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We recall first the analytic theory of the Hilbert modular varieties of level $\Gamma_1(\mathfrak{c},\mathfrak{n})$ and their compactifications. We construct arithmetic toroidal compactifications of the universal Hilbert-Blumenthal abelian variety (and of its fiber products), over the compactifications of the Hilbert modular variety constructed by the first author. We give several applications related to Hodge theory, Hilbert-Jacobi modular forms and Hilbert modular forms of half-integer weight., Comment: 60 pages, in french, final version

We study surfaces constructed from groups of units in quaternion orders $\Lambda$ over the integers in real quadratic fields k. A short presentation of some general theory of such surfaces is given, in particular, we construct certain ``modular curves'' on the surfaces and examine their properties. We apply our results to some surfaces related to the case $k=Q(\sqrt{13})$ and $d(\Lambda)=(3)$, and find their place in the Kodaira classification of algebraic surfaces., Comment: 15 pages

Let V_D be the Shimura curve over \Q attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of V_D and prove that, in many cases, it is the Atkin-Lehner group. Moreover, we determine the family of bielliptic Shimura curves over \bar \Q and over \Q and we use it to study the set of rational points on V_D over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin-Lehner quotients of V_D., To appear in Compositio Mathematica

This paper is a continuation of our paper math.AG/0006222. We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" $p$-adic integral models of these Shimura varieties and study their 'etale local structure. In particular, we exhibit a stratification of their (singular) special fibers and give a partial calculation of the sheaf of nearby cycles., LaTeX, 51 pages, to appear in the Duke Math. Journal

This is a report on results and methods in the reduction modulo p of Shimura varieties with parahoric level structure. In the first part, the local theory, we explain the concepts of parahoric subgroups, of the mu-admissible and mu-permissible subsets of the Iwahori-Weyl group, of the corresponding union of affine Deligne-Lusztig varieties and of local models. In the second part, the global theory, we use these concepts to formulate conjectures on the points in the reduction modulo p of Shimura varieties with parahoric level structure., AMS-TeX, 40 pages

We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements., Comment: AMS-LaTeX, 8 pages, submitted

We review previous methods of computing the modular degree of an elliptic curve, and present a new method (conditional in some cases), which is based upon the computation of a special value of the symmetric square L-function of the elliptic curve. Our method is sufficiently fast to allow large-scale experiments to be done. The data thus obtained on the arithmetic character of the modular degree show two interesting phenomena. First, in analogy with the class number in the number field case, there seems to be a Cohen--Lenstra heuristic for the probability that an odd prime divides the modular degree. Secondly, the experiments indicate that {\small $2^r$} should always divide the modular degree, where r is the Mordell--Weil rank of the elliptic curve. We also discuss the size distribution of the modular degree, or more exactly of the special L-value which we compute, again relating it to the number field case.

We prove a purity theorem for abelian schemes in arbitrary unramified mixed characteristic (0,p). The case p=2 is completely new and the case p>2 fixes several errors in the literature., 11 pages

A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part of J_0(N) with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and each integer g at least 2, the set of genus g curves over k dominated by a Fermat curve is finite and computable., Comment: 53 pages; minor revisions made

A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions., 25 pages, published version

We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced., 31 pages, LaTeX

Let g_0(N) be the genus of the modular curve X_0(N). We record several properties of the sequence {g_0(N)}. Even though the average size of g_0(N) is 1.25N/pi^2, a random positive integer has probability zero of being a value of g_0(N). Also, if N is a random positive integer then g_0(N) is odd with probability one. The smallest non-negative integer not occuring as g_0(N) for any N is 150; the smallest such odd integer is 49267., Comment: 9 pages; minor changes in v2

Let g_0(N) be the genus of the modular curve X_0(N). We record several properties of the sequence {g_0(N)}. Even though the average size of g_0(N) is 1.25N/pi^2, a random positive integer has probability zero of being a value of g_0(N). Also, if N is a random positive integer then g_0(N) is odd with probability one. The smallest non-negative integer not occuring as g_0(N) for any N is 150; the smallest such odd integer is 49267., 9 pages; minor changes in v2

