Dr Jayanta Manoharmayum
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Teaching:
Research:
Past grants, as Principal Investigator |
Modularity and Galois Respresentation of Totally Real Fields |
Nuffield |
Research interests:
The absolute Galois group of the rationals is my primary
interest. It contains almost all arithmetic information: eg,
solutions to explicit diophantine equations (as in Fermat's Last
Theorem). The whole group in general is rather too large an
object to study; a better way of understanding the Galois group
is through its representations, and this brings out deep
connections with other mathematical objects (such as modular
forms). For example, given a two dimensional representation of the
Galois group satisfying `usual conditions', there should be a
modular form whose Fourier coefficients are related to the traces
of the representation. The precise correspondences are
conjecturally given by the conjectures of Artin (complex
representations), Fontaine and Mazur (p-adic representations), and
Serre (finite characteristic). It is aspects of these conjectures,
both over the rationals and in the setting of totally real number
fields, that I am most interested in.