Amanda Folsom
We provide an exact formula for the complex exponents in the modular product expansion of the modular units, and deduce a characterization of the modular units in terms of the growth of these exponents, answering a question of W. Kohnen.
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Recently, D. Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $Γ_0(N)$. Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $X_0(N)$ is $0$ or $1$.
In this note, we prove multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a general statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents.
We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. We further find a necessary and sufficient condition for a Siegel modular function of degree $g$ to have neither zero nor pole on the domain when restricted to certain subset of the Siegel upper half-space $\mathbb{H}_g$.
We show that all but 5 of the zeros of the period polynomial associated to a Hecke cusp form are on the unit circle.
Let $p$ be a prime. We prove that if a modular unit has a $p^{th}$ root that is again a modular unit then the level of that root is at most $p$ times the level of the original unit.
Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.
In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood $q$-series that appear in the generalized Rogers-Ramanujan type identities of [6]. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp $0$ generate a subgroup of t…
Parametrizing elliptic curves by modular units
本文证明有理数域上只有有限多条椭圆曲线可由模单位参数化，回答了祖迪林的问题，并列出导子不超过1000的所有此类曲线，最后提出开放问题。
In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross-Stark units and Stark-Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Brumer-Stark units in narrow Hilbert class fields of rea…
The modular discriminant $Δ$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $Δ_N$ at level $N$ which enables one to structure the sequence $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$ in an identical way. We will apply this result to the bases search for each of the spaces $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$.\\ This article is the first in a series of three. In the second part we will propose explicit bases of $(M_{2k}(Γ_0(N)))_{k\in \; \mathbb{N}^*}$ for $1…
The Shintani--Faddeev modular cocycle: Stark units from $q$-Pochhammer ratios
本文将与实二次域相关的Stark单位解释为模上闭链的实乘法值，并证明精化的Shintani的Kronecker极限公式将Stark类不变量平方根与该上同调不变量联系起来。
Modular Units on $X_{1}( p)$ and Quotients of the Cuspidal Group
本文研究素数级 p≥5 时 X_1(p) 上的模单元群，给出其显式基，并利用该基数值研究尖点群结构及确定其一个特定的大商群。
Invariants for the Weil representation and Modular Units for Orthogonal Groups of Signature (2,2)
该研究证明了含自迷向子群的判别群的Weil表示不变量空间由自迷向子群特征函数张成，并应用该结果通过Borcherds乘积构造了符号(2,2)正交群的模单位。
The Multiplicity One Defect and Congruence-Modular Exponent Ratios
本文引入重数一缺陷定量衡量Shimura曲线雅可比重数一性质的失效程度，推导其显式上界，证明其约束模阿贝尔簇同余与模指数比，并给出水平126的Gorenstein Hecke代数实例。
A function field analogue of Ligozat's theorem for Drinfeld modular units
本文研究了函数域中Drinfeld模单位的性质，并通过构造映射建立了有理尖点divisor class group的指数上界。
Modular Functions and the Monstrous Exponents
本文通过模函数方法解释怪物有限简单群阶的素数分解中的指数问题。
Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors
本文通过定义并计算酉 Shimura 簇上除子的算术生成级数及其算术相交数，推导了酉 Shimura 曲线的模高公式及其上 CM 点的高度公式。
Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem
通过比较混合 Eisenstein-theta 级数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。
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…
In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\ell$ by relatin…
This is a revised version of ANT-0049. Given an elliptic curve E --> B over a base B with zero section i, we denote, letting E':= E - i(B), by L(E) the Q-vector space with basis ({s}, s \in E'(B)). Assume that B is smooth and separated over a field of characteristic 0. On the lowest step, the weak version of the elliptic Zagier conjecture predicts the existence of a homomorphism ϕfrom the kernel of a certain differential d on L(E) to the vector space O*(B) \otimes Q of units on B. This homomorphism should behave functorially with respect to change of the base B, and it should satisfy a certain…
Under the generalized Lindelöf hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.
We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah \cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.
We present a proof of the formula, due to Mellit and Brunault, which evaluates an integral of the regulator of two modular units to the value of the $L$-series of a modular form of weight 2 at $s=2$. Applications of the formula to computing Mahler measures are discussed.
Let $φ(τ)=η((τ+1)/2)^2/\sqrt{2π}e^\frac{πi}{4}η(τ+1)$ where $η(τ)$ is the Dedekind eta-function. We show that if $τ_0$ is an imaginary quadratic number with $\mathrm{Im}(τ_0)>0$ and $m$ is an odd integer, then $\sqrt{m}φ(mτ_0)/φ(τ_0)$ is an algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $θ_K$ be an element of $K$ with $\mathrm{Im}(θ_K)>0$ wh…
Comparison of integral structures on the space of modular forms of full level N
本文研究有理数域上全阶N模形式空间两种整结构商的零化子指数，通过结合Brian Conrad方法与Klein形式构造，得到匹配的上下界，最终精确算出该指数。
In this article, we consider the group $F_1^\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\infty(p^n)$ for …
We give two explicit sets of generators of the group of invertible regular functions over QQ on the modular curve Y1(N). The first set of generators is very surprising. It is essentially the set of defining equations of Y1(k) for k <= N/2 when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials. The second set of generators is explicit in terms of classical anal…
Let $q:=e^{2 πiz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Γ_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$…
Given a modular form which is not a cusp form $M_k(z)=\sum_{n=0}^{\infty}r_ne^{2πinz}$ of weight $k \geq 4$, we define the series $M_{k,s}(x)=\sum_{n=1}^{\infty}\frac{r_n}{n^s}\sin(2πnx),$ which converges for all $x\in\mathbb{R}$ when $s>k$. In this paper, we compute the Hölder regularity exponent of $M_{k,s}$ at irrational points. In our analysis we apply wavelets methods proposed by Jaffard in 1996 in the study of the Riemann series. We find that the Hölder regularity exponent at a point $x$ is related to the fine diophantine properties of $x$, in a very precise way.
Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=\text{PSL}(2,\mathbb{R})/\text{PSL}(2,\mathbb{Z})$ of the modular orbifold $\text{PSL}(2,\mathbb Z)$. In fact, the canonical projection $\mathfrak{p}:M\to\mathbb{H}/\text{PSL}(2,\mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $y\to0$: for every smooth fu…
We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.
There are at most finitely many singular moduli that are S-units
本文证明对任意有限素数集S，至多有限多个奇异模数是S-单位，并推广至相关模函数，且特化出无穷多个代数单位的模函数必为弱模单位。
A Lyapunov exponent attached to modular functions
本研究针对满足特定非负条件的SL(2,Z)弱全纯模函数，构造推广已知形式的GL(2,Z)不变李雅普诺夫指数映射，证明其值域性质、在马尔可夫无理数上的递增凸性，及纯周期二次无理数处的值与模曲面周期积分的关系，关联Kaneko关于j函数的猜想。
On the $K_4$ group of modular curves
本文利用权3多对数复形和由Weierstrass $\wp$函数除法值交比构造的新模单位，统一构造模曲线$K_4$群元素，并推测其与Beilinson元素成比例。
We investigate a conjecture to describe the characters of large families of RCFT's in terms of contour integrals of Feigin-Fuchs type. We provide a simple algorithm to determine the modular S-matrix for arbitrary numbers of characters as a sum over paths. Thereafter we focus on the case of 2, 3 and 4 characters, where agreement between the critical exponents of the integrals and the characters implies that the conjecture is true. In these cases, we compute the modular S-matrix explicitly, verify that it agrees with expectations for known theories, and use it to compute degeneracies and multipl…
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