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In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using \'etale cohomology of X with Q_p(r) and Z_p(r)-coefficients., Comment: 62 pages. Remark 2.2 has been added. Proposition 7.3 has been modified. Some details of the proof of Lemma 7.9 have been improved, and many typos have been corrected. To appear in Journal of Number Theory

We give necessary and sufficient conditions for log smoothness of a proper regular arithmetic surface with smooth geometrically connected generic fibre over a discrete valuation ring with perfect residue field. As an application, we recover known criteria for log smooth reduction of minimal normal crossings models of curves.

We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing., Comment: 42 pages

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of P^1 over a discretely valued field to be regular., Final version, to appear in the Journal of Algebraic Geometry. 31 pages

In the paper, a description of the Grothendieck-Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of theta-invariants.

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring--Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and distribution results such as ergodic theorems.

On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles.

We explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces π : X → Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X. Further, we derive a direct sum decomposition of Rπ*Z(n) into motivic degree components.

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface., 21 pages; minor changes; to appear in Izvestiya: Mathematics

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve., Comment: 9 pages

We study the eigenfunctions of the quantized cat map, desymmetrized by Hecke operators. In the papers (Olofsson in Ann Henri Poincare 10(6):1111–1139, 2009; Math Phys 286(3):1051–1072, 2009) it was observed that when the inverse of Planck’s constant is a prime exponent N = pn, with n > 2, half of these eigenfunctions become large at some points, and half remains small for all points. In this paper we study the large eigenfunctions more carefully. In particular, we answer the question of for which q the Lq norms remain bounded as N goes to infinity. The answer is q ≤ 4.

In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface., Comment: To appear in J. Algebra

In this paper, the theory of e-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant µ(X ,G , V) associated to certain symplectic motives of weight one. The relationship between this invariant and w2(π) (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Nogues, Erez and Taylor) is also discussed. In his paper (4), Deligne used elements in the Brauer group of Q and their relation- ship with certain e-constants to give a proof of the Frohlich-Queyrut theorem. In particular, he showed that certain global orthogonal root numbers are equal to one, by interpreting the associated local orthogonal root numbers as Stiefel-Whitney classes and then using the local root numbers to define an element of order two in the Brauer group of Q. This idea was furthered by Saito (in (13), for example) and others who defined Brauer group invariants associated to situations which can be interpreted as motives that are orthogonal and of even weight. In this paper, we define a Brauer group invariant associated to certain motives that are symplectic and have weight one. In order to construct the relevant motives, we first define X to be an arithmetic surface of dimension two which is flat, regular, and projective over Z. Throughout this paper, we assume that f : X− →Spec(Z) is the structure morphism. Let G be a finite group that acts tamely on X. In other words, for each closed point x ∈ X, the order of the inertia group of x is relatively prime to the residue characteristic of x .L etY be the quotient scheme X /G, which we assume is regular, and assume that for all finite places v, the fiber Yv =( Xv)/G = Y⊗ Z (Z/p(v)) has normal crossings and smooth irreducible components with multiplicities relatively prime to the residue characteristic of v. Finally, let V be a virtual representation of G over

Let Open image in new window be a regular arithmetic surface. Assume that for all irreducible curves Open image in new window there are given open normal subgroups of π1(C), which fulfill a compatibility condition at all closed points x ∈ Open image in new window . We then show that these data uniquely determine a normal subgroup of π1( Open image in new window ). This is used to construct abelian class field theory for arithmetic surfaces using only K0 and K1 groups of local and global fields.

A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.

Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle., Comment: 20 pages

The aim of this paper is the study of the genus 3 curves C n : Y 4 = X 4 - (4n - 2) X 2 + 1, from the Arakelov viewpoint. The Jacobian of the curves C n splits as a product of elliptic curves, and this fact gives enough arithmetical datum to determine the stable model and the canonical sheaf of the curves. We use this information to look for explicit expressions of the modular height and the self-intersection of the dualizing sheaf of the curves C n .

