Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2011


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12.04.2011 Vorbesprechung
19.04.2011  Anil Aryasomayajula (HU Berlin)
Estimates of the Canonical Green's Function
Abstract: In this talk, I will try to present the estimates obtained for the canonical Green's function associated to the canonical metric, on non-compact finite volume hyperbolic Riemann surfaces of genus g > 1, in terms of invariants from hyperbolic geometry.
26.04.2011 Hartwig Mayer (HU Berlin)
On Zhang's admissible intersection theory
Abstract: In 1993, S.-W. Zhang introduced (based on previous work of T. Chinburg and R. Rumely) an intersection theory for smooth, irreducible curves $X/K$ over a local field $K$ by defining a potential theory on the dual of the reduction graph $R(X)$ of $X$. This theory was used to give first answers to the Bogomolov conjecture. In the first part of this talk, we present Zhang's intersection theory and its arithmetic implication in case of a modular curve. In the second part, we present recent developments in extending Zhang's potential theory to Berkovich curves, and end with some open questions concerning a higher-dimensional analogue, which would be of great interest in the context of Arakelov theory.
03.05.2011 Dennis Eriksson (Univ. Göteborg)
Multiplicities of discriminants
Abstract: I will discuss some recent formulas for multiplicities of discriminants of polynomials, generalising in one direction those of Ogg's formula for the discriminant of elliptic curves. In the particular case of discriminants of planar curves we obtain more precise information, and we can relate it to finite contributions of Arakelov intersection numbers.
10.05.2011 Lenny Taelman (Univ. Leiden)
Special values in characteristic p
Abstract: I will present a theorem which is a characteristic-p-valued function field analogue of the class number formula and the Birch and Swinnerton-Dyer conjecture. In these special value formulas the multiplicative group (for the CNF) and elliptic curves (for BSD) are replaced by Drinfeld modules. Reference: [http://arxiv.org/abs/1004.4304].
17.05.2011 Vincenz Busch (Univ. Hamburg)
Beilinson's conjecture for K_2 of a superelliptic curve
Abstract: The Beilinson conjecture generalize and unify multiple theorems and conjectures in arithmetic geometry, e.g. the class number formula and the BSD conjecture. In the case of K_2 of algebraic curves the effects of the Beilinson conjectures can actually be observed in concrete calculations. In this talk, we will cover an example of such calculations in the case of a superelliptic curve.
24.05.2011 Aaron Greicius (HU Berlin)
Images of Galois representations attached to l-Tate modules
Abstract: Let A/K be a principally polarized abelian variety with trivial endomorphism ring. In many cases (for example, when dim(A) is 2,6 or odd), the various l-adic Galois representations to GSp associated to the l-Tate modules of A/K are surjective for all but finitely many primes l. In such situations one hopes to find an algorithm for finding, or at least bounding, the finite set of exceptional primes where the Galois representation fails to be surjective. We will consider existing algorithms for elliptic curves and abelian surfaces over *Q* and endeavor to extend these to more general number fields. Time permitting, we will also examine the situation when the endomorphism ring of A is larger than *Z*.
31.05.2011 Gérard Freixas i Montplet (Univ. Paris 6)
Arithmetic Riemann-Roch theorem and Jacquet-Langlands correspondence
Abstract: In this talk we will review the arithmetic Riemann-Roch theorem for pointed curves and we will show how to combine it with the Jacquet-Langlands correspondence, in order to get equalities between certain arithmetic self-intersection numbers on modular and Shimura curves.
07.06.2011 Ananyo Dan (HU Berlin)
Noether-Lefschetz locus on projective hypersurfaces
Abstract: Noether's theorem states that a generic degree $d$ smooth hypersurface in $\mathbb{P}^3$ has picard number 1. We define the Noether-Lefschetz to be the smooth hypersurfaces in $\mathbb{P}^3$ of degree $d$. In my talk we study the geometry of this locus including when is an irreducible component of this locus non-reduced.
14.06.2011 George Walker (Bristol)
Zeta functions of plane curves
Abstract: The problem of computing zeta functions of varieties over finite fields has received considerable interest in recent years, particularly for the case of curves. I shall outline a new algorithm, based on $p$-adic cohomology and the work of Lauder, which may be applied to a wide range of classes of smooth curves. In particular, it matches in complexity Kedlaya's algorithm for hyperelliptic curves and its generalizations to superelliptic curves. I shall focus on its application to smooth plane curves, where the algorithm improves on the complexity of the previously best known algorithm almost by a factor of $g$, the genus.
21.06.2011 kein Seminar
28.06.2011 Christine Robinson (Univ. of Illinois at Chicago)
On a converse theorem and a Saito-Kurokawa lift for Siegel wave forms
05.07.2011 kein Seminar
Freitag
08.07.2011 Bjorn Poonen (MIT)
Random maximal isotropic subspaces and Selmer groups
Abstract: We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The random model is consistent with Delaunay's heuristics for Sha[p], and predicts that the average rank of elliptic curves is at most 1/2. This is joint work with Eric Rains.
Ort: BMS Loft, Urania             Beginn: 10.15 Uhr
        (An der Urania 17, 10787 Berlin)
12.07.2011 Doug Ulmer (Georgia Inst. of Technology)
Arithmetic of the Legendre curve in a Kummer tower
Abstract: Let k be a finite field of odd characteristic, K=k(t), and K_d=k(t^{1/d}). We consider the arithmetic of the Legendre elliptic curve E: y^2=x(x-1)(x-t) over the fields K_d. A remarkable elementary construction gives many points on E over K_d for suitable values of d. Less elementary considerations lead to interesting problems and results on the full Mordell-Weil group E(K_d), on heights, and on the Tate-Shafarevich group of E over K_d.