Special parameters

An old paper (Über hypergeometrische Funktionen, deren letztes Element speziell ist) by W. Heymann (from a talk by Wadim Zudilin) discusses the value of an ${}_2F_1$ at special $t$’s. He does $t=-1/3,1/4,1/5$ and related values and mentions Gauss for $t=1,-1,1/2$ and Kummer for $t=-1/8,1/9,8/9$. For example we have this identity $$ F_h = F \left( -\frac{h}{2}, -\frac{h}{2} + \frac{1}{2}, h + \frac{3}{2}, -\frac{1}{3} \right) = \frac{2}{3} \cdot \left( \frac{8}{9} \right)^h \cdot \frac{\Gamma \left( \frac{1}{3} \right) \cdot \Gamma \left( h + \frac{3}{2} \right)}{\sqrt{\pi} \cdot \Gamma \left( h + \frac{4}{3} \right)} $$ If we take $h=1/2$ we get an HGM defined over $\mathbb Q$, we get the gamma vector $[-4,1,1,2]$ and corresponding Weierstrass model $$ y^2+xy=x^3+\frac t{64}x.

September 5, 2024

MFO

I have been going to Oberwolfach for a number of years now, twice for two weeks at a time in the Research in Pairs scheme. What an incredible place, a real gem. MFO

July 19, 2024

HGM

This project got started at a fabulous workshop held in Benasque, Spain (in the mountains, a few hours drive from Barcelona) in 2009. I wrote this post in my blog after returning from another workshop there on p-adic analysis in 2007. In my own notes I added: The place is truly amazing. Mountains falling over you as you walk down the street. The plan of the 2009 workshop was to explore ways to compute exotic L-functions, not arising from elliptic curves or usual modular forms.

July 19, 2024

Hamilton and the Discovery of Quaternions

Towards the end of my Ph.D. years at The Ohio State University together with a bunch of graduate students we thought of publishing our own magazine. We struggled to come up with a good name for it so after a lot of back and forth we settled on “The Empy Set”. The OSU graduate advisor found this too nihilistic and wanted us to pick a different name. We nevertheless wrote what inevitably ended up being the only issue of “The Empty Set” and circulated it among the department.

July 25, 2007

Benasque

While I wait for my brain strings to settle after a long summer shuttling about I make a quick core dump before it’s all gone. I spent two weeks in Benasque, Spain for the workshop P-adic analysis, Periods and Physics . We ended up being a very small group of participants, unfortunate in a way but the result was a very charming workshop. And what a place! Benasque is in a valley in the Spanish Pyrenees not far from the border with France, a small mountain town where you walk everywhere, surrounded by amazing mountains full of unbelievably beautiful hikes.

April 1, 2002

K-theory and L-functions

These are notes (taken by M. Lalín) for a course on $K$-theory and $L$-functions I gave at Harvard University in the Spring of 2002. Notes for the first few lectures were taken by S. Valverde. The main goal of the course was to explain how to frame conceptually the numerical examples of D. Boyd relating the Mahler measure of certain two variable polynomials $A(x,y)$ and $L(E,2)$ for $E$ the elliptic curve defined by the zeros of $A$.

June 7, 1999

La Función Zeta de Hasse-Weil (in Spanish)

These are notes for a course in Spanish on the Hasse-Weil zeta function given en Mérida, Venezuela in 1999.

June 7, 1989

Bernoulli Polynomials

These are notes on approaching Bernoulli polynomials as eigenvalues of certain operators acting on polynomials. It is my re-interpretation of work of E. H. Lehmer. (A new approach to Bernoulli polynomials, Amer. Math. Monthly 95(1988), 905-911.) I wrote this note at the end of my Ph.D. studies around 1989.