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Irregular Primes and Dedekind Zeta Functions
Posted:
Dec 19, 2008 3:00 PM
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In 1981 D. Weisser proved that a prime not congruent to 1 mod 8 and >=
7 is irregular if and only if the rational number Zeta_K(-1) is p-
adically integral, that is has a denominator not divisible by p, where
K is the maximal real subfield of the cyclotomic field of p'th roots
of unity. His proof was very indirect, depending upon a formula for
the arithmetic genus of the Hilbert Modular Variety of this field.
A number of questions come to mind. Is the restriction on the
congruence mod 8 really necessary? Is there a more direct approach to
the same result? How does this result connect with other results
about regular and irregular primes? Has it been followed up - I
haven't seen any references?
I would greatly appreciate any answers to the above questions or any
other information about it.
Thanks,
Achava
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