Robert Osburn
We extend a result of Ahlgren and Ono on congruences for traces of singular moduli of level 1 to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.
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We address a question posed by Ono, prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results coincides with a recent result of Edixhoven, and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.
We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.
After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $Γ_0(N)$.
Congruences for traces of singular moduli and Hurwitz - Kronecker class numbers
本文研究了模数迹与Hurwitz-Kronecker类数之间的同余关系，通过U-算子作用下的模形式同余，揭示了模11下的特殊同余条件。
Computations and observations on congruence covering systems
该研究针对同余覆盖系领域，完成了模数不超过10的不同模覆盖系的完整分类，并据此得出所有模大于2的不同模覆盖系的最小基数为11的结论。
Zagier proved that the generating series for the traces of singular moduli is a \textit{weakly holomorphic} modular form of weight 3/2 on $Γ_0(4)$. Bruinier and Funke extended the results of Zagier to modular curves of arbitrary genus. Zagier also showed that the twisted traces of singular moduli are generated by a weakly holomorphic modular form of weight 3/2. In this paper, we study the extension of Zagier's result for the twisted traces of singular moduli to congruence subgroups $Γ_0(N)$. As an application, we study congruences for the twisted traces of singular moduli of weakly holomorphic…
Small solutions of ternary quadratic congruences with averaging over the moduli
本文研究了三次二次同余式的解的存在性，通过模数平均方法，证明了在特定条件下解的高度上限为Q^{3/8+ε}α₂^{ε}。
We will generalize Osburn's work about a congruence for traces defined in terms of Hauptmodul associated to certain genus zero groups of higher levels.
Small solutions of generic ternary quadratic congruences to general moduli
本文研究一般奇数模数下三元二次同余方程的小解，利用Burgess和Heath-Brown的特征和估计突破指数1/2障碍，在平均意义下对几乎所有系数得到解的渐近公式，并提出最小解大小的精细猜想。
Hilbert modular Eisenstein congruences of local origin
本文研究了希尔伯特模形式与欧拉特征级数之间的欧拉特征 congruences，探讨了其模数来源及新形式满足的条件。
We determine couples of singular moduli which have rational products
We prove a refinement of the results of Gross and Zagier on prime factorizations of singular moduli.
Congruences for Siegel modular forms of nonquadratic nebentypus mod $p$
本文证明了非二次nebentypus的Siegel模形式同余时权重满足同余关系，由此得出不存在模$p$奇异形式，并利用Eisenstein级数构造了模$\ rak{p}$奇异形式的例子。
Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In the classical case, the maps can be given analytically. However, in the Shimura curve cases, no such analytical expansion is possible. Fortunately, in both cases there are known algorithms for algebraically computing the rational norms of the singular moduli. We demonstrate a method of using these norm algorithms to algebraically determine the minimal polynomial of the singular moduli below a discriminant threshold. We then use these mini…
Let ${\mathbf F}_q$ denote a finite field of characteristic $p$ and let $n$ be an effective divisor on the affine line over ${\mathbf F}_q$ and let $v$ be a point on the affine line outside $n$. In this paper, we get congruences between ${\mathbb Q}_l$-valued weight two $v$-old Drinfeld modular forms and $v$-new Drinfeld modular forms of level $vn$. In order to do this, we shall first construct a cokernel torsion-free injection from a full lattice in the space of $v$-old Drinfeld modular forms of level $vn$ into a full lattice in the space of all Drinfeld modular forms of level $vn$. To get th…
We show that the field $\mathbb{Q}(x,y)$, generated by two singular moduli~$x$ and~$y$, is generated by their sum ${x+y}$, unless~$x$ and~$y$ are conjugate over~$\mathbb{Q}$, in which case ${x+y}$ generates a subfield of degree at most~$2$. We obtain a similar result for the product of two singular moduli.
We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ),j(τ') such that the numbers 1, j(τ)^m and j(τ')^n are linearly dependent over $\mathbb{Q}$ for some positive integers m,n, must be of degree at most 2. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number.
