good fibrations – math is art

Honda, Taira, On the Formal Structure of the Jacobian Variety of the Fermat Curve over a $p$-adic Integer Ring

On the Formal Structure of the Jacobian Variety of the Fermat Curve over a $p$-adic Integer Ring, Symposia Mathematica Volume XI (Volume 11).

This paper of Honda was very hard for me to track down. It was not previously available digitally. I found it in a retired library volume in a thrift store in England. Hopefully, this digital copy will make it easier for others to enjoy Honda’s incredible insight and understanding of how to create power series in one variable with high amounts of arithmetic information, and in general of higher height formal group laws.

jacobian_of_fermat_honda

If the pdfviewer isn’t working, here is the link.

Overview of the Classic Theory of p-Divisible groups

Here are the texed notes for my 1 hour oberwolfach talk (the key thing I left out is Cartier duality). I included Manin’s original approach, and remark on some details I find left out of most sources.

officialober

If the pdf viewer doesn’t work, here is a link to the paper.

Hidden Structure

Beauty wilts
and love will flucture
what remains
is hidden structure

blushing primes
sly chiding knots
intertwine
betwixt our thoughts

creation
lives to create
chipping from
the ob’lisk slate

talk amoungst
our little selves
speculating
filling shelves

search we will
and see we’ll not
blindness is
our earthly lot

but the voice
of the void is vast
notice it
through the nets we cast

Automorphisms of the Jacobian

Read our arxiv paper: https://arxiv.org/abs/1811.07007

Here, $A$ is any abelian variety. This post consists of the backstory of this paper, and something interesting I learned about the relationship between the size of $Aut(A)$, and the number of principal polarizations $A$ has. Continue reading Automorphisms of the Jacobian

Fiber Bundles of Formal Disks

Here is an incomplete proof that varieties are fiber bundles of formal disks over their deRham Stacks. The fact makes intuitive sense, the deRham stack is the variety without infinitesimal data, and then by adding the infinitesimal data (formal disks) back in, you recover the result. However, the fact that you can build anything non infinitesimal out of formal disks fills me with confusion and awe.

Acknowledgements: This is the result of a working group with Dan Fletcher, Adam Holeman, and me as part of the Northwestern Homotopy Working Seminar (started by Matthew Weatherly, Grigory Kondyrev, and me). The working session in which Adam and I figured out the proof, Dan was not there, which is why his name is not mentioned, but he was very helpful in understanding the claim. This proof is the result of Yaroslav Khromenkov coming up with the idea of it, and Adam and I understanding and correcting his solution during a working session.

formalgroup

If the pdf viewer doesn’t work, here is a link to the paper.