P-adic Numbers, p-adic Analysis, and Zeta-Functions (Graduate Texts in Mathematics, Volume 58)

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Artin, H.

Tohoku Mathematical Journal

Kraft, and R. Remmert, eds. Menezes and S. A survey of number theory and cryptography, in R.

Neal Koblitz

Bambah, V. Dumir, and R. Cryptography, in B. Engquist and W. Schmid, eds. Good and bad uses of elliptic curves in cryptography, Moscow Math. Journal, Vol. The p-adic approach to solutions of equations over finite fields, The American Mathematical Monthly, Vol. Mathematics as propaganda, in L.

Neal Koblitz - Wikipedia

Steen, ed. Why study equations over finite fields?

Mathematics and the external world, an interview with A. Fomenko, The Mathematical Intelligencer, Vol.

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\(p\)-adic modular forms

These lecture notes are intended as an introduction to p-adic analysis on the elementary level. For this reason they presuppose as little background as possi ble. Besides about three semesters of calculus, I presume some slight exposure to more abstract mathematics, to the extent that the student won't have an adverse reaction to matrices with entries in a field other than the real numbers, field extensions of the rational numbers, or the notion of a continuous map of topolog ical spaces.

The purpose of this book is twofold: to develop some basic ideas of p-adic analysis, and to present two striking applications which, it is hoped, can be as effective pedagogically as they were historically in stimulating interest in the field. The first of these applications is presented in Chapter II, since it only requires the most elementary properties of Q ; this is Mazur's construction by p means of p-adic integration of the Kubota-Leopoldtp-adic zeta-function, which "p-adically interpolates" the values of the Riemann zeta-function at the negative odd integers.

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