Rational points on curves and higher dimensional varieties (ARITH)
Organised by Siksek, Colliot-Th{\'e}l{\`e}ne, Skorobogatov and Stoll

Many strands of modern number theory relate intimately to algebraic
geometry. Varieties over number fields are analogous to varieties over
function fields; the spectacular recent methods for rational curves on
varieties over the complexes should be relevant to rational points on
varieties over arithmetic fields (finite fields, local fields, global
fields); there are deep relations between the geometry of varieties
over finite and p-adic fields and analytic aspects of complex geometry
(involving both coherent and etale cohomology). In many cases geometry
(notably the classification of varieties) is involved in the very
statement of the arithmetic problem: rationally connected varieties,
Abelian varieties, K3 surfaces, ramification of elements of the Brauer
group, and so on. There are also many common technical tools or
approaches, such as descent, torsors, moduli spaces, Chow groups or
cohomology. Number theory also has much more specific links to other
components of WAG07-08, notably HD MMP, COM and MOD (rational curves,
Fanos, K3s, Brauer groups and geometry of torsors or quasi-homogeneous
varieties, etc.).

This activity will spread over most of the Warwick summer term, from
late Apr to late Jun, with a concentrated workshop scheduled for Mon
16th--Sat 21st Jun. ARITH will also be a component of the suite of
concluding workshops in Jul 2008.

Participants. Confirmed: Colliot-Th{\'e}l{\`e}ne, Cremona, Fisher
O'Neil, Poonen, Reid, Siksek, Silverman, Skorobogatov, Stoll, van
Luijk,

Targetted: Batyrev, Berkovich, Bogomolov, Borovoi, Bourqui, Browning,
Campana, Caporaso, Darmon, de Jong, de la Bret{\`e}che, Derenthal,
Ducros, Ellenberg, Esnault, Frossard, Gille, Harari, Harris, Hassett,
Heath-Brown, Kirwan, Koll{\'a}r, Kresch, Lieblich, Livne, Madore,
McCallum, Peyre, Popov, Raskind, Rotger, Salberger, Saltman, Satge,
Shepherd-Barron, Starr, Swinnerton-Dyer, Szamuely, Tschinkel, van
Hamel, Venkatesh, Verrill, Voloch, Wittenberg, Wooley, Yafaev,

============================

<< Could put one or two more specific research areas/directions/recent
achievements in place of the woffly "deep relations"? >>

Esnault has proved that geometric or cohomological conditions on the
Chow group imply the the existence of rational points over a finite
field.

Graber--Harris--Starr, Koll\'ar,  de Jong and Hassett--Tschinkel have
obtained result arithmetic by applying methods from rational
connectedness in higher dimensional geometry: extensions of Tsen's
theorem, weak approximation for rationally connected varieties over a
function field in one variable, the proof of the index = exponent
conjecture for central simple algebras over a two-dimensional function
field.

============================

Notes from original e-mails

Colliot: May or June 2008 are fine. I would have a very slight
preference for May or early June.
Skoro: May or Jun would work, Mon 16th--Sat 21st Jun
(week 9 of term, good time for conference)

Here are some topics:

1. Close to complex geometry: things having to do with HD MMP and
classification, rationally connected varieties, 1-rationally connected
varieties, Mori and Campana classification, various versions of the Lang
conjectures on higher analogues of the Mordell conjecture.

2. Brauer-Manin and torsors: Skorobogatov has already given a list. For
Miles, in your report if need be you may mention that Poonen and Voloch
have recently made great progress on the study of the Brauer-Manin
condition for rational points on curves -- in the function field case.
Let me make some of the suggestions more precise: Harari, Wittenberg.

3. The "counting people": Manin-Batyrev "conjectures", circle method,
methods a la Bombieri-Pila-Heath-Brown: some have already been mentioned
by Alexei (in particular Wooley). Let me already add: .

There is now a group in Bristol (you may discuss the matter with Timothy
Browning).

4. Rational points on K3 surfaces. This is still very much an
experimental topic, but  quite interesting. Some names: .

5. Possible connexions with model theory?

6. Do we want to touch upon the arithmetic of zero-cycles and things
such as higher class field theory? There is interesting work of
Jannsen, Shuji Saito, Kanetomo Sato.

Siksek: Dear all, I've been in touch with Stoll who is keen to be
involved {we can put him as a coorganiser if we want to}. Poonen is
interested, but does not want to be involved in organization.

Of course as I am at Warwick, so all the dates should be fine with me. I
will let others have their say. All the best, Samir

Dear Jean-Louis, Miles, and Samir,

I am often away during April, but May and June are both fine.
The conference in the week Mon 16th--Sat 21st Jun looks good,
but other weeks are good as well.

I can think of two special themes:

-- explicit descriptions of torsors (this has links with analytic number
theory, and some of UK number theory people may want to come - e.g.
Heath-Brown, Browning, but also with homogeneous spaces of algebraic
groups, so people who know about Lie algebras and representations and
root systems and singularities and GIT may be interested as well,
possible names are Batyrev, Popov, Tschinkel, Hassett, Derenthal, Peyre,
but maybe also Kirwan?)

-- rational points on curves (Samir will be able to advise here, one
thinks of Stoll and his colleagues)

I am sure that Jean-Louis will come up with more ideas. Alexei