Given the orthonormal basis of Hecke eigenforms in S2k(Γ(1)), Luo established an associated probability measure dμk on the modular surface Γ(1)\H that tends weakly to the invariant measure on Γ(1)\H. We generalize his result to the arithmetic surface Γ0(N)\H where N⩾1 is square-free

Let $S$ be a Dedekind scheme with perfect residue fields at closed points, let $f: X\rightarrow S$ be a minimal regular arithmetic surface of fibre genus at least $2$ and let $f': X'\rightarrow S$ be the canonical model of $f$. It is well known that $\omega_{X'/S}$ is relatively ample. In this paper we prove that $\omega_{X'/S}^{\otimes n}$ is relative very ample for all $n\geq 3$., Comment: 10 pages, no figures

We generalise a formula of Shou-Wu Zhang, which describes local arithmetic intersection numbers of three Cartier divisors with support in the special fibre on a a self-product of a semi-stable arithmetic surface using elementary analysis. By an approximation argument, Zhang extends his formula to a formula for local arithmetic intersection numbers of three adelic metrized line bundles on the self-product of a curve with trivial underlying line bundle. Using the results on intersection theory from arXiv:1404.1623 [math.AG] we generalize these results to d-fold self-products for arbitrary d. For the approximations to converge, we have to assume that d satisfies the vanishing condition 4.7 from arXiv:1404.1623 [math.AG], which is true at least for $d\in \{2,3,4,5\}$., Comment: 31 pages

Les invariants d'Arakelov des surfaces arithmetiques sont bien connus pour le genre 1 et 2 ([4], [2]). Dans cette note, nous etudions la hauteur modulaire et la self-intersection d'Arakelov pour une famille de courbes de genre 3 possedant beaucoup d'automorphismes, a savoir C n : Y 4 = X 4 - (4n - 2) X 2 Z 2 + Z 4 . La theorie d'Arakelov fait intervenir a la fois des calculs arithmetiques et des calculs analytiques. Nous exprimons les periodes de C n en termes d'integrales elliptiques. Les substitutions utilisees dans les integrales fournissent une decomposition de la jacobienne de C n en produit de trois courbes elliptiques. En utilisant l'isogenie correspondante, nous determinons le modele stable de la surface arithmetique definie par C n . Une fois calcules les periodes et le modele stable de C n , nous sommes en mesure de determiner la hauteur modulaire et la self-intersection du modele canonique. Nous donnons une bonne estimation de cette hauteur modulaire, traduite par son comportement logarithmique. Nous donnons egalement une minoration de la self-intersection qui montre qu'elle peut etre arbitrairement grande. Nous presentons ici nos calculs presque sans demonstrations. Les details peuvent etre lus dans [5].

We prove an arithmetic analogue of the so-called Lefschetz theorem which asserts that, if D is an effective divisor in a projective normal surface X which is nef and big, then the inclusion map from the support |D| of D in X induces a surjection from the (algebraic) fondamental group of |D| onto the one of X. In the arithmetic setting, X is a normal arithmetic surface, quasi-projective over Spec Z, D is an effective divisor in X, proper over Spec Z, and furthermore one is given an open neighbourhood Ω of |D|(C) on the Riemann surface X(C) such that the inclusion map |D|(C)↪Ω is a homotopy equivalence. Then we may consider the equilibrium potential g D,Ω of the divisor D(C) in Ω and the Arakelov divisor ( D,g D,Ω ), and we show that if the latter is nef and big in the sense of Arakelov geometry, then the fundamental group of |D| still surjects onto the one of X. This results extends an earlier theorem of Ihara, and is proved by using a generalization of Arakelov intersection theory on arithmetic surfaces, based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L12.

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.

On construit une hauteur sur l'espace des fibres stables de rang et de determinant fixes sur une courbe sur un corps de nombres, dans le cas ou le rang et le degre sont premiers entre eux et la courbe a partout une bonne reduction. Cette hauteur est definie en utilisant la theorie d'Arakelov.

For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genusg, for line bundles of degreeg equivalence is shown to the height on the Jacobian defined by Θ. We recover the classical formula due to Faltings and Hriljac for the Neron-Tate height on the Jacobian in terms of the intersection pairing on the arithmetic surface.

We prove an effective upper bound on the number of effective sections of a hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert--Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.

In this paper, we give a numerical characterization of nef arithmetic R-Cartier divisors of C^0-type on an arithmetic surface. Namely an arithmetic R-Cartier divisor D of C^0-type is nef if and only if D is pseudo-effective and deg(D^2) = vol(D)., Comment: 26 pages, Rewrite Section 3

One of the driving motivations to develop $\F_1$-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields, which might result in a proof of the classical Riemann hypothesis. The underlying idea is that the spectrum of $\Z$ should find an interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$ analogous to a curve over a finite field. The hope is that intersection theory for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$ will allow to mimic Weil's proof. It turns out that it is possible to define an object $\bar{\Spec\Z}$ from the viewpoint of blueprints that has certain properties, which come close to the properties of its analogs in positive characteristic. This shall be explained in the following note, which is a summary of a talk given at the Max Planck Institute in March, 2012., Comment: This note is the summary of a talk given at the Max Planck Institute in March, 2012. 11 pages

