Arithmetic of L-Functions by Cristian Popescu, Karl Rubin, Alice Silverberg

Number Theory

By Cristian Popescu, Karl Rubin, Alice Silverberg

The final topic of the 2009 IAS/PCMI Graduate summer time college was once connections among designated values of $L$-functions and mathematics, specially the Birch and Swinnerton-Dyer Conjecture and Stark's Conjecture. those conjectures are brought and mentioned extensive, and development remodeled the final 30 years is defined. This quantity includes the written types of the graduate classes introduced on the summer time university. it might be an appropriate textual content for complicated graduate subject matters classes at the Birch and Swinnerton-Dyer Conjecture and/or Stark's Conjecture. The publication also will function a reference quantity for specialists within the box. Titles during this sequence are co-published with the Institute for complicated Study/Park urban arithmetic Institute. participants of the Mathematical organization of the US (MAA) and the nationwide Council of academics of arithmetic (NCTM) obtain a 20% from record cost.

Show description

Read Online or Download Arithmetic of L-Functions PDF

Best number theory books

Multiplicative Number Theory I. Classical Theory

A textual content in response to classes taught effectively over a long time at Michigan, Imperial university and Pennsylvania kingdom.

Mathematical Problems in Elasticity

This quantity beneficial properties the result of the authors' investigations at the improvement and alertness of numerical-analytic tools for usual nonlinear boundary price difficulties (BVPs). The tools into account provide a chance to resolve the 2 very important difficulties of the BVP idea, particularly, to set up lifestyles theorems and to construct approximation recommendations

Iwasawa Theory Elliptic Curves with Complex Multiplication: P-Adic L Functions

Within the final fifteen years the Iwasawa concept has been utilized with striking luck to elliptic curves with complicated multiplication. a transparent but common exposition of this concept is gifted during this book.

Following a bankruptcy on formal teams and native devices, the p-adic L capabilities of Manin-Vishik and Katz are built and studied. within the 3rd bankruptcy their relation to type box idea is mentioned, and the purposes to the conjecture of Birch and Swinnerton-Dyer are handled in bankruptcy four. complete proofs of 2 theorems of Coates-Wiles and of Greenberg also are offered during this bankruptcy which could, additionally, be used as an creation to the more moderen paintings of Rubin.

The e-book is essentially self-contained and assumes familiarity in simple terms with primary fabric from algebraic quantity thought and the speculation of elliptic curves. a few effects are new and others are offered with new proofs.

Extra resources for Arithmetic of L-Functions

Example text

These sub-branches reflect, either t he study of t he properties of t he integers from different points of view, or techniques used to solve t he problems in number theory. For example, probabilistic number t heory makes extensive use of probabilistic methods, whilst analytic number theory employs deep results in mathematical analysis in solving number-theoretic problems. Note that arithmetic algebraic geometry is a brand new subject of modern number theory, which is the study of arit hmetic properties of elliptic (cubic) curves.

Ln aq = (2 5695 ... ) ln aq ln2 ln2 a= { ~ where · q if q = 3 (mod 4) ifq:=1 (mod4). 2. Let qn be the nth prime such that Mqn is a Mersenne prime. 9. Numbers of the form Fn = 22 n + 1, whether prime or composite, are called Fer·mat numbers. A Fermat number is called a prime Fermat number if it is prime. A Fermat number is called a composite Fermat number if it is composite. 58) In other words, Fn+L - 2 is divisible by all lower Fermat numbers: Fn-k I (Fn+i - 2), 1 < k < n. 2 Theory of Divisibility 37 but Euler in 1732 found that the fifth Fermat number is not a prime, since F 5 = 225 + 1 is the product of two primes 641 and 6700417.

2 Theory of Divisibility of the qj (j = 1, 2, · · · , s ). Next, there is no loss of generality in presuming that p 1 < q1 , and we define the positive integer N as N = (q1 - pl)q2q3 · · · qs = P1 (p2p3 · · · Pr- q2q3 · · · qs)· < < n, so N is uniquely factorable into primes. However, and q1 is prime. Hence one of the above expressions for N contains p 1 and the other does not. This contradiction proves the result: ' there cannot be any exceptions to the theorem. D Certainly 1 P1 f (q1 - N P1), since p 1 < q1 Note that if n is prime, then the product is, of course, n itself.

Download PDF sample

Rated 4.71 of 5 – based on 40 votes