We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of -schemes to interpret conceptually the spectral realization of zeros of L-functions.

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

We generalise results of Buzzard, Taylor and Kassaei on analytic continuation of p-adic overconvergent eigenforms over ℚ to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F, under the assumption that p splits completely in F. This includes weight-one forms and has applications to generalisations of Buzzard and Taylor’s main theorem. Next, we follow an idea of Kassaei’s to generalise Coleman’s well-known result that ‘an overconvergent Up-eigenform of small slope is classical’ to the case of p-adic overconvergent Hilbert eigenforms of Iwahori level.

In Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.

We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

We show an arithmetic generalization of the recent work of Lazarsfeld–Mustaţǎ which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.

We prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

We investigate the special fibres of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz–Rapoport (KR) stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne–Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.

On Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic 0. This extends prior work in the one-dimensional case of Christol, Dwork, Robba, Young, et al. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds.

Let K be a local field of equal characteristic p>2, let XK/K be a smooth proper relative curve, and let ℱ be a rank 1 smooth l-adic sheaf (l≠p) on a dense open subset UK⊂XK. In this paper, under some assumptions on the wild ramification of ℱ, we prove a conductor formula that computes the Swan conductor of the etale cohomology of the vanishing cycles of ℱ. Our conductor formula is a generalization of the conductor formula of Bloch, but for non-constant coefficients.

In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

Desingularized fiber products of semi-stable elliptic surfaces with Hetale3=0 are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of arithmetic degenerations of rigid Calabi–Yau threefolds.

Let and be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

We construct general type surfaces in mixed characteristic whose geometric genera can be made to jump by an arbitrarily prescribed positive amount under specialization. We then show that this phenomenon of jumping geometric genus presents itself in some compact Shimura surfaces. Finally, we find a set of conditions, met by the latter Shimura surfaces, that forces the higher plurigenera to remain constant in reduction modulo p.