| Talk_id | Date | Speaker | Title |
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19675
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Wednesday 10/9
3:00 PM
|
Romyar Sharifi, UCLA
|
Cocycles valued in motivic cohomology
- Romyar Sharifi, UCLA
- Cocycles valued in motivic cohomology
- 10/09/2019
- 3:00 PM - 4:00 PM
- C304 Wells Hall
I will describe joint work in progress with Akshay Venkatesh on the construction of 1-cocycles on $\mathrm{GL}_2(\mathbb{Z})$ valued in a limit of second motivic cohomology groups of open subschemes of the square of (1) the multiplicative group over the rationals and (2) a universal elliptic curve. I’ll explain how these cocycles specialize to homomorphisms taking modular symbols to special elements in second cohomology groups of cyclotomic fields and modular curves in the respective cases.
|
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19676
|
Wednesday 10/23
3:00 PM
|
Sumit Chandra Mishra, Emory University
|
Local-global principle for norms over semi-global fields
- Sumit Chandra Mishra, Emory University
- Local-global principle for norms over semi-global fields
- 10/23/2019
- 3:00 PM - 3:50 PM
- C304 Wells Hall
Let $K$ be a complete discretely valued field with
residue field $\kappa$.
Let $F$ be a function field in one variable over $K$
and $\mathscr{X}$ a regular proper model of $F$
with reduced special fibre $X$ a union of regular curves
with normal crossings.
Suppose that the graph associated to
$\mathscr{X}$ is a tree (e.g. $F = K(t)$).
Let $L/F$ be a Galois extension of degree $n$ with Galois group $G$
and $n$ coprime to char$(\kappa)$.
Suppose that $\kappa$ is algebraically closed field or
a finite field containing a primitive $n^{\rm th}$ root of unity.
Then we show that an element in $F^*$ is a norm
from the extension $L/F$ if it is a norm from the
extensions $L\otimes_F F_\nu/F_\nu$
for all discrete valuations $\nu$ of $F$.
|
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20689
|
Tuesday 11/5
3:00 PM
|
Carl Wang-Erickson, University of Pittsburg
|
Bi-ordinary modular forms
- Carl Wang-Erickson, University of Pittsburg
- Bi-ordinary modular forms
- 11/05/2019
- 3:00 PM - 4:00 PM
- C304 Wells Hall
It is known that p-ordinary cuspidal Hecke eigenforms give rise to 2-dimensional global Galois representations which become reducible after restriction to a decomposition group at p. For which such forms is this restriction not only reducible but also splittable? Complex multiplication (CM) forms satisfy this p-local property, but is such a restrictive global property as CM necessary? In classical weights at least 2, it is expected that this is the case. We present a construction of "bi-ordinary" p-adic modular forms, which can measure exceptions to this expectation. We also give evidence that there are non-CM but p-locally splittable forms in p-adic weights. This is joint work with Francesc Castella.
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20709
|
Wednesday 11/20
3:00 PM
|
Patrick Allen, UIUC
|
On the modularity of elliptic curves over imaginary quadratic fields
- Patrick Allen, UIUC
- On the modularity of elliptic curves over imaginary quadratic fields
- 11/20/2019
- 3:00 PM - 4:00 PM
- C304 Wells Hall
The Langlands-Tunnell theorem is an important starting point in Wiles's proof of the modularity of semistable elliptic curves over the rationals. Over imaginary quadratic fields it is unclear how to similarly use the Langlands-Tunnell theorem and Wiles's strategy runs into problems right from the start. I will motivate and explain the subtle but fundamental issues that arise, and indicate how they can be circumvented in many cases. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work with Chandrashekhar Khare and Jack Thorne.
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