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Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

Information:

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Workshop
30th March 2020 to 3rd April 2020
Organisers: 
James D. Lewis University of Alberta, University of Alberta
Rob de Jeu Vrije Universiteit Amsterdam

Workshop theme:

It is a classical problem to understand the value of the Riemann zeta-function at n=2, 3, 4, ..... For n even this value is π<sup>n</sup> times a non-zero rational number, but for n odd much less is known. In the 70s Borel proved that, for a number field k, and n≥2, up to some simple factors and a non-zero rational number, ζ<sub>k</sub>(n) equals the regulator R<sub>n</sub>(k) associated to the K-group K<sub>2n−1</sub>(k) defined earlier by Quillen. Because ζ<sub>Q</sub>(s)=ζ(s), and K<sub>3</sub>(Q), K<sub>7</sub>(Q), ... are finite whereas K<sub>5</sub>(Q), K<sub>9</sub>(Q), ... have rank one, this explains the behaviour of ζ(n).

Using the functional equation of ζ<sub>k</sub>(n), Borel’s result gives the main ingredient of the first non-zero coefficient in the Taylor expansion at s=1−n of ζ<sub>k</sub>(s), but the Lichtenbaum conjecture is more precise: up to sign and a power of 2, this coefficient should equal |K<sub>2n−1</sub>(O<sub>k</sub>)<sub>tor</sub>|<sup>-1</sup>|K<sub>2n−2</sub>(O<sub>k</sub>)| R<sub>n</sub>(k), where O<sub>k</sub> is the ring of algebraic integers in k. Nowadays there are various conjectures on special values of zeta-functions and their generalisations. Techniques to prove results have involved Iwasawa theory, as well as more explicit descriptions of the K-groups involved in terms of complexes of algebraic cycles or formal generators and relations.

One of those conjectures (predating Borel’s result and Lichtenbaum’s conjecture, and, in fact, one of the Millenium Prize Problems of the Clay Mathematical Institute) is the conjecture by Birch and Swinnerton-Dyer, on the behaviour at s=1 of L(E, s) for an elliptic curve E over a number field, where L(E, s) is the analogue of the zeta-function in this context. Its statement involves another K-group, K<sub>0</sub>(E), as well as periods of 1-forms on the associated Riemann surface, a height pairing, and some more refined arithmetic invariants. Such periods tie up with Hodge theory, one of the other areas covered by this workshop.

Deadline for applications: 5th January 2020

KAH programme participants DO NOT need to apply, programme participants with visit dates during KAHW03 will automatically be added to the attendee list.

Please note members of Cambridge University are welcome to turn up and sign in as a non-registered attendee on the day(s) during the workshop and attend the lecture(s). Please note that we cannot provide you with any support including name badge, meals or accommodation.

In addition to visiting the INI, there are multiple ways in which you can participate remotely.

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Fees

Registration Only    
  • Registration Package: £220
  • Student Registration Package: £170

The Registration Package includes admission to all seminars, lunches and refreshments on the days that lectures take place (Monday - Friday), wine reception and formal dinner, but does not include other meals or accommodation.

Registration and Accommodation
  • Accommodation Package: £588

The Accommodation Package includes a registration fee, bed and breakfast accommodation at Churchill College from the evening of Sunday to breakfast on Saturday, together with lunches and refreshments during the days that lectures take place (Monday - Friday). The formal dinner is also included, but no other evening meals.

Formal Dinner Only
  • Formal Dinner: £50

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event. For all remaining participants who would like to attend, such as programme participants, the above charge will apply.

Accommodation

Accommodation in single study bedrooms with shared facilities and breakfast are provided at Churchill College,

Meals

Lunch

Lunch timings and location will be confirmed with timetable.

Evening Meal

Participants are free to make their own arrangements for dinner.

Formal Dinner

The Formal Dinner location and date is to be confirmed. Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons