# Oberseminar Zahlentheorie (WS 2013/14)

Werner Bley - Ulrich Derenthal - Cornelius Greither - Andreas Rosenschon

Winter 2013/14, Wednesdays, 16:30, room B 251 (Mathematisches Institut, Theresienstr. 39, 80333 München)

Previous semester: Summer 2013

Date | Speaker | Title | Remarks |
---|---|---|---|

16.10.13 | Vorbesprechung | ||

23.10.13 | Masanori Asakura (Hokkaido University) | Real regulator on K_1 of a fibration of curves having a totally degenerate semistable fiber | |

30.10.13 | Stéphane Viguié (LMU) | Iwasawa's main conjecture over a quadratic field | |

6.11.13 | Marco Hien (Universität Augsburg) | Stokes Strukturen und Fourier Transformation meromorpher Zusammenhänge | |

13.11.13 | Dasheng Wei (LMU) | Strong approximation with Brauer-Manin obstruction | |

20.11.13 | Thomas Jahn (LMU) | Higher Brauer Groups | |

27.11.13 | Henri Johnston (Exeter) | Applications of hybrid p-adic group rings and hybrid Iwasawa algebras to arithmetic conjectures | |

4.12.13 | Norbert Hoffmann (Limerick) | Rational families of instanton bundles on P^{2n+1} | |

29.1.14 | Rachel Newton, MPI Bonn | The transcendental Brauer group of a product of CM elliptic curves |

## Abstracts

### Masanori Asakura (Hokkaido University): *Real regulator on K_1 of a fibration of curves having a totally degenerate semistable fiber * (23.10.13)

I give a certain method to compute the Beilinson regulator on K_1 of a fibration of curves which contains at least one totally degenerate semistable fiber. As an application, I give lots of examples of regulator indecomposable elements for a surface defined over Q with arbitrary large p_g.

### Stéphane Viguié (LMU): *Iwasawa's main conjecture over a quadratic field * (30.10.13)

Let k be an imaginary quadratic field, p a prime number which splits in k, and k_\infty the unique Z_p2-extension of k.

Let K_\infty be a finite extension of k_\infty, abelian over k, and let G be the torsion subgroup of Gal(K_\infty/k).

Using Euler systems, we will present a version of the Iwasawa's main conjecture which holds for each irreducible C_p-character \chi of G. It relates the characteristic ideals of the \chi-quotient of the projective limit of global units modulo elliptic units and the characteristic ideal of the \chi-quotient of the projective limit of the p-part of the ideal class groups.

### Marco Hien (Universität Augsburg) *Stokes Strukturen und Fourier Transformation meromorpher Zusammenhänge * (6.11.13)

Die lokale Klassifikation meromorpher Zusammenhänge auf einer Kurve zerfällt in zwei Teilprobleme, nämlich der formalen und der analytischen Klassifikation. Dies führt zum Begriff der Stokes Struktur. In höheren Dimensionen sind diese Probleme noch nicht vollständig gelöst, es gibt jedoch durch Resultate von T. Mochizuki und K. Kedlaya aktuelle Ansätze. Eine wichtige Frage ist die, wie sich die Stokes Strukturen in einer Dimension unter der Fourier Transformation verhalten. Hierzu gibt es neue Ergebnisse aus einer Zusammenarbeit mit C. Sabbah, über die ich berichten möchte.

### Dasheng Wei (LMU) *Strong approximation with Brauer-Manin obstruction * (13.11.13)

Some varieties over a number field satisfy strong approximation with Brauer-Manin obstruction. On these varieties, the existence of integral points on their integral models can be determined by the Brauer-Manin obstruction. In this talk, I will discuss such varieties, which is joint work with Ulrich Derenthal.

### Thomas Jahn (LMU) *Higher Brauer Groups* (20.11.13)

Let X be a smooth projective variety over a finite field. The cohomological Brauer group Br(X) is related to the Zeta function on X and the Tate conjecture. There is the notion of higher Brauer groups Br^r(X) which admit generalizations of those links In this talk we define higher Brauer groups and sketch the links mentioned. Moreover, we outline the proof of a recent result regarding the order of l-primary subgroups Br^r(X)(l) of higher Brauer groups of certain varieties.

### Henri Johnston (Exeter) *Applications of hybrid p-adic group rings and hybrid Iwasawa algebras to arithmetic conjectures* (27.11.13)

We introduce the closely related notions of hybrid p-adic group ring and hybrid Iwasawa algebra. When certain group-theoretic criteria are met, hybrid p-adic group rings allow one to reduce the proof of the equivariant Tamagawa number conjecture for a particular Galois extension of number fields to the proofs of two easier conjectures; this leads to unconditional proofs in certain cases. A similar story holds for hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture for totally real fields, though there are some important differences. This is joint work with Andreas Nickel (Bielefeld).

### Norbert Hoffmann (Limerick) *Rational families of instanton bundles on P^{2n+1}* (4.12.13)

Instanton bundles are algebraic vector bundles of rank 2n on complex projective (2n+1)-space having a particular Chern polynomial and with certain cohomology groups vanishing. The talk is about two natural irreducible loci in the moduli space of symplectic instanton bundles. One result is that these loci are often rational. I will also present some evidence for Ottaviani's conjecture that one of the loci is an irreducible component for n > 1, and deduce that the moduli space is then reducible. This is joint work with L. Costa, R.M. Miro-Roig and A. Schmitt.

### Rachel Newton, MPI Bonn: *The transcendental Brauer group of a product of CM elliptic curves * (29.1.14)

In 1971, Manin showed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the existence of points everywhere locally on X despite the lack of a global point is sometimes explained by non-trivial elements in Br(X). Since Manin's observation, the Brauer group has been the subject of a great deal of research.

The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. In contrast, until recently the transcendental part of the Brauer group had not been computed for a single variety. The transcendental part of the Brauer group is known to have arithmetic importance: it can obstruct the Hasse principle and weak approximation.

I will use class field theory together with results of Ieronymou, Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the product ExE for an elliptic curve E with complex multiplication. The results for the odd-order torsion descend to the Kummer surface Kum(ExE).