On singular moduli for higher rank Drinfeld modules
本研究推广Dorman的工作，估计高秩Drinfeld模的奇异模，给出带复乘的Drinfeld模奇异模赋值的下界，并计算3秩情形的实例，拓展了相关数论研究。
We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight $3/2$ whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla-Millson theta lift of Bruinier and Funke to meromorphic modular functions.
We compute the divisor of the modular equation on the modular curve $Γ_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $Γ_0(N)$ of genus zero. We also introduce the notions and properties of $Γ$-equivalence and $Γ$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.
For a fixed singular modulus $α$, we give an effective lower bound of norm of $x-α$ for another singular modulus $x$ with large discriminant. We then generalize this result for $Φ_m(x,α)$, where $Φ_m(X,Y) \in \Z[X,Y]$ is the $m$-th modular polynomial.
We prove that $|x-y|\ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the "primitive element problem" for two singular moduli. In a previous article Faye and Riffaut show that the number field $\mathbb Q(x,y)$, generated by two singular moduli $x$ and $y$, is generated by $x-y$ and, with some exceptions, by $x+y$ as well. In this article we fix a rational number $α\ne0,\pm1$ and show that the field $\mathbb Q(x,y)$ is generated by $x+αy$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field o…
Riffaut (2019) conjectured that a singular modulus of degree $h\ge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.
We prove that only finitely many $j$-invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Siegel's Theorem on integral points of algebraic curves.
We determine the asymptotic behavior of twisted traces of singular moduli with a power-saving error term in both the discriminant and the order of the pole at $i\infty$. Using this asymptotic formula, we obtain an exact formula for these traces involving the class number and a finite sum involving the exponential function evaluated at CM points.
Let $p$ be a prime for which the congruence group $Γ_0(p)^*$ is of genus zero, and $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $Γ_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and extend Zagier's trace formula for singular values of $j_p^*$. Further we investigate the twisted traces of sigular values of $j_p^*$ and construct Borcherds products re…
Gross and Zagier proved a formula for the absolute norm N(j(α_1) - j(α_2)) of a difference of singular values of the modular function j. We formulate and prove the analogues of their result for a number of functions of level 2 and 3.
Antisymmetry of real quadratic singular moduli
本文通过分析四变量分裂正交群的刚性解析余弦，证实了实二次奇异模的反称性，并证明了Kudla-Millson除数生成函数的模性。
This article is an overview of Zagier's and Kim's work on traces of singular moduli. We give more detailed or new proofs to some of their results and also describe some algorithms to compute spaces of Jacobi forms and weight $3/2$ modular forms. Also, for $p$ a prime dividing the order of the Monster group, we give an explicit construction of the Hauptmodul $j_p$ of $Γ_0(p)^*$ using Rademacher series and Rademacher sums.
We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincaré series with evaluations of Niebur-Poincaré series as Fourier coefficients. This allows us to study twisted traces of singular moduli in an integral weight setting. In particular, we recover explicit series expressions for twisted traces of singular moduli and extend algebraicity results by Bengoechea to the weight $2$ case. We also compute regularized inner products of these …
Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.
We show that two distinct singular moduli $j(τ),j(τ')$, such that for some positive integers $m, n$ the numbers $1,j(τ)^m$ and $j(τ')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree $3$. The purpose of this article is to treat these two remaining cases.
Heights and singular moduli of Drinfeld modules
本文发展了Drinfeld模的高技术，证明了在$q>5$为奇数时，秩为2的$\mathbb{F}_q[t]$-Drinfeld模中至多仅有有限个奇异模为代数单位。
Let $a \in \mathbb{Z}_{>0}$ and $ε_1, ε_2, ε_3 \in \{\pm 1\}$. We classify explicitly all singular moduli $x_1, x_2, x_3$ satisfying either $ε_1 x_1^a + ε_2 x_2^a + ε_3 x_3^a \in \mathbb{Q}$ or $(x_1^{ε_1} x_2^{ε_2} x_3^{ε_3})^{a} \in \mathbb{Q}^{\times}$. In particular, we show that all the solutions in singular moduli $x_1, x_2, x_3$ to the Fermat equations $x_1^a + x_2^a + x_3^a= 0$ and $x_1^a + x_2^a - x_3^a= 0$ satisfy $x_1 x_2 x_3 = 0$. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establ…
We establish an effective version of the André-Oort conjecture for linear subspaces of $Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$. Apart from the trivial examples provided by weakly special subvarieties, this yields the first algebraic subvarieties in a Shimura variety of dimension $> 1$ whose CM-points can be (theoretically) determined.
