1,666 research outputs found
A modular description of
As we explain, when a positive integer is not squarefree, even over
the moduli stack that parametrizes generalized elliptic curves
equipped with an ample cyclic subgroup of order does not agree at the cusps
with the -level modular stack defined by
Deligne and Rapoport via normalization. Following a suggestion of Deligne, we
present a refined moduli stack of ample cyclic subgroups of order that does
recover over for all . The resulting modular
description enables us to extend the regularity theorem of Katz and Mazur:
is also regular at the cusps. We also prove such regularity
for and several other modular stacks, some of which have
been treated by Conrad by a different method. For the proofs we introduce a
tower of compactifications of the stack that
parametrizes elliptic curves---the ability to vary in the tower permits
robust reductions of the analysis of Drinfeld level structures on generalized
elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor
Probability Theory of Random Polygons from the Quaternionic Viewpoint
We build a new probability measure on closed space and plane polygons. The
key construction is a map, given by Knutson and Hausmann using the Hopf map on
quaternions, from the complex Stiefel manifold of 2-frames in n-space to the
space of closed n-gons in 3-space of total length 2. Our probability measure on
polygon space is defined by pushing forward Haar measure on the Stiefel
manifold by this map. A similar construction yields a probability measure on
plane polygons which comes from a real Stiefel manifold.
The edgelengths of polygons sampled according to our measures obey beta
distributions. This makes our polygon measures different from those usually
studied, which have Gaussian or fixed edgelengths. One advantage of our
measures is that we can explicitly compute expectations and moments for
chordlengths and radii of gyration. Another is that direct sampling according
to our measures is fast (linear in the number of edges) and easy to code.
Some of our methods will be of independent interest in studying other
probability measures on polygon spaces. We define an edge set ensemble (ESE) to
be the set of polygons created by rearranging a given set of n edges. A key
theorem gives a formula for the average over an ESE of the squared lengths of
chords skipping k vertices in terms of k, n, and the edgelengths of the
ensemble. This allows one to easily compute expected values of squared
chordlengths and radii of gyration for any probability measure on polygon space
invariant under rearrangements of edges.Comment: Some small typos fixed, added a calculation for the covariance of
edgelengths, added pseudocode for the random polygon sampling algorithm. To
appear in Communications on Pure and Applied Mathematics (CPAM
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