Posts Tagged ‘modular forms’

I’ve been reading a lot of mathematics lately, but I don’t feel capable of explaining most of what I’ve been reading about, so I’m not sure what to blog about these days. Fortunately, SPUR will be starting soon, so I’ll start focusing on relevant material for my project eventually. Until then, here are some more random updates.

  • Martin Gardner and Walter Rudin both recently passed away. They will be sorely missed by the mathematical community, although I can’t say I’m particularly qualified to eulogize about either.
  • For my number theory seminar with Scott Carnahan I wrote a paper describing an important corollary of the Eichler-Shimura relation in the theory of modular forms. The actual relation is somewhat difficult to state, but the important corollary relates the number of points on certain elliptic curves E over finite fields to the Fourier coefficients of certain modular forms of weight 2. You can find the paper here. Although the class is over, corrections and comments are of course welcome. (Though I hope Scott doesn’t change my grade if someone spots a mistake he missed!)
  • If you’re at all interested in the kind of mathematics where planar diagrams are used instead of traditional algebraic notation for computation, you should read Joachim Kock’s excellent book Frobenius Algebras and 2D Topological Quantum Field Theories. The book is much less intimidating than its title might suggest, and it is full of enlightening pictures and discussions. You might also be interested in a related MO question.

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The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field F. The group of fractional linear transformations PGL_2(F) acts triple transitively on \mathbb{P}^1(F) and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on \mathbb{P}^1(\mathbb{C}) preserving the upper half plane \mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \} is PSL_2(\mathbb{R}).

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).


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