Heights and moduli spaces June 10 14, Lorentz Center, Leiden

Transcription

1 Heights and moduli spaces June 10 14, Lorentz Center, Leiden Abstracts Jean-Benoît Bost, Université Paris-Sud 11 Pluripotential theory, heights, and algebraization on arithmetic threefolds Abstract: We present a construction of algebraic vector bundles on projective surfaces over number fields, starting from formal and analytic data along some ample divisor. It involves pluripotential theory and complex analysis on pseudoconcave domains, and the use of hermitian vector bundles of infinite rank. Jan Hendrik Bruinier, Technische Universität Darmstadt Heights of Kudla-Rapoport divisors and derivatives of L-functions Abstract: We describe how Borcherds regularized theta correspondence can be used to construct arithmetic theta lifts from harmonic Maass forms to arithmetic Chow groups. We discuss the Gross-Zagier formula and recent joint work with Ben Howard and Tonghai Yang on heights of Kudla-Rapoport divisors on unitary Shimura varieties in this setting. François Brunault, ENS Lyon On Zagier s conjecture for Q-curves Abstract: Let E be an elliptic curve defined over a number field. Zagier s conjecture predicts that L(E, 2) can be expressed in terms of the Bloch elliptic dilogarithm evaluated at algebraic points of E. It also predicts very precisely which divisors on E give rise to a relation with the L-value. In this talk I will explain how to prove the first part of this conjecture in the case of Q-curves satisfying certain technical conditions. José Ignacio Burgos, CSIC-ICMAT The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve Abstract: A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, provided with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key to being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford s result remains valid for line bundles on mixed Shimura varieties. In this talk we will examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We will show that Mumford s result can not be extended to this case and that a new interesting kind of singularities appear. We will discuss some preliminary results. This is joint work with G. Freixas, J. Kramer and U. Kühn. 1

2 Stephan Ehlen, Technische Universität Darmstadt CM values of regularized theta lifts and harmonic weak Maass forms of weight one Abstract: We show that the values of regularized theta lifts on Shimura varieties of orthogonal type at certain CM points of discriminant D < 0 are closely related to harmonic weak Maass forms of weight one. We relate the coefficients of the holomorphic parts of these forms to special cycles on the moduli stack of elliptic curves with complex multiplication by the ring of integers in the imaginary quadratic field of discriminant D. As a corollary of our result, we obtain a new proof of the modularity of the generating series of these cycles. Javier Fresán, Université Paris-Nord 13 Periods of connections and special values of the Gamma function Abstract: Motivated by his algebraic proof of the Chowla-Selberg formula, B. H. Gross conjectured at the end of the 70s that periods of geometric Hodge structures with complex multiplication by an abelian field are always products of special values of the Gamma function. In this short talk I will explain how to use a product formula for periods of regular singular connections, due to Saito and Terasoma, in order to prove some new results in the direction of the Gross conjecture. Wojtek Gajda, Adam Mickiewicz University Monodromies for abelian varieties and independence of l-adic Galois representations Abstract: In the frst part of the talk I will explain some of recent computations of the image of the absolute Galois group of a field, acting on torsion points of abelian varieties. In the second part of the talk we switch attention to Galois representations attached to l-adic cohomologies of schemes, and discuss some new independence (of the prime l) results on the image of the absolute Galois. This will be a report on my joint works with S. Arias-de-Reyna, G. Boeckle and S. Petersen. Walter Gubler, Universität Regensburg Normal toric varieties over valuation rings of rank 1 Abstract: Tropicalizations lead to the study of normal toric varieties over valuation rings of rank 1. In this talk, some of their properties are discussed including their classification in terms of fans. Ariyan Javan Peykar, Universiteit Leiden and Université Paris-Sud 11 Bounding stable Arakelov invariants of curves Abstract: Let X be a curve over a number field. We study invariants of X such as the Belyi degree and the Faltings height. Our main result is an explicit inequality relating these invariants. 2

3 As a first application, we deduce a conjecture of Edixhoven-de Jong-Schepers. Secondly, we give an algorithmic application. Finally, we give a Diophantine application: Szpiro s small points conjecture is true for hyperelliptic curves. Johannes Kolb, Universität Regensburg An analytic description of local intersection numbers on products of semi-stable curves Abstract: Let R be a complete discrete valuation ring with algebraically closed residue field and X a regular strict semi-stable curve over Spec(R) with reduction graph Γ. The Gross-Schoen desingularization yields a regular strict semistable scheme W whose generic fiber W η is the d-fold product Xη d. Intersection theory with support and a local degree map allow to associate with d + 1 Cartier divisors on W with support in the special fibre a local intersection number. We may identify these Cartier divisors with piecewise integral affine functions on the d-fold product Γ d of the graph Γ. This allows to interprete the local intersection pairing as a pairing on piecewise integral affine functions on Γ d. In the cases d = 2, d = 3 and for all d which satisfy a certain vanishing condition, I give a formula which computes this pairing and involves only elementary analysis on the product of the reduction graphs. This generalizes work of S. Zhang who proved such a formula in the case d = 2. Jürg Kramer, Humboldt Universität zu Berlin Uniform sup-norm bounds on average for cusp forms of higher weights Abstract: Let Γ PSL 2 (R) be a Fuchsian subgroup of the first kind acting on the upper half-plane H. Consider the d-dimensional space of cusp forms Sk Γ of weight 2k for Γ, and let {f 1,..., f d } be an orthonormal basis of Sk Γ with respect to the Petersson inner product. In our talk we will show that the sup-norm of the quantity Sk Γ(z) := d j=1 f j(z) 2 Im(z) 2k is bounded as O Γ (k) in the cocompact setting, and as O Γ (k 3/2 ) in the cofinite case, where the implied constants depend solely on Γ. We also show that the implied constants are uniform if Γ is replaced by a subgroup of finite index. Such bounds are useful for effectively bounding various Arakelov invariants. Ulf Kühn, Universität Hamburg Upper and lower bounds for ω 2 Abstract: Recent results on upper and lower bounds for ω 2 on arithmetic surfaces will be presented. These bounds will be made explicit in various examples. Christophe Mourougane, Université de Rennes 1 - IRMAR High order jet spaces and height inequalities on complex function fields Abstract: I will explain how the study of effectiveness of some natural line bundles on jet spaces, 3

