The Bielefeld Arithmetic Geometry Seminar 2015 (winter term)

Welcome to the website of the 6th edition of the Bielefeld arithmetic geometry seminar! Below you will find a list of the upcoming talks, hover with your mouse over abstract to see the abstract. The seminar is related to projects B5 and C7.
The seminar is supported by the SFB CRC 701: Spectral Structures and Topological Methods in Mathematics.

Talks

Wednesday 28 October, 2015
16:15	C01-142	Valentina Di Proietto (FU Berlin) The homotopy exact sequence for the log algebraic fundamental group: a way to define a monodromy action [Abstract] The classical Riemann-Hilbert correspondence relates the fundamental group of a complex analytic variety and the differential equations we can define on it. The definition of the fundamental group given in terms of homotopy classes of loops does not generalize easily to algebraic varieties defined over an arbitrary field. But exploiting this link we can give another definition that makes sense in very general contexts: it is called the algebraic fundamental group. We prove the homotopy exact sequence for the algebraic fundamental group for a fibration with singularities with normal crossings, result of a collaboration with A. Shiho. We explain how via this exact sequence we can define monodromy action: if time permits we present also a work in progress with B. Chiarellotto and A. Shiho in which we study deeply this action.
Wednesday 11 November, 2015
16:15	C01-142	Cornelius Greither (Munich) A question of Hayes concerning integrality of Brumer elements [Abstract] To every abelian Galois extension $K/k$ of number fields with Galois group $G$ and every finite set $S$ of places of $k$ containing all infinite and all ramified places, one can associate the so-called Brumer element $\theta_{K/k}\in \mathbb Q[G]$. In fact this element is almost integral, in the sense that $I \theta_{K/k}\subset \mathbb Z[G]$, where $I$ is the annihilator ideal of the module $\mu_K$ of roots of unity. In particular if $w_K$ denotes the number of roots of unity in $K$, then $w_K\theta_{K/k}$ has integer coefficients. The Brumer element can be thought of as a $G$-equivariant generalization of the class number $h_K$; it plays the lead role in the Brumer conjecture which predicts that $I \theta_{K/k}$ annihilates the class group of $K$. It is quite natural to ask under what circumstances the Brumer element is $p$-integral. David Hayes addressed the interesting case $p\|w_K$ by formulating a certain set of hypotheses, which is somewhat sharper than demanding $p\|h_K$ (which is in some sense a necessary condition), and asking: do these conditions imply that $\theta_{K/k,S}$ is $p$-integral? Barry Smith gave a positive answer to a related question in a fairly restricted setting. In this talk we answer Hayes' question in the negative by a systematic construction of counterexamples. First we exhibit "approximate counterexamples" with base field $\mathbb Q$ (Hayes' question happens never to apply when the base field is $\mathbb Q$), and then we obtain the desired counterexamples by a fairly simple shift of base field.
Wednesday 25 November, 2015
15:00	C01-142	Andreas Langer (Exeter) $p$-adic deformation of motivic Chow-groups [Abstract] In their recent work on $p$-adic deformation of algebraic cycle classes Bloch, Esnault and Kerz give an equivalent Hodge-theoretic condition on the crystalline chern class when an algebraic cycle class lifts from char $p$ to a pro-class in the continous Chow-group of a formal lifting. In my talk I will present a relative version of their work. Starting with a projective smooth variety $X$ defined over the ring $R = n$-truncated Witt-vectors of a perfect field the classical Chow-group is replaced by a motivic Chow-group which is defined using a mixed characteristic version of the Suslin-Voevodsky complex and is still related to Milnor $K$-Cohomology. I explain that the main results of Bloch-Esnault-Kerz formally hold in this relative setting as well. One can give a Hodge-theoretic condition when an element in the motivic Chow-group lifts to a pro-class on a formal lifting of $X$ over $W(R)$. In the proof the relative de Rham-Witt complex and a relative version of syntomic cohomology play a crucial role.
16:15	C01-142	Steffen Müller (Oldenburg) Quadratic Chabauty [Abstract] I will introduce a method to $p$-adically approximate integral points on a hyperelliptic curve defined over the rationals whose genus is equal to the Mordell-Weil rank of its Jacobian. The method relies on $p$-adic heights and iterated Coleman integrals. If time permits, I will also discuss how this approach can be used to provably find all integral points and how it can be extended to curves defined over number fields. This is joint work with Jennifer Balakrishnan and Amnon Besser.
Wednesday 27 January, 2016
15:00	C01-142	Yara Elias (MPI Bonn) On the Selmer group associated to a modular form twisted by a ring class character [Abstract] Kolyvagin constructs an Euler system based on Heegner points and uses it to bound the size of the Selmer group of certain (modular) elliptic curves $E$ over imaginary quadratic fields $K$ assuming the non-vanishing of a suitable Heegner point. In particular, this implies that $\mathrm{rank}(E(K))=1$, and the Tate-Shafarevich group of $E$ over $K$ is finite. Bertolini and Darmon adapt Kolyvagin's descent to Mordell-Weil groups over ring class fields. Nekovar develops the method of Euler systems in the setting of modular forms of higher even weight to describe the image by the Abel-Jacobi map of Heegner cycles on the associated Kuga-Sato varieties. In this talk, we combine these two approaches to study modular forms of higher even weight twisted by ring class characters of imaginary quadratic fields assuming the non-vanishing of a suitable Heegner cycle. Our motivation stems from the Beilinson-Bloch conjecture which is a generalization of the Birch and Swinnerton-Dyer conjecture.
16:15	C01-142	Georg Tamme (Regensburg) $K$-theory for non-archimedean algebras and spaces [Abstract] In this talk I will report on joint work with Moritz Kerz and Shuji Saito. We propose an "analytic" $K$-theory for affinoid algebras and rigid spaces over a complete non-archimedean field. This $K$-theory satisfies analogs of the Eilenberg-Steenrod axioms from topology, and is closely related to continuous $K$-theory of formal models. This gives a hint that this $K$-theory could be useful in the study of the variational Hodge conjecture.
Wednesday 10 February, 2016
16:15	C01-142	Lin Jie (Jussieu) An automorphic version of the Deligne conjecture [Abstract] The Deligne conjecture relates special values of a motivic L-function and the Deligne period defined as a determinant with respect to certain rational structures. In this talk, we will present an analogue of this conjecture for tensor products of automorphic representations.

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Seminars

Conference

The Bielefeld Arithmetic Geometry Seminar 2015 (winter term)

Talks

Wednesday

28 October, 2015

Wednesday

11 November, 2015

Wednesday

25 November, 2015

Wednesday

27 January, 2016

Wednesday

10 February, 2016