Symplectic Lagrangian fibrations

In this work we investigate the deformation theory of pairs of an irreducible symplectic manifold X together with a Lagrangian subvariety Y in X, where the focus is on singular Lagrangian subvarieties. Among other things, Voisin's results [Voi92] are generalized to the case of simple normal crossing subvarieties; partial results are also obtained for more complicated singularities.rnAs done in Voisin's article, we link the codimension of the subspace of the universal deformation space of X parametrizing those deformations where Y persists, to the rank of a certain map in cohomology. This enables us in some concrete cases to actually calculate or at least estimate the codimension of this particular subspace. In these cases the Lagrangian subvarieties in question occur as fibers or fiber components of a given Lagrangian fibration f : X --> B. We discuss examples and the question of how our results might help to understand some aspects of Lagrangian fibrations.

L-2-Kohomologie von Calabi-Yau-Familien über Kurven

Ist $f: X \to S$ eine glatte Familie von Calabi-Yau-Mannigfaltigkeiten der Dimension $m$ über einer quasiprojektiven Kurve, so trägt nach einem Resultat von Zucker die erste $L^2$-Kohomologiegruppe $H^1_{(2)}(S, R^m f_* \mathbb{C}_X)$ eine reine Hodgestruktur vom Gewicht $m+1$. In dieser Arbeit berechnen wir die Hodgezahlen solcher Hodgestrukturen für $m= 1, 2, 3$ und verallgemeinern dabei Formeln aus einem Artikel von del Angel, Müller-Stach, van Straten und Zuo auf den Fall, in dem die lokalen Monodromiematrizen bei Unendlich nicht unipotent, sondern echt quasi-unipotent sind. Wir verwenden dazu den $L^2$-Higgs-Komplex nach Jost, Yang und Zuo. Für Familien von Kurven führt dies auf eine bereits bekannte Formel von Cox und Zucker. Schließlich wenden wir die Ergebnisse im Fall $m=3$ auf 14 Familien von Calabi-Yau-Mannigfaltigkeiten an, die eine Rolle in der Spiegelsymmetrie spielen, sowie auf eine von Rohde konstruierte Familie ohne Punkte mit maximal unipotenter Monodromie.

Hearing Delzant Polytopes From the Equivariant Spectrum

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu asked whether the spectrum of the Laplace operator $\Delta_g$ on $\mathcal{C}^\infty(M)$ determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of M^4 is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of M and the spectrum of its associated real manifold M_R determine its polygon, up to translation and a small number of choices. For M of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of M.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

Integrality of Tate-cycles

We explain a technical result about p-adic cohomology and apply it to the study of Shimura varieties. The technical result applies to algebraic varieties with torsion-free cohomology, but for simplicity we only treat abelian varieties.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

On Selmer groups and factoring p-adic L-functions

Thesis (Ph.D.)--University of Washington, 2016-06

On realizability of p-groups as Galois groups

2000 Mathematics Subject Classification: 12F12, 15A66.

Classifying Rational G-Spectra for Finite G

We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore, the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model.

Rational O(2)–Equivariant Spectra

The category of rational O(2)-equivariant cohomology theories has an algebraic model A(O(2)), as established by work of Greenlees. That is, there is an equivalence of categories between the homotopy category of rational O(2)-equivariant spectra and the derived category of the abelian model DA(O(2)). In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. This Quillen equivalence is also compatible with the Adams short exact sequence of the algebraic model.

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

MSC 19L41; 55S10.