Marc Reversat
We compare modular forms of characteristic $p>0$ (i.e. Drinfeld's modular forms) and automorphic forms. We prove that spaces of these modular forms (which are of characteristic $p$) can be described by function spaces of characteristic zero, close to those of automorphic forms.
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We consider mod p Hilbert modular forms associated to a totally real field of degree d in which p is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a d-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren-Oort stratification on mod p Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.
We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic $p$, generalizing the author's prior work with Sasaki to the case where $p$ is ramified in the totally real field $F$. In particular we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial $Θ$-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore we describe the kernels of partial $Θ$-operators in terms of images of…
We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.
Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a "geometric" variant of the "algebraic" Serre weight conj…
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
We report on a systematic computation of weight one cuspidal eigenforms for the group $Γ_1(N)$ in characteristic zero and in characteristic $p>2$. Perhaps the most surprising result was the existence of a mod 199 weight~1 cusp form of level 82 which does not lift to characteristic zero.
We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.
By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.
We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…
We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diam…
The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over F_p^bar of parallel weight 1 and level prime to p is unramified above p. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic p embed into the ordinary part of parallel weight p forms in two different ways per prime dividing p, namely via `partial' Frobenius operators.
Let $\goth E(\goth p)$ denote the Eisenstein ideal in the Hecke algebra $\Bbb T(\goth p)$ of the Drinfeld modular curve $X_0(\goth p)$ parameterizing Drinfeld modules of rank two over $\Bbb F_q[T]$ of general characteristic with Hecke level $\goth p$-structure, where $\goth p\triangleleft\Bbb F_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\Bbb F_q$ does not divide the order of the quotient $\Bbb T(\goth p)/\goth E(\goth p)$ and the Eisenstein ideal $\goth E(\goth p)$ is locally principal.
Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are also obtained. By using our theorems, some key theorems of Igusa [1] can be recovered.
Euler characteristic of crepant resolutions of specific modular quotient singularities
本文研究正特征下特殊线性群有限子群商奇点双解析解欧拉示性数的McKay对应推广，利用有限域上wild McKay对应证明其对特定半直积群成立，并提出模情形下的猜想。
Theta characteristics and noncongruence modular forms
本文研究模曲线上 theta 特征与模形式的关系，证明不同于 Hodge 丛的 theta 特征及其奇次幂截面通常为非同余模形式，并构造了到 Siegel 三重流形的扭转对角周期映射，回答了模曲线 Brill–Noether 理论的若干问题。
We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. Using the interplay between singular (a.k.a. CM) j-invariants in characteristic zero and supersingular ones in characteristic p, we obtain such congruences in which F is the minimal polynomial of a CM j-invariant, and give a sufficient condition for G to be a constant polynomial in these congruences.
Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a quotient of any X(p^e;N) by any subgroup of ((Z/p^e Z)^* x \SL_2(Z/NZ))/{+-1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X_0(N) for p not dividing N. We give an application to the function field a…
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field $\mathbb{F}_q(θ)$ with same conductor are isogenous. In…
We discuss certain representations of GL 2 Fq[T] in equal characteristic and associated vectorial modular forms
Lifting ethereal Bianchi modular forms
本文研究模 p Bianchi 模形式的提升问题，开发高效算法验证特征值同余，并借助模性定理证明某些情形下提升存在，实验表明多数模形式可提升。
In this paper we deal with Drinfeld modular forms, defined and taking values in complete fields of positive characteristic. Our aim is to study a sequence of families of Drinfeld modular forms depending on a parameter t that produces, for certain values of t, several kinds of Eisenstein series considered by Gekeler. We obtain formulas involving these functions. To obtain our results, we introduce and discuss tau-linear recurrent sequences and deformations of vectorial modular forms in this setting.
