Monsky描述的模2水平3模形式空间分次部分的Hecke代数，并给出特征p下仍为模形式代数自同构的重整化Atkin-Lehner对合版本。

模曲线和志村曲线商中具有良好约化的意外素数

Unexpected primes of good reduction in quotients of modular and Shimura curves

研究权重2、平方自由层数N且有固定阿特金 - 勒纳符号模式的新形式零维空间，借此对平方自由层数的模曲线和志村曲线阿特金 - 勒纳商中良好约化的意外素数进行分类。

Existence of $η$-Quotients For Squarefree Levels

In this paper, we investigate which modular form spaces can contain $η$-quotients, functions of the form $f(τ) = \prod_{δ\mid N} η(δτ)^{r_δ}$. For $k\geq 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(Γ_1(N))$ to contain $η$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.

Fermat's Last Theorem and modular curves over real quadratic fields

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.

Hecke eigenvalues of Klingen--Eisenstein series of squarefree level

We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $χ$ by using formulas for the action of the Hecke operators on Fourier expansions. Using an explicit description of the Satake compactification of $Γ_0^{(n)}(N) \backslash \mathfrak{H}_n$ when $N$ is squarefree we extend this to give intertwining relations for each cusp. As an application we give formulas for the action of Hecke operators on the space of Klingen--Eisenstein series of squarefree level $N$, for primes $p \nmid N$.

Factorization tests and algorithms arising from counting modular forms and automorphic representations

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $Γ_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $Γ_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic repre…

Higher congruences between newforms and Eisenstein series of squarefree level

Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $Γ_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert-…

The Eisenstein ideal with squarefree level

We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that "multiplicity one" fails for the modular Jacobian in these…

Sign of Fourier coefficients of modular forms of half integral weight

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

可约模$\\,\ell$伽罗瓦表示的某些无平方因子水平

Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations

本文确定了模$\ell$表示$1 \oplus \overline{\chi}_\ell^{k-1}$来自权$k$、水平为两素数乘积且具有指定Atkin-Lehner特征值的新形式的充要条件，证明了Billerey-Menares猜想并给出大量应用实例。

Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.

A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on $GL_2(\mathbb F_q(T))$ with squarefree level

The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension of the $i$th stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection-theoretic problem of bounding the "polar multiplicities" of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newfor…

Equidistribution of Signs for Modular Eigenforms of Half Integral Weight

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it f…

A reduction of integer factorization to modular tetration

Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] In this paper, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to compute the prime factorization of $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

Periods of modular forms on $Γ_0(N)$ and products of Jacobi theta functions

Generalizing a result of~\cite{Z1991} for modular forms of level~one, we give a closed formula for the sum of all Hecke eigenforms on $Γ_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that for $N=2$,~3 and~5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on $Γ_0(N)$.

Rational torsion of generalised modular Jacobians of odd level

We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group structure of the rational torsion $J_{0}(N)_{\mathbf{m}}(\mathbb{Q})_{tor}$ up to $2$-primary and $l$-primary parts for any prime $l$ dividing $N$. Our results extend those of Wei--Yamazaki for squarefree levels and Yamazaki--Yang for prime-power levels.

Subconvexity for modular form L-functions in the t aspect

Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus resolving a problem that has been open for 35 years. A key innovation in our proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree.

Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when $n$ runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $n\le x$ such that $a(n)>n^{-α}$ where $x$ and $α$ are positive and $f$ is not necessarily a Hecke eigenform.

On the Convex Hull of the Points on Modular Hyperbolas

Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the convex closure of $\Hm$, and show that uniformly over all $a$ with $\gcd(a,m)=1$ we have $v_a(m) \le m^{1/2 + o(1)}$ and furthermore, we have $v_a(m) \le m^{5/12 + o(1)}$ for $m$ which are almost squarefree.

