Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $Δ$-quotients, where $Δ$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$.
We show that the Drinfeld modular forms with $A$-expansion that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with $A$-expansions that remain modular after hyperdifferentiation. In addition, we show that Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with $A$-expansions are Drinfeld-Poincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen…
In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(Γ_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.
Let $ \mathfrak{C}^+_{ns}(p) $ be the Cuspidal Divisor Class Group of the modular curves $X^+_{ns}(p) $ associated to the normalizer of a non-split Cartan subgroup of level $ p$. I study the $ p-$primary part of $ \mathfrak{C}^+_{ns}(p) $ and estimate the order of growth of $ |\mathfrak{C}^+_{ns}(p)| $.
Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.
Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a square-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}]\neq 0$. To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$ contained in $\mathfrak{m}$ by computing the order of a cuspidal divisor annihilated by $\mathcal{I}$.
Suppose we are given a Drinfeld Module $ϕ$ over $\mathbb{F}_q(t)$ of rank $r$ and a prime ideal $\mathfrak{l}$ of $\mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $\mathfrak{l}$ Galois representation $${\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\rightarrow {\rm{Aut}}(ϕ[\mathfrak{l}])\cong {\rm{GL}}_r(\mathbb{F}_\mathfrak{l})$$ gives a bound on the degree of $\mathfrak{l}$ which depends only on the rank $r$ of Drinfeld module $ϕ$ and the minimal degree of place $\mathcal{P}$ where $ϕ$ has good reduction at $\mathcal{P}$. Then, we apply this reducibility bound t…
Consider the Drinfeld modular curve $X_0(\mathfrak{p})$ for $\mathfrak{p}$ a prime ideal of $\mathbb{F}_q[T]$. It was previously known that if $j$ is the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$, then the reduction of $j$ modulo $\mathfrak{p}$ is a supersingular $j$-invariant. In this paper we show the converse: Every supersingular $j$-invariant is the reduction modulo $\mathfrak{p}$ of the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$.
Let $ψ: A \to F\{τ\}$ be a Drinfeld $A$-module over $F$ of rank 2 and without complex multiplication, where $A = {\mathbb{F}}_q[T]$, $F = {\mathbb{F}}_q(T)$, and $q$ is an odd prime power. For a prime $\mathfrak{p} = p A$ of $A$ of good reduction for $ψ$ and with residue field ${\mathbb{F}}_{\mathfrak{p}}$, we study the growth of the absolute value $|Δ_{\mathfrak{p}}|$ of the discriminant of the ${\mathbb{F}}_{\mathfrak{p}}$-endomorphism ring of the reduction of $ψ$ modulo $\mathfrak{p}$. We prove that for all $\mathfrak{p}$, $|Δ_{\mathfrak{p}}|$ grows with $|p|$. Moreover, we prove that for a…
The present paper deals with Atkin-Lehner theory for Drinfeld modular forms. We provide an equivalent definition of $\mathfrak{p}$-newforms (which makes computations easier) and commutativity results between Hecke operators and Atkin-Lehner involutions. As applications we show a criterion for a direct sum decomposition of cusp forms, we exibit $\mathfrak{p}$-newforms arising from lower levels and we provide $\mathfrak{p}$-adic Drinfeld modular forms of level greater than 1.
Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $ϕ$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $ϕ$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $ϕ\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(ϕ\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_…
We introduce the notion of Drinfeld modular forms with $A$-expansions, where instead of the usual Fourier expansion in $t^n$ ($t$ being the uniformizer at `infinity'), parametrized by $n \in \mathbb{N}$, we look at expansions in $t_a$, parametrized by $a \in A = \Fq [T]$. We construct an infinite family of eigenforms with $A$-expansions. Drinfeld modular forms with $A$-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) the computation of the eigensyste…
In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{BV19a}) for $S_{k,l}(Γ_0(T))$ when $\mathrm{dim}\ S_{k,l}(\mathrm{GL}_2(A))\leq 2$. Then, we frame this conjecture for prime, higher levels, and provide some evidence in favour of it. For any square-free level $\mathfrak{n}$, we define oldforms $S_{k,l}^{\mathrm{old}}(Γ_0(\mathfrak{n}))$, newforms $S_{k,l}^{\mathrm{new}}(Γ_0(\mathfrak{n}))$, and investigate their properties. These properties depend on the commutativity of the (partial) Atkin-Lehner operators with the $U_\mathfrak{p}$-operators. Finally, we show…
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 $|\mathfrak{C}^+_{ns}(p^k)|$.
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 …
For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(\mathfrak{n})$ with some square-free level $\mathfrak{n}$. This is an analogue of Elkies' well-known result on $\mathbb{Q}$-curves.
In this paper, let $ϕ$ be the Drinfeld module over $\mathbb{F}_{q}(T)$ of prime rank $r$ defined by $$ϕ_T=T+τ^{r-1}+T^{q-1}τ^r.$$ We prove that under certain condition on $\mathbb{F}_q$, the adelic Galois representation $$ρ_ϕ:{\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\longrightarrow \varprojlim_{\mathfrak{a}}{\rm{Aut}}(ϕ[\mathfrak{a}])\cong {\rm{GL_r}}(\widehat{A})$$ is surjective.
Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope zero for a suitably defined Hecke operator $\mathrm{U}_{\mathfrak{p}}$. Second, we show the existence in the finite slope case of families of Drinfeld modular forms varying continuously with respect to the weight. Finally, we show a classicity result: an overconvergent Drinfeld modular form of sufficiently small slope with respect to the weight is a classical D…
Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is prime to $\wp$. Let $k\geq 2$ and $r\geq 1$ be integers. We denote by $S_k(Γ_1(\mathfrak{n}\wp^r))$ the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$ for $A$. Let $n\geq 1$ be an integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}\wp$ has a prime factor of degree one and the generalized eigenspace in $S_k(Γ_1(\mathfrak{n}\wp^r…
We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.
In this paper, we consider automorphic forms on $\mathrm{Sp}_4(\mathbb{A}_\mathbb{Q})$ which generate large discrete series representations of $\mathrm{Sp}_4(\mathbb{R})$ as $(\mathfrak{sp}_4(\mathbb{R}),K_\infty)$-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.
Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld modules over $\mathbf{F}_q[t]$ and, using it, develop a geometric theory of $\wp$-adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at $\infty$ modulo $\wp^n$, their weights are also congruent modulo $(q^d-1)p^{\lceil \log_p(n)\rcei…
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 …
We consider cuspidal representations in spaces of automorphic forms for the congruence subgroup $Γ_0(I)$ of Hilbert modular groups for some number field $F$. To each such representation are associated the eigenvalue $λ_j$ of the Casimir operator at each real place $j$ of $F$, and the number $\ld_{\mathfrak p}$ parametrizing the eigenvalue of the Hecke operator $T_{\mathfrak p^2}$ at each finite place $\mathfrak p$ outside the ideal $I$. We study the joint distribution of the $λ_j$ for all real places $j$, and the $\ld_{\mathfrak p}$ for finitely many $\mathfrak p$ outside $I$, over the cuspida…
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 study the Eisenstein ideal of Drinfeld modular curves of small levels, and the relation of the Eisenstein ideal to the cuspidal divisor group and the component groups of Jacobians of Drinfeld modular curves. We prove that the characteristic of the function field is an Eisenstein prime number when the level is an arbitrary non square-free ideal of $\mathbb{F}_q[T]$ not equal to a square of a prime.
For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…
Let $f$ be a Drinfeld modular form for $Γ_0(\mathfrak{p})$. From such a form, one can obtain two forms for the full modular group $\operatorname{GL}_2(A)$: by taking the trace or the norm from $Γ_0(\mathfrak{p})$ to $\operatorname{GL}_2(A)$. In this paper we show some connections between the arithmetic modulo $\mathfrak{p}$ of the coefficients of the $u$-series expansion of $f$ and those of a form closely related to its trace, and of the coefficients of $f$ and those of its norm.
Given a cuspidal Hilbert modular eigenform $π$ of parallel weight 2 and a nonarchimedian place $\mathfrak p$ of the underlying totally real field such that the local component of $π$ at $\mathfrak p$ is the Steinberg representation, one can associate two types of $\mathscr L$-invariants, one defined in terms of the cohomology of arithmetic groups and the other in terms of the Galois representation associated to $π$. We show that the $\mathscr L$-invariants are the same.
Given $\mathbf{F}$ a number field with ring of integers $\mathcal{O}_{\mathbf{F}}$ and $\mathfrak{p},\mathfrak{q}$ two squarefree and coprime ideals of $\mathcal{O}_{\mathbf{F}}$, we prove a reciprocity relation for the first moment of the triple product $L$-functions $L(π\otimesπ_1\otimesπ_2,\frac{1}{2})$ twisted by $λ_π(\mathfrak{p})$, where $π_1$ and $π_2$ are a fixed unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ with $π_1$ cuspidal and $π$ runs through unitary automorphic representations of conductor dividing $\mathfrak{q}$. The method uses adelic integral…
Let $π$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at $\mathfrak{p}$ and the local component $π_\mathfrak{p}$ of $π$ at $\mathfrak{p}$ is the Steinberg representation. Assuming that the representation is non-critical at $\mathfrak{p}$ we construct automorphic $\mathcal{L}$-invariants for the representation $π$. If the number field $F$ is totally real, we show that these automorphic $\mathcal{L}$-invariants agree with the Font…
Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $τ$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to $Γ_\infty \backslash PSL(2,{\mathbb Z})$. The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg…
Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial endomorphism ring, with $k= \mathbb{Q}$, $A = \mathbb{Z}$ in the former case and $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case, for any nonzero ideal $\mathfrak{a} \lhd A$ we prove best possible estimates in the norm $|\mathfrak{a}|$ for the degrees over $K$ of the subfields of the $\mathfrak{a}$-div…
A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations
Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；CNRS, IMJ-PRG（法国国家科学研究中心，IMJ-PRG）；School of Mathematics, Southampton University（南安普顿大学数学学院）；School of Mathematics, Manchester University（曼彻斯特大学数学学院）
Let $π$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $π$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(\frac{1}{2},π\timesχ)$, with a certain automorphic character $χ$ of ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results…
