Author Archives: Jim Stankewicz

The torsion on CM elliptic curves over prime degree number fields

It’s good to be back! This weekend I’m going to Paris to give a talk in the London-Paris Number Theory seminar so I’m going to give a preview of that talk, based on joint work with Pete Clark and Abbey Bourdon. … Continue reading

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The sound you hear is another conjecture in birational geometry dropping like a fly

These are interesting times to look over the algebraic geometry arxiv postings. Just over a week ago, there was a posting by Tanaka which claimed the minimal model program was false in characteristic two. Then yesterday at the top of … Continue reading

Posted in Algebraic Geometry, Uncategorized | 1 Comment

A particularly lousy version of academic dishonesty

Last semester, a number of emails like this circulated: Subject: Math Requirements From: To: jstankewatbutIdon’ Dear James Stankewicz, My name is Firstname Lastname and I am a graduate student at the University of Prestigious Institution studying for my ph … Continue reading

Posted in Math Culture | 8 Comments

Local Fields

Dear Readers, Let’s examine the role of topology in the study of fields and arithmetic. A topology on a field compatible with the field operations is given by an absolute value, which in turn defines a metric. Outside of number … Continue reading

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Shih’s Theorem

Dear Readers, In spite of orals closing in a little more every day, I clearly haven’t been updating so much recently. I’d started a post about using Minkowski’s geometry of numbers to think about class numbers and unit groups and … Continue reading

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Dedekind Domains in Finite Galois Extensions

Last time we examined Dedekind domains in finite separable field extensions. One advantage to using a separable field extension that we did not use is that we can base extend to a finite Galois extension, where as we see the … Continue reading

Posted in Algebraic Number Theory, Curves, Number Fields, Uncategorized | Leave a comment

Dedekind Domains in finite separable extensions

Last time we took a look at Dedekind domains with fraction fields and found that if was any finite field extension of that the integral closure of in is Dedekind. The proof in this case is somewhat involved, but becomes … Continue reading

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Dedekind Domains, Krull-Akizuki and the Picard Group

As has been hinted in many previous posts, many facts about algebraic number theory tell us about geometric objects like elliptic curves. For instance, if you are working on a problem which primarily uses the affine geometry of a curve … Continue reading

Posted in Curves, Uncategorized | 5 Comments

Endomorphisms of Elliptic Curves and the Tate module

Dear Readers, We’ve now talked about quaternion algebras, and today I’ll talk about the surprisingly close connection between quaternion algebras and elliptic curves. We first recall a fundamental fact about isogenies:

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Quaternion Algebras and Modular Forms

Dear readers, I know I promised a post  on modular curves, but I had to devote more time to my end of semester project. Since it’s strongly related to the topic of modular curves and I present on it tomorrow, … Continue reading

Posted in Talks | 4 Comments