当$N$为非无平方因子数时$X_0(N)^*$上的有理点

Rational points on $X_0(N)^*$ when $N$ is non-squarefree

该研究针对非无平方因子$N$对应的正亏格模曲线商$X_0(N)^*$，证明相关模雅可比簇零秩商的存在性，建立非尖点有理点$j$-不变量的整性结果，完成1到5亏格下有限有理点的分类，并发现两个新的例外有理点，推进了Elkies猜想的相关研究。

Sign changes of a product of Dirichlet character and Fourier coefficients of half integral weight modular forms

Let $f\in S_{k+1/2}(N,χ)$ be a Hecke eigenform of half integral weight $k+1/2\,(k\geq 2)$ and the real nebentypus $χ=\pm 1$ where the Fourier coefficients $a(n)$ are reals. We prove that the sequence $\{χ(p^ν)a(tp^{2ν})\}_{ν\in\N}$ has infinitely many sign changes for almost all primes $p$ where $t$ is a squarefree integer such that $a(t)\neq 0$. The same result holds for the sequences of Fourier coefficients $\{a(tp^{2(2ν+1)})\}_{ν\in\N}$ and $\{a(tp^{4ν})\}_{ν\in\N}$.

Bias of Root Numbers for Modular Newforms of Cubic Level

Let $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $φ(N)$ and the class number of $\mathbb{Q}(\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.

数域上的广义费马方程

Generalized Fermat equation over number fields

在两个标准数论猜想假设下，利用模方法研究数域上广义费马方程的渐近行为，建立基于$S_K'$-单位方程解的渐近判别准则，针对一类虚二次域证明方程无渐近解，且该类无平方因子整数相对密度为5/6。

Fermat's Last Theorem over some small real quadratic fields

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,χ_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $χ_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic p…

Products of Eisenstein series and Fourier expansions of modular forms at cusps

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.

A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

Let f be a modular form of weight k and Nebentypus $ψ$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes ξ)$, where $ξ$ is a Dirichlet character. When s=k-1 and $ξ=ψ^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implie…

Equidistribution of cusp forms in the level aspect

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincaré measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show that newforms of level 1 and large weight have equidistributed mass. …

关于素数树的有限性及其与模形式的关系

On the finiteness of prime trees and their relation to modular forms

本文研究了素数树的有限性及其与模形式的关系，证明了在特定条件下模形式可由两个特定艾森斯坦级数的线性组合表示。

On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $Γ_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, where $\mathcal{C}(N)$ is the …

算术子群的形式西格尔模形式

Formal Siegel modular forms for arithmetic subgroups

本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。

On the modularity level of modular abelian varieties over number fields

Let f be a weight two newform for Gamma_1(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A_f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N_L(B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N_L(B) belongs…

Non annulation des fonctions L automorphes au point central

The question about modular forms have recently received a lot of attention; concerning the non-vanishing of automorphic L-functions Michel, Kowalski and Vanderkam proved (among others results) that there's positive proportion of non-vanishing of primitives forms at the critical point. This result was proved by these authors in the prime level case; on the othe hand, Iwaniec, Luo and Sarnak showed the same result for the squarefree level case. In order to understand the influence of the arithmetic shape of level forms in their vanishing, this paper studies a generalisation to the primitives for…

扭曲L值的精确双重平均值

Exact double averages of twisted L-values

本文研究虚二次扩张扭曲的模形式中央L值的双重平均值，在权为2时给出精确公式，并证明其在稳定范围内的稳定性，推广了已知的单重平均值并得到非消失的有效结果。

$Γ_0(N)$上权为一的Jacobi形式

Jacobi forms of weight one on $Γ_0(N)$

本文推广了Skoruppa等人的结果，通过改进其方法并分析Weil表示的局部不变量，确定了使$J_{1,m}(N)=0$的所有层级$N$，建立了显式维数公式，并证明了部分二阶Siegel模形式的消失性。

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions…

Galois变形环的算术统计

Arithmetic statistics for Galois deformation rings

研究椭圆曲线Galois变形环的无障碍性，证明固定无复乘曲线对几乎所有素数无障碍，并基于启发式与计算表明随素数增大变形环光滑比例接近100%。

最小负 Hecke 特征值的 Omega 结果

An omega result for the least negative Hecke eigenvalue

本文证明存在许多大权重的全纯 Hecke 本征形式，其最小负 Hecke 特征值对应的整数至少为 (log k)^{1-o(1)}，改进了 Kowalski 等人的结果，并推广到平方自由水平。