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Dr Dan Fretwell

Number Theory

I am mainly interested in modular forms, in particular congruences between Hecke eigenvalues of particular types of modular form. Perhaps the most famous of these is Ramanujan's 691 congruence (linking the Ramanujan tau function with the 11th power divisor sum modulo 691).

One such congruence that I am interested in is between the Hecke eigenvalues of genus 2 and genus 1 Siegel modular forms. Such a congruence was predicted by Harder in 2002 and remains unproven. Even finding evidence of such congruences is tough and is the scope of my current research.

In order to find evidence for these congruences I have been using algebraic modular forms. These are a special type of modular form that is algebraically defined (there is no need for analytical axioms due to the locally symmetric space being a finite set of points).

I also like to study the representation theoretic phenomenon attached to such congruences...both in Galois representation and Automorphic representation guises. 

Research keywords

  • Modular Forms
  • Galois Representations
  • Automorphic Forms/Representations
  • Elliptic Curves.