3:45: The Picard group of the moduli stack of elliptic curves
Martin Olsson
I will report on joint work with William Fulton.
If E is an elliptic curve (i.e. a genus 1 curve with one marked point)
over a field, then the space
H0(E, Ω 1E) of global differential
forms on E is a vector space of rank 1. More generally, if S
is a scheme and f:E --> S is a flat family of 1-pointed curves of
genus 1, then f*Ω 1E/S is a
line bundle on S called the
"Hodge bundle" of E. In this way we
obtain for any scheme S a "rule" associating to any family of elliptic
curves over S a line bundle on S, and this rule is functorial in
S. In a beautiful 1965 paper, Mumford called such a rule a "line bundle
on the moduli problem of elliptic curves" and showed that
if one restricts to k-schemes S for a field k not of
characteristic 2
or 3 then the Hodge bundle essentially defines the only possible rule.
In modern language he showed that the Picard
group of the moduli stack M1,1,k is canonically
isomorphic to Z/(12) with generator the Hodge bundle.
In light of Mumford's work, two natural questions arise:
(1) compute the Picard group of the moduli stack M1,1,S
of elliptic curves over a general base scheme S,
(2) compute the Picard group of the canonical
compactification of M1,1,S.
In this talk I will answer these questions.
I will not assume any familiarity with stacks and the first part of the
talk will be to explain what is meant by a line bundle on
M1,1,S.
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