Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used.

There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way.
 1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems.
 1 As a direct source of new theorems, when a sequence arises in two different contexts.
 1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed.
 

The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet.

The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned.

Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences".

Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

The On-Line Encyclopedia of Integer Sequences.

We discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

/pdf/use-of-elliptic-curves-in-cryptography-credwc2ehh.pdf

Use of Elliptic Curves in Cryptography

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

/pdf/the-on-line-encyclopedia-of-integer-sequences-okduuz5t6g.pdf

The On-Line Encyclopedia of Integer Sequences

The Theory of Jacobi Forms

Heegner points and derivatives of L-series.

Let Fg 1, g> 1, be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface F of genus g. Let )~(~1) be its Euler characteristic in the sense of Wall, that is Z(F~I)= [Fgl: F] l z(E/F), where F is any torsion free subgroup of finite index in F~ 1 and E is a contractible space on which F acts freely and properly discontinuously. An example of such a space is the Teichmiiller space ~-~1, and g(F~ ~) can be interpreted as the orbifold Euler characteristic of ~-~'/F~I =JOin, the moduli space of curves of genus g with base point. The purpose of this paper is to prove the following formula for )~(F~I):

http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01390325/fulltext.pdf

The Euler characteristic of the moduli space of curves

Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem that ζ(k) = rational number × π k (k > 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction of p-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)

Values of Zeta Functions and Their Applications

https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1016/0040-9383(85)90004-7/fulltext.pdf

Don Zagier

Papers

The Theory of Jacobi Forms

Heegner points and derivatives of L-series.

The Euler characteristic of the moduli space of curves

Values of Zeta Functions and Their Applications

Volumes of hyperbolic three-manifolds