A Dirichlet series expansion for the p-adic zeta-function

Abstract

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion

Frobenius Map for Quintic Threefolds

“…Our calculations are very explicit and do not rely on any deep machinery from the theory of motives. It follows from our computations that the natural conjecture that all the off-diagonal entries of the Frobenius matrix at the boundary point are zero is false; in fact we obtain an explicit expression for the remaining undetermined entry and its nonvanishing can be verified by means of a computer calculation 3 . More precisely, the remaining Frobenius matrix entry can be expressed as a p-adic series in terms of the coefficients B n of the Dwork exponential B n z n := exp(z p /p + z).…”

“…More precisely, the remaining Frobenius matrix entry can be expressed as a p-adic series in terms of the coefficients B n of the Dwork exponential B n z n := exp(z p /p + z). In fact it was pointed out to us by V. Vologodsky that from certain conjectures of the theory of motives one can derive that the above entry is a rational multiple of ζ p (3), where the latter is a p-adic Riemann zeta value (see [4] for example). We carried out (by computer) some calculations 4 that strongly point to the truth of the above.…”

On -invariants attached to cyclic cubic number fields

“…The key ingredient is to utilise the p-adic approximations developed by the first author in [2,3], which seem well suited to resolving problems of type (I) and (II) (see previous page), relatively quickly. In total, the computations in this paper took approximately five months to run on PARI/GP.…”

“…Firstly, suppose that we fix the character χ and allow p to vary over the set of prime numbers. Assuming that the λ-invariant is less than or equal to p, by the previous result, one computes c (2) j (F χ,β ) until one hits a value of j (namely, j = λ p (χω 1+β )) for which c…”

“…Cassou-Noguès. It would be a worthwhile project to combine Roblot's methods with the Dirichlet series expansions (only found over Q thus far) from [2,3], and, consequently, extend the algorithm presented here to the totally real case.…”