## Friday, 7 March 2008

### Want a group named after you?

People have stars named after them, craters on the moon, even comets...but how about having a symmetrical object in hyperspace named after you.

Part of Finding Moonshine narrates my discovery of some new symmetrical objects that have interesting connections with objects in number theory called elliptic curves. During my presentations there is the chance to win one of these groups and have the group named after you. I have created infinitely many of these groups so they won't run out! (I did think of selling them for a dollar a group, like the guy who sold a million pixels for a dollar a pixel and became a millionaire. With infinitely many groups that could make me very rich.)

Here are the groups that have been named so far. Each group specifies what values [C[1], C[2], C[3], C[4]] should take in the presentation of the groups pictured at the top of this post.

The Vandewalle Group. Won in Belgium on 14 December 2007. Set [C[1], C[2], C[3], C[4]]=[1,1,0,-2] Corresponds to an elliptic curve of conductor 102.

The Mitrokhin Group. Won at Imperial College on 19 February 2008. Set [C[1], C[2], C[3], C[4]]=[1,0,0,-8] Corresponds to an elliptic curve of conductor 114.

The Lane Group. Won at the Royal Society on 21 February 2008. Set [C[1], C[2], C[3], C[4]]=[0,1,0,-20] Corresponds to an elliptic curve of conductor 120.

The Sutton Group. Won at the Bath Literary Festival on 23 February 2008. Set [C[1], C[2], C[3], C[4]]=[1,0,1,2] Corresponds to an elliptic curve of conductor 122.

The Chiodo Group. Won in Cambridge on 26 February 2008. Set [C[1], C[2], C[3], C[4]]=[0,-1,1,1] Corresponds to an elliptic curve of conductor 131.

The Course Group. Won at Wanstead Library in the east end of London on 28 February 2008. Set [C[1], C[2], C[3], C[4]]=[0,1,0,3] Corresponds to an elliptic curve of conductor 132.

The Arnott Group. Won at Brighton Science Festival on 2 March 2008. [C[1], C[2], C[3], C[4]]=[0,1,0,-4] Corresponds to an elliptic curve of conductor 136.

The Jones and Rippington Group. Won at Didcot Girl's School as part of Oxford University's Science Week Roadshow on 12 March 2008. The first group to be named after two winners. But after all, many of the sporadic groups are also have two names associated with them, e.g. The Harada-Norton Group. [C[1], C[2], C[3], C[4]]=[0,-1,1,-1] Corresponds to an elliptic curve of conductor 141.

The Cork Group. Won at the Wadham College Mathematics Reunion on 15 March 2008. [C[1], C[2], C[3], C[4]]=[0,1,0,-12] Corresponds to an elliptic curve of conductor 168.

The Vivi Group Won at the Oxford Literary Festival on 3 April 2008 by a young man who very romantically dedicated the group to his girlfriend sitting next to him. [C[1], C[2], C[3], C[4]]=[0,0,1,6] Corresponds to an elliptic curve of conductor 171.

The Stevens Group Won at the Oxford-North American Reunion in New York on the 5 April 2008. [C[1], C[2], C[3], C[4]]=[0,0,1,-3] Corresponds to an elliptic curve of conductor 189.

The McBride Group Won at the Swindon Literary Festival on 15 May 2008. [C[1], C[2], C[3], C[4]]=[0,-1,1,2] Corresponds to an elliptic curve of conductor 201.

The Harris Group Won at the Hay Literary Festival on 31 May 2008. [C[1], C[2], C[3], C[4]]=[1,1,0,2] Corresponds to an elliptic curve of conductor 206.

The Cousins Group Won at the Oxford Science Centre on 18 June 2008. [C[1], C[2], C[3], C[4]]=[1,0,1,1] Corresponds to an elliptic curve of conductor 214.

The Falk Group Won at ESOF 2008 Barcelona on 20 July 2008. [C[1], C[2], C[3], C[4]]=[1,1,1,-5] Corresponds to an elliptic curve of conductor 235.

The Wallis Group Won at The Oxford UK Summer School in Theoretical Chemistry, 11 September 2008. [C[1], C[2], C[3], C[4]]=[1, 1, 0, 16] Corresponds to an elliptic curve of conductor 222.

The Morris Group Won at M500 The Open University's 34th Revision Weekend at Aston University in Birmingham, 13 September 2008. [C[1], C[2], C[3], C[4]]=[0 1, 0, 2] Corresponds to an elliptic curve of conductor 224.

The Keller Group Won at Manchester Grammar School, 16 October 2008. [C[1], C[2], C[3], C[4]]=[0 -1, 1, 1] Corresponds to an elliptic curve of conductor 131.

The Barge and Wright Group Won at a talk given to the Liverpool Mathematics Society, 16 October 2008. [C[1], C[2], C[3], C[4]]=[0, 1, 0, 3] Corresponds to an elliptic curve of conductor 132.

The Stewart-Price-Hawkins Group Won at a talk given to the Kellogg College Gaudy, 24 January 2009. [C[1], C[2], C[3], C[4]]=[0, -1, 1, -1] Corresponds to an elliptic curve of conductor 141.

The Ozaki Group Won at a talk given at the Royal Academy of Arts, 30 January 2009. [C[1], C[2], C[3], C[4]]=[0, 1, 0, -12] Corresponds to an elliptic curve of conductor 168.

The Banga Group Won at the Geary Lecture given at City University, London, 20 March 2009. [C[1], C[2], C[3], C[4]]=[0, 0, 1, 6] Corresponds to an elliptic curve of conductor 171.

If you would like to explore a little bit more of the mathematical significance of these groups then these two papers are where the first groups I constructed are explained:

A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups, Israel J. of Math 126 (2001), 269-288.

Counting subgroups in nilpotent groups and points on elliptic curves, J. Reine Angew. Math. 549 (2002) 1-21.

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