4 as developed by D ly, lead to some height inequalities for algebraic points on varieties defined over a complex function field. Using ideas of Clemens, Voisin and Siu, I will show how to prove a statement toward Lang s conjecture on the paucity of rational points on generic hypersurfaces of large degree and large variation defined over a complex function field. Steffen Müller, Universität Hamburg p-adic heights and integral points on hyperelliptic curves Abstract: Let X be a hyperelliptic curve of genus g over the rationals. I will describe a method to p-adically approximate integral points on X when the Mordell-Weil rank of the Jacobian J of X is equal to g. For this method we express certain p-adic heights on J in terms of iterated Coleman integrals and arithmetic intersection multiplicities on a regular model of X. This is joint work with Jennifer Balakrishnan and Amnon Besser. Fabien Pazuki, Institut de Mathématiques de Bordeaux Moret-Bailly models and applications Abstract: Using the notion of Moret-Bailly model of an abelian variety A over a number field k, we will study the differential height of A/k: first we will derive an explicit comparison between the differential height and the theta height of A. As a second application we will focus on a local-global formula for the differential height. Anna von Pippich, Humboldt Universität zu Berlin An arithmetic Riemann Roch theorem for weighted pointed curves Abstract: In this talk, we report on work in progress with G. Freixas generalizing the arithmetic Riemann Roch theorem for pointed stable curves to the case where the metric is allowed to have conical singularities at the marked points. One main ingredient of the proof is a Mayer-Vietoris type formula for the singular hyperbolic metric. This formula requires the explicit computation of the regularized determinants for hyperbolic cusps and cones. Furthermore, we show that the related local index theorem involves an analogue of the Takhtajan Zograf metric employing so-called elliptic Eisenstein series. Lukas Pottmeyer, Universität Regensburg Fields with the Bogomolov property Abstract: We say that a field F Q has the Bogomolov property relative to a height function h if and only if h(α) is either zero or bounded from below by a positive constant for all α F. This property is in general not preserved under finite field extensions. Schinzel proved in 1973 that Q tr, the maximal totally real extension of Q, has the Bogomolov property relative to the standard Weil-height h. However, the maximal CM-field Q tr (i) is known to have elements of arbitrarily 4

5 small Weil-height. We will classify all rational functions f Q(x) such that Q tr has the Bogomolov property relative to the canonical height h f introduced by Call and Silverman. Using this result we will show that Q tr (i) is essentially the only finite extension of Q tr where the Bogomolov property relative to h gets lost. Gaël Rémond, Institut de Mathématiques de Bordeaux Isogenies and polarizations Abstract: In a joint work with Éric Gaudron, we give several explicit estimates for the geometry of abelian varieties over number fields. In particular, we establish the existence of a small polarization, whose degree is bounded in terms of the Faltings height of the variety, its dimension and the degree of the base field. We further improve and make explicit the isogenies and factorization estimates of Masser and Wüstholz. The proofs are based on some geometry of numbers arguments in euclidean lattices of morphisms between abelian varieties, together with a period theorem. Damian Rössler, Institut de Mathématiques de Toulouse On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic Abstract: Let K be the function field of a smooth and proper curve S over an algebraically closed field k of characteristic p > 0. Let A be an ordinary abelian variety over K. Suppose that the Néron model A of A over S has a closed fibre A s, which is an abelian variety of p-rank 0. We show that in this situation the group A(K perf )/Tr K k (A)(k) is finitely generated (thus generalizing a special case of the Lang-Néron theorem). Here K perf = K p is the maximal purely inseparable extension of K. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification constructed by Faltings-Chai. Maryna Viazovska, Universität zu Köln CM values of Green s functions Abstract: Higher Green functions are real-valued functions of two variables on the upper half-plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, and satisfy the equation f = k(1 k)f, where is a hyperbolic Laplace operator and k is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier Heegner points and derivatives of L-series (1986). In particular, it was conjectured that the values of higher Green s functions at CM points are equal to the logarithms of algebraic numbers. In this talk we will present a proof of the conjecture for any k and any pair of CM points lying in the same imaginary quadratic field. Our proof has two parts. First, we show that the regularized Petersson scalar product of binary theta series with a weight one weakly holomorphic cusp form is equal to the logarithm of an algebraic number. Second, we prove that the special values of weight k Green s function occurring at the conjecture of Gross and Zagier can be written as Petersson products of this type. We also give a formula for the factorization of the algebraic number obtained. 5

6 Hang Xue, Columbia University On a canonical quadratic point in the Jacobian of a genus four curve Abstract: We construct a quadratic point in the Jacobian of a non-hyperelliptic curve of genus four over a global field. We then compute the Neron-Tate height of this point in terms of the self-intersection of the dualizing sheaf and some canonically defined local invariants. We show that the height of this point satisfies the Northcott property. We also give some estimates of the local invariants that appear in the height computation. When the reduction of the curve is simple, we compute explicitly the local invariants. 6