WEIL RECIPROCITY IN RIGID ANALYTIC GEOMETRY


Abstract. Weil reciprocity has been a pivot point for several modern areas of research. 

For instance, when working over an algebraically closed field K, by phrasing Weil reciprocity in
terms of a tame symbol, one is able to recover the equivalence between Qcoh(LocSys_{G_m}(D^*_K)) 
and D-mod(LG_m(K)), which is a version of categorical local geometric class field theory. 


By appealing to a Contou-Carrere pairing and tools from complex-analytic homological algebra,
Deligne proved a version of Weil reciprocity for bounded open complex analytic curves. 

Then, using techniques in infinite-dimensional linear algebra, Garcıa Lopez unified Deligne’s result
with a version of Weil reciprocity that applies to smooth formal lifts of affine subvarieties
of a projective curve defined over F_p. 

In this talk we will describe how one can use similar methods to derive a version of Weil reciprocity that works for a large class of (relative) rigid analytic curves. 

The central object in our work is the Robba ring R_X of a partially proper curve X, which can be viewed as a certain ind-pro object of Banach spaces. We will describe a determinant gerbe attached to the Robba ring, 
which receives Pic and GL(R_X) actions.

In particular we can define a central extension of GL(R_X), from which one is able to define
a symbol on R^*_X × R^*_X that is equal to 1 on pairs of global invertible analytic functions. 

If time permits, we will discuss the relationship between our constructions and recent work of
Matthew Emerton, future work on global duality, and reconciling our definition of Robba
rings with the notion of Tate objects in Quillen exact categories. 

This talk is based on the speaker’s PhD thesis, done at the University of Texas at Austin.