Drinfeld singular moduli, hyperbolas, units
该研究针对有限域多项式环上的秩2Drinfeld模，证明低次多项式对应的双曲线上无Drinfeld奇异模坐标点（一类有效André-Oort定理），还证固定素数幂下Drinfeld奇异模中代数单位有限并给判别式有效界，附录推广了经典模形式值的不可分性准则。
Let $g$ be a principal modulus with rational Fourier coefficients for a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ between $Γ(N)$ or $Γ_0(N)^†$ for a positive integer $N$. Let $K$ be an imaginary quadratic field. We give a simple proof of the fact that the singular value of $g$ generates the ray class field modulo $N$ or the ring class field of the order of conductor $N$ over $K$. Furthermore, we construct primitive generators of ray class fields of arbitrary moduli over $K$ in terms of Hasse's two generators.
At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $α_n$. All those results were proved by Berndt et. al by employing Weber-Ramanujan's class invariants. In this paper, we initiate to derive the explicit evaluations formula for $α_{9n}$ and $α_{n/9}$ by involving class invariant. For this purpose, we establish several new $P-Q$ mixed modular equations involving theta-functions. Further application of these modular equations, we derive a new formula to explicit evaluation of Ramanujan- Selberg continued fraction.
Effective estimates for traces of singular moduli
本文针对奇异模的广义扭曲迹，利用 $Γ_0(4)$ 上 Kloosterman 和之和的显式界，建立了具有有效误差界的渐近公式，改进了 Andersen 和 Duke 的相关估计。
A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.
We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature (n,2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can occur in this factorization. One remarkable phenomenon we observe is that the regularized theta lift of a weakly holomorphic modular form is always finite.
The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over $\mathbb{Q}$. This paper shows that the coordinate maps for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to…
In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and $p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$ and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva and Sellers …
In this paper, we partially extend recent results of Wan concerning the relationship between the zeta functions of a Calabi-Yau hypersurface and its (singular) mirror variety.
An archimedean approach to singular moduli on Shimura curves
本文通过评估Shimura曲线上的Green函数在CM点上的值，提供了Gross-Zagier工作在Shimura曲线上的新证明，强调了与p-进方法的异同。
We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)
The Gross-Zagier formula on singular moduli for Shimura curves
研究志村曲线上关于奇异模的格罗斯 - 扎吉尔公式，通过证明将其在模曲线积分模型上的相关结果推广到志村曲线，得到了该公式的一种推广情形。
We study the field of moduli of singular abelian and K3 surfaces. We discuss both the field of moduli over the CM field and over $\Q$. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an explicit approach to the computation of the field of moduli.
Given a singular modulus $j_0$ and a set of rational primes $S$, we study the problem of effectively determining the set of singular moduli $j$ such that $j-j_0$ is an $S$-unit. For every $j_0 \neq 0$, we provide an effective way of finding this set for infinitely many choices of $S$. The same is true if $j_0=0$ and we assume the Generalized Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a "uniformity conjecture" for singular $S$-units.
Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.
In this note, we provide evidence for new (super) moonshines relating the Monster and the Baby monster to some weakly holomorphic weight 1/2 modular forms defined by Zagier in his work on traces of singular moduli. They are similar in spirit to the recently discovered Thompson moonshine.
We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric i…
We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of $\text{SL}_2(\mathbb{Z})$ on these level structures. If $Γ$ is a stabilizer of this action, then the quotient of the upper half plane by $Γ$ parametrizes isomorphism classes of elliptic curves equipped with $G$-structures. When $G$ is "sufficiently" nonabelian, the stabilizers $Γ$ are noncongruence. As a result we realize noncongruence modular curves as moduli spaces of elli…
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。
We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay--Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Tho…
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