In this paper, we generalize Dorman's work to estimate singular moduli for higher rank Drinfeld modules. In particular, we give a lower bound on the valuation of singular moduli for Drinfeld modules with complex multiplication by an imaginary field extension over the rational function field. Furthermore, we compute several examples for rank-$3$ case.
We construct a compactification of the moduli space of Drinfeld modules of rank $r$ and level $N$ as a moduli space of $A$-reciprocal maps. This is closely related to the Satake compactification, but not exactly the same. The construction involves some technical assumptions on $N$ that are satisfied for a cofinal set of ideals $N$. In the special case $A={\mathbb F}_q[t]$ and $N=(t^n)$ we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect the same results in gener…
We construct certain rational functions (modular units) on the moduli stack of Drinfeld shtukas. The divisors of these rational functions are supported on horospherical divisors of the moduli stack. The key to our construction is a vanishing theorem for shtukas with zeros and poles satisfying certain conditions. Using deformation theory, we calculate the divisors of these rational functions.
Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroup…
In this paper I'm going to study the intersection of two Heegner-Drinfeld cycles coming from two different nonsplit tori on the Yun-Zhang moduli stack of $PGL_2$ Drinfeld stukas with Iwahori level structure. We will see that the intersection number is related to a certain period integral. It is an extension of the result by Howard-Shinidman to the Iwahori case.
We study $\mathscr{D}$-elliptic sheaves in terms of their associated modules, which we call Drinfeld-Stuhler modules. We prove some basic results about Drinfeld-Stuhler modules and their endomorphism rings, and then examine the existence and properties of Drinfeld-Stuhler modules with large endomorphism algebras, which are analogous to CM and supersingular Drinfeld modules. Finally, we examine the fields of moduli of Drinfeld-Stuhler modules.
Motivated by the question of constructing certain rational functions (modular units) on the moduli stack of Drinfeld shtukas, we introduce the notion of toy shtukas. We prove basic properties of the moduli scheme of toy shtukas. Analogously to horospherical divisors on the moduli stack of Drinfeld shtukas, there are toy horospherical divisors on the moduli scheme of toy shtukas. We describe the space of principal toy horospherical divisors. There is a canonical morphism from the moduli stack of Drinfeld shtukas to the moduli scheme of toy shtukas. Our main result is a description of the space …
Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $V$ be a vector space over $\mathbb{F}_q$ of dimension $n>0$. Let $Ω_V$ be the Drinfeld period domain over $\mathbb{F}_q$. This is an affine scheme of finite type over $\mathbb{F}_q$, and its base change to $\mathbb{F}_q(t)$ is the moduli space of Drinfeld $\mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder's smooth compactification $B_V$ of $Ω_V$. Let $\hat V$ be the set $V\cup\{\infty\}$ for a new symbol $\infty$. We define the notion of a $V…
A new proof of an old theorem of Drinfeld concerning the representability of the moduli problem of special formal $\mathcal{O}_{D}$-modules by Deligne's $p$-adic formal model of Drinfeld's upper half-plane is given for $d=2.$ The display associated to the universal object is constructed explicitly.
We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\F_q[T]$-modules with a structure of level $N \in \F_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N)$, the projective variety associated with the graded ring ${\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{\MF}_r(N)$. Modular forms are characterized…
Torsion of Drinfeld modules and equicharacteristic unimodular Galois covers
本文证明了群PSL（r，q^d）可以F_q（T）-正则地实现为纯超越域F_q（T）（t_1，.，t_{r-1}）如果r是偶数并且r/2与q^d-1互质。该方法是使用Drinfeld模的扭模方案（a la Shih）。如果人们对模形式有足够的控制，就可以把它变成一种算法来求显式方程。对于PSL（2，q^2），我们使用Drinfeld模形式来实现这一点。
We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-…
Let $F$ be the field of rational functions on a smooth projective curve over a finite field, and let $π$ be an unramified cuspidal automorphic representation for $\mathrm{PGL}_2$ over $F$. We prove a variant of the formula of Yun and Zhang relating derivatives of the $L$-function of $π$ to the self-intersections of Heegner-Drinfeld cycles on moduli spaces of shtukas. In our variant, instead of a self-intersection, we compute the intersection pairing of Heegner-Drinfeld cycles coming from two different quadratic extensions of $F$, and relate the intersection to the $r$-th derivative of a produc…
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.
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.
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 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.
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 …
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.
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.
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.
