Emmy Noether Lectures

2013 Lecturer: Raman Parimala

A Hasse principle for quadratic forms over function fields

The term `Hasse principle’ stands for the idea of finding integer solutions to equations by piecing together solutions modulo different prime powers. Solutions modulo prime powers lead to solutions in the completions of the field of rational numbers at p-adic valuations. A classical theorem of Hasse-Minkowski states that a quadratic form over rational numbers admits a nontrivial solution if it does over all p-adic completions and is indefinite over real numbers. There are more general theorems on the obstructions to the Hasse principle for the existence of rational points on homogeneous spaces under linear algebraic groups defined over number fields. We shall discuss formulations of the Hasse principle over function fields and some consequences for quadratic forms.
Brief Biography
Raman Parimala is the Arts and Sciences Distinguished Professor of Mathematics at Emory University and has been selected as the 2013 Noether Lecturer for her fundamental work in algebra and algebraic geometry with significant contributions to the study of quadratic forms, hermitian forms, linear algebraic groups and Galois cohomology.
Parimala received her Ph.D. from the University of Mumbai (1976). She was a professor at the Tata Institute of Fundamental Research in Mumbai for many years before moving in 2005 to Emory University in Atlanta, Georgia. She has also held visiting positions at the Swiss Federal Institute of Technology (ETH) in Zurich, the University of Lausanne, University of California-Berkeley, University of Chicago, Ohio State, and the University of Paris at Orsay.
Parimala has won many awards in recognition of her accomplishments. She gave a plenary address at the 2010 International Congress of Mathematicians (ICM) in Hyderabad and a sectional address at the 1994 ICM in Zurich. By 1992 she was a Fellow of the Indian Academy of Sciences, the Indian National Science Academy and the National Academy of Sciences India. In 2005, she was awarded the prize in mathematics by the Academy of Sciences for the Developing World, making her the first woman to receive that honor. Parimala has also received the Srinavasa Ramanujan Medal of the Indian National Science Academy in 2006, an honorary doctorate from the University of Lausanne in 1999, and the Bhatnagar prize in 1987.
In the seventies, Parimala’s examples of nontrivial quadratic spaces over an affine plane came as a surprise to experts in contrast to the affirmative solution of Serre’s question on triviality of algebraic vector bundles over an affine space by Quillen and Suslin. Parimala is perhaps best known for proving Serre’s Conjecture II for classical groups (Bayer-Fluckiger & Parimala, 1995). This well-known conjecture on the Galois cohomology of linear algebraic groups was formulated in the early 1960s. The problem is of continued interest and has yet to be solved for many exceptional groups. Another of her significant contributions to the theory of quadratic forms concerns the u-invariant of a field‚ a measure of the maximum dimension of an anisotropic quadratic form over that field. In a 2010 paper, Parimala proved that the u-invariant of a function field of a nondyadic p-adic curve is exactly 8, settling a conjecture made nearly 30 years earlier (Parimala & Suresh, 2010).