x FOREWOR D

Schottky uniformizatio n provide s a visualizatio n o f Arakelov' s ge -

ometry a t arithmeti c infinity , whic h serve s a s th e mai n motivatio n o f

Chapter 4 .

Among th e mos t tantalizin g development s i s th e recurren t emer -

gence of patches of common ground fo r numbe r theor y an d theoretica l

physics.

In fact , on e can presen t i n this ligh t th e famou s theore m o f youn g

Gauss characterisin g regula r polygon s tha t ca n b e constructe d usin g

only rule r an d compass . I n hi s Tagebuch entry o f Marc h 3 0 h e an -

nounced tha t a regular 17-gon has this property .

Somewhat modernizin g hi s discovery, on e can presen t i t i n the fol -

lowing way.

In th e comple x plane , root s o f unit y o f degre e n for m vertice s o f

a regula r n-gone . Henc e it make s sens e t o imagin e tha t w e stud y th e

ruler and compass constructions as well not in the Euclidean, but i n the

complex plane. Thi s has an unexpected consequence: w e can character-

ize the set of all points constructible i n this way as the maximal Galoi s

2-extension of Q. I t remains to calculate the Galois group of Q(e 2m^17):

since i t i s cyclic of order 16, this roo t o f unity i s constructible. More -

over, th e sam e i s tru e fo r al l p-gon s wher e p i s a prim e o f th e for m

2n + 1 but no t fo r othe r primes .

A remarkabl e featur e o f thi s resul t i s th e appearanc e o f a hidde n

symmetry group , I n fact , th e definition s o f a regula r n-go n an d rule r

and compas s construction s ar e initiall y formulate d i n term s o f Eu -

clidean plan e geometry an d sugges t tha t th e relevan t symmetr y grou p

must b e tha t o f rigid rotation s SO (2) , eventuall y extende d b y reflec -

tions and shifts. Thi s conclusion turns out to be totally misleading: in -

stead, on e should rely upon Ga l (Q/Q). Th e actio n of the latter grou p

upon root s o f unit y o f degre e n factor s throug h th e maxima l abelia n

quotient an d i s give n b y £ H- ( k

}

wit h k runnin g ove r al l k mo d n

with (kjU) = 1, whereas th e actio n o f th e rotatio n grou p i s give n b y

C ^ Co C withC o running ove r al l n-t h roots . Thus , th e Gal(Q/Q) -

symmetry doe s not conserv e angles between vertice s which seem to b e

basic fo r th e initia l problem . Instead , i t i s compatibl e wit h additio n

and multiplication o f complex numbers, and this property prove s to b e

crucial.

With som e stretc h o f imagination , on e ca n recogniz e i n th e Eu -

clidean avata r o f thi s pictur e a physic s flavor (puttin g i t somewha t

pompously, i t appeal s t o th e kinematic s o f 2-dimensiona l rigi d bodie s