Let $p\ge 5$ be a prime. We show that the space of weight one Eisenstein series defines an embedding into $\PP^{(p-3)/2}$ of the modular curve $X_1(p)$ for the congruence group $\Gamma_1(p)$ that is scheme-theoretically cut out by explicit quadratic equations., Comment: AMS-LaTeX, 15 pages

In this paper we generalize Cherednik's method and prove that certain Shimura varieties corresponding to groups of unitary similitudes and automorphic vector bundles over them have p-adic uniformization. This is proved for Shimura varieties, uniformized by the complex unit ball, when the central simple algebra over a CM-field defining the group of unitary similitudes has Brauer invariant 1/d at p. In this case, Shimura varieties can be uniformized by Drinfeld's covers of p-adic upper half-spaces., 67 pages

In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety. Moreover, we show that difficult theorems of Yau and Kottwitz appearing in [V] may be avoided., 30 pages

This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of PEL-type, associated to a certain unitary group, whose connected components are isomorphic to Mumford's fake projective plane. We also give the canonical model, Shimura field and number of connected components of this Shimura surface. As a consequence of the construction we prove that the Mumford's fake projective plane is an arithmetic complex unit-ball quotient, and has the 7-th cyclotomic field as a field of definition. In Tohoku Math. J. 50 (1998), 537-555, Masanori Ishida and the author discussed two other possible fake projective planes. The author has also obtained the corresponding results as above for these fake projective planes and has written them in the above preprint, which is available in the web page: http://www.mpim-bonn.mpg.de/html/preprints/preprints.html, 11 pages

We have identified explicitly the dual of the hypersurface I5 defined as the unique quintic hypersurface in $P^5\funnyC$ invariant under the natural action of the Weyl group $W(\Ee)$ of the lattice $\Ee$.

We investigate the existence and non-existence of modular forms of low weight with a character with respect to the paramodular group $\Gamma_t$ and discuss the resulting geometric consequences. Using an advanced version of Maa\ss\ lifting one can construct many examples of such modular forms and in particular examples of weight 3 cusp forms. Consequently we find many abelian coverings of low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces which are not unirational. We also determine the commutator subgroups of the paramodular group $\Gamma_t$ and its degree 2 extension $\Gamma^+_t$. This has applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$. Finally we prove non-existence theorems for low weight modular forms. As one of our main results we obtain the theorem that the maximal abelian cover ${\Cal A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or 5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7., Comment: 30 pages, AMS-TeX, 2 diagrams

We briefly discuss the relationship between several characterizations of the Hasse--Witt invariant of curves in characteristic $p$ with the goal of computing its value in concrete instances. We study its asymptotic behaviour when dealing with the geometric fibres of curves of genus $\geq 2$ defined over the rationals. Numerical evidence gathered for several modular curves supports certain conjectural distribution laws.

We verify that the descent maps provided by Langlands's Conjugacy Conjecture do satisfy the continuity condition necessary for them to be effective. Thus Langlands's conjecture does imply the existence of canonical models. This replaces an earlier version of the paper --- the proof in this version is simpler, and the exposition more detailed., Comment: 6 pages

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the ��tale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures., Added a section on Harder's conjectural congruences. Some minor changes. 16 pages

According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety then its L-function must capture substantial part of the arithmetic properties of A. The smallest number field L where A has all its endomorphisms defined must also have a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties A_f/Q associated to weight 2 newforms for the modular group Gamma_1(N). Specifically, our goal is to relate the order of L(A_f/Q,s) at s = 1 with Euler products cropped by the set of primes that split completely in L. The results we obtain for the case when f has complex multiplication are complete, while in the absence of CM, our results depend on the rate of convergence in Sato-Tate distributions., 24 pages