In this paper, we consider the relation between the simplest types of arithmetic theta series, those associated to the cycles on the moduli space $\Cal C$ of elliptic curves with CM by the ring of integers $\OK$ in an imaginary quadratic field $\kay$, on the one hand, and those associated to cycles on the arithmetic surface $\M$ parametrizing 2-dimensional abelian varieties with an action of the maximal order $O_B$ in an indefinite quaternion algebra $B$ over $\Q$, on the other. We show that the arithmetic degree of the pullback to $Cal C$ of the arithmetic theta function of weight 3/2 valued in $\hat CH^1(\M)$ can be expressed as a linear combination of arithmetic theta functions of weight 1 for $\Cal C$ and unary theta series. This identity can be viewed as an arithmetic seesaw identity. In addition, we show that the arithmetic theta series of weight 1 coincide with the central derivative of certain incoherent Eisenstein series for SL(2)/Q, generalizing earlier joint work with M. Rapoport for the case of a prime discriminant., 36 pages

We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed., Update to previous version of paper: most important change is a proof of the horizontal reciprocity law on arithmetic surfaces

Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height of X. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

Let K be a number field and OK the ring of integers of K. Let (E, h) be a hermitian finitely generated flat OK-module. For an OK-submodule F of E, let us denote by hF↪→E the submetric of F induced by h. It is well known that the set of all saturated OK-submodules F with deg(F, hF↪→E) ≥ c is finite for any real numbers c (for details, see [4, the proof of Proposition 3.5]). In this note, we would like to give its generalization on a projective arithmetic variety. Let X be a normal and projective arithmetic variety. Here we assume that X is an arithmetic surface to avoid several complicated technical definitions on a higher dimensional arithmetic variety. Let us fix a nef and big C∞-hermitian invertible sheaf H on X as a polarization of X. Then we have the following finiteness of saturated subsheaves with bounded arithmetic degree, which is also a generalization of a partial result [5, Corollary 2.2].

We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry., Comment: 31 pages, 2 figures. Added some applications to tropical geometry and an appendix by Brian Conrad

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac's arithmetic Hodge index theorem for an arithmetic surface.

We prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range. In fact it is close to having a Poisson behavior. This fact was discovered numerically by Schmit, Bogomolny, Georgeot and Giannoni. It has its origin in the high degeneracy of the length spectrum, first observed by Selberg.

We treat the openp-adic string world sheet as a coset spaceF=T/Γ, whereT is the Bruhat-Tits tree for thep-adic linear groupGL(2, ℚ p ) and Γ⊂PGL(2, ℚ p ) is some Schottky group. The boundary of this world sheet corresponds to ap-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset spaceF. The tachyon amplitudes expressed in terms ofp-adic θ-functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach top-adic algebraic curves. The connection of the problem of closedp-adic strings with the considered topics is discussed.

Let K be a number field and OK be its ring of integers. Let f:X→Spec(OK) be an arithmetic surface and let L̄ be an arithmetically nef hermitian line bundle over X. The hermitian structure on L̄ defines a natural structure of hermitian OK-module on H0(X;L̄⊗ωX/OK). A closed point P∈X is said to be a fixed point for the adjoint system of L̄ if, for every D∈H0(X;L̄⊗ωX/OK) such that ‖D‖sup⩽1 we have that D∣P=0. We prove that the existence of a fixed point for the adjoint system of L̄ imposes some (Arakelov) numerical condition on L̄. We prove also an arithmetic analogue of Cayley–Bacharach criterion for the existence of an hermitian vector bundle of rank two over X with prescribed arithmetic Chern classes and a section vanishing only on a fixed closed point. In the last part we apply this to find an arithmetic analogue of Reider’s Theorem on fixed points of the adjoint system of L̄.

We incorporate the string theory into the number theoretic formulation based on arithmetic geometry. The string theory is generalized p-adically and interpreted on an arithmetic surface. A p-adic multi-loop scattering amplitude is constructed.

The purpose of this chapter is to recall briefly the intersection theory that we shall need. This first arose in Shafarevich [Sh], and also Lichtenbaum [Li], which is a useful reference for other types of theorems than those in Shafarevich, for instance, the projective imbedding of certain schemes, which we shall use if and when necessary.

In [A2] and [A3], Arakelov introduces an intersection calculus for arithmetic surfaces, that is, for stable models of curves over a number field. In this paper we intend to show that his intersection product has a lot of useful properties. More precisely, we show that the following properties from the theory of algebraic surfaces have an analogue in our situation