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
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