This article is divided in two parts. In the first part we endow a certain ring of ``Drinfeld quasi-modular forms'' for $\GL_2(\FF_q[T])$ (where $q$ is a power of a prime) with a system of "divided derivatives" (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in \cite{Ge}, and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of famous Ramanujan's differential system relating to the first derivatives of the classical Eisenstein series of weights 2, 4 and 6. In the second part of this article we prove th…
Wild Stacky Curves and Rings of Mod p Modular Forms
本文扩展了Voight等人的工作，计算野栈曲线的对数典范环，从而计算模p模形式环，并发现特征2和3中存在无穷多水平N的权2模形式不可提升。
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing
本文通过构造特征 p 下 Hilbert 模簇的几何 Jacquet-Langlands 关系，建立模 p Hilbert 模形式与四元数模形式之间的联系，推广 Serre 的经典结果，并证明上同调消失定理以关联 Galois 表示。
In this article we will show how to compute $U_p$ acting on spaces of overconvergent $p$-adic modular forms when $X_0(p)$ has genus 1. We first give a construction of Banach bases for spaces of overconvergent $p$-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the $U_p$ operator and eigenvectors of finite slope for $U_p$, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.
Hyperelliptic and trigonal modular curves in characteristic $p$
针对N级中间模曲线X_Δ(N)，研究其在特征p下为超椭圆或三角曲线的素数p的确定问题，通过Castelnuovo-Severi不等式建立N的界并研究典范理想明确满足条件的p，还证明其不会是光滑平面五次曲线。
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius …
Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X \subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-mo…
Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the "admissible" $p$-adic $L$-functions to characteristic ideals of "finite-slope" Selmer modules constructed by the second author. As a key ingredient, we improve Rubin's result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.
Modularity of trianguline Galois representations
本文通过伪刚性空间上的三角线模理论，证明了特定三角线伽罗瓦表示的模性提升定理，并构造了修补的四元数本征簇。
We prove that there are only finitely many modular curves of $D$-elliptic sheaves over $\mathbb{F}_q(T)$ which are hyperelliptic. In odd characteristic we give a complete classification of such curves.
We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.
An algorithm to obtain equations between theta functions with integral characteristics evaluated at $τ$ and $pτ$ for $g>1$ is presented.
Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve. We then prove a formula that allows us to evaluate this modular parametrization at cusps and we produce an explicit method to compute these values. Finally we illustrate our results with several examples in characteristic 2 and 3.
Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.
Let X be a noetherian scheme defined over an algebraically closed field of positive characteristic p, and G be a finite group, of order divisible by p, acting on X. We introduce a refinement of the equivariant K-theory of X to take into account the information related to modular representation theory. As an application, in the 1-dimensional case, we generalize a modular Riemann-Roch theorem given by S.Nakajima, extending the link between Galois modules and wild ramification.
We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$-theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $Θ_χ$ and study the variation of the filtration under these operators. In particular, we prove that every ordinary eigenform has filtration in a prescribed box of weights. Our methods are geometric -- comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are sup…
Just-likely intersections on Hilbert modular surfaces
本文证明了正特征下 Hilbert 模曲面上曲线的恰似相交猜想情形，并计算了带实乘法阿贝尔曲面在 p-幂同源下 Faltings 高度的变化。
We prove a vanishing theorem for one forms on the moduli stack of principally polarized abelian varieties of genus g>1 with level structure N over fields of characteristic p different from two. This is used to compute the Picard groups of these stacks. As an application we obtain results one congruences between integral Siegel modular forms.
Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…
We compute the stable reduction of the Lubin-Tate space $X(π^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(π^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(π^2)$. Our method is very explicit and elementary.
To every $k$-dimensional modular invariant vector space we associate a modular form on $SL(2,\mathbb{Z})$ of weight $2k$. We explore number theoretic properties of this form and find a sufficient condition for its vanishing which yields modular identities (e.g., Ramanujan-Watson's modular identities). Furthermore, we focus on a family of modular invariant spaces coming from suitable two-dimensional spaces via the symmetric power construction. In particular, we consider a two-dimensional space spanned by graded dimensions of certain level one modules for the affine Kac-Moody Lie algebra of type…
Hilbert surfaces, modular forms, and Siegel-Veech constants
本文研究了翻译表面中普林姆特征形式定位的希伯特-维克常数，证明其在不同定位间一致，并揭示了相关希伯特模曲面的欧拉特征关系。
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