## Number Theory 3: Iwasawa Theory and Modular Forms by Nobushige Kurokawa, Masato Kurihara, Takeshi Saito

By Nobushige Kurokawa, Masato Kurihara, Takeshi Saito

This is often the 3rd of 3 similar volumes on quantity idea. (The first volumes have been additionally released within the Iwanami sequence in glossy arithmetic, as volumes 186 and 240.) the 2 major issues of this booklet are Iwasawa concept and modular kinds. The presentation of the speculation of modular kinds begins with numerous attractive family members stumbled on through Ramanujan and ends up in a dialogue of numerous very important constituents, together with the zeta-regularized items, Kronecker's restrict formulation, and the Selberg hint formulation. The presentation of Iwasawa idea makes a speciality of the Iwasawa major conjecture, which establishes far-reaching family members among a p-adic analytic zeta functionality and a determinant outlined from a Galois motion on a few excellent type teams. This publication additionally includes a brief exposition at the mathematics of elliptic curves and the evidence of Fermat's final theorem through Wiles. including the 1st volumes, this booklet is an effective source for a person studying or instructing glossy algebraic quantity concept.

**Read or Download Number Theory 3: Iwasawa Theory and Modular Forms PDF**

**Similar number theory books**

In the event you significant in mathematical economics, you come back throughout this booklet many times. This booklet contains topological vector areas and in the neighborhood convex areas. Mathematical economists need to grasp those subject matters. This publication will be an outstanding support for not just mathematicians yet economists. Proofs aren't challenging to persist with

**Game, Set, and Math: Enigmas and Conundrums**

A suite of Ian Stewart's leisure columns from Pour los angeles technological know-how, which display his skill to carry smooth maths to existence.

From July 25-August 6, 1966 a summer season university on neighborhood Fields used to be held in Driebergen (the Netherlands), equipped via the Netherlands Universities beginning for foreign Cooperation (NUFFIC) with monetary help from NATO. The medical organizing Committl! e consisted ofF. VANDER BLIJ, A. H. M.

The hot variation of this thorough exam of the distribution of leading numbers in mathematics progressions deals many revisions and corrections in addition to a brand new part recounting fresh works within the box. The ebook covers many classical effects, together with the Dirichlet theorem at the life of top numbers in arithmetical progressions and the concept of Siegel.

- Number Theory Related to Fermat's Last Theorem
- Arithmetic of Algebraic Curves (Monographs in Contemporary Mathematics)
- Area, lattice points, and exponential sums
- Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society
- Defects of Properties in Mathematics: Quantitative Characterizations

**Additional resources for Number Theory 3: Iwasawa Theory and Modular Forms**

**Example text**

We start from the 22 9. MODULAR FORMS formula oo 2 S in (7 T z )= 7 r Z n (l-y . n=l Taking the logarithmic derivative of both sides, we have , 1 ^ 7rCOt(7r2) = - + z ^ n=l Letting q = 2z 1 —T? z 1 — -------- = - + \ 1 \ I ----------- 1------ 1---- ) . ^n=l \ z —n z + n/ for Im( 2:) > 0, we have gZTTZ . cosi'Kz) 7rCOt(7TZ) = 'K—;—,--- r = '7T sin(7T2;) / + e 2z . 14) f ; (. - n )-‘ = f ; (This is called the Lipschitz formula, and it can also be obtained by the Poisson summation formula. 2. -in CW ¿ 1 (fc - 1)!

9. MODULAR FORMS 38 A lternative proof of C{s) ^ 0 on Re(s) = 1. Suppose C (l+ • Then, in the formula ito) = 0 {to ^ 0). Put So = E{so, z) = C(2so)y*« + C(2so oo m^°~^ai-2so{‘m)y/yK^^_i{2nmy) cos{2Trmx), + 4^ m =l we have C ( 25 o ) = C (2 s o - C ( l + i io ) = 1) = C (iio ) = 0, C (1 - ito) = C ( 1 + ito) = 0 . Here, we used the functional equation of C(s) and the reflection prin ciple. Thus, we have CX) m^'>~i

Note that the number 12 is significant. 6(b)). 4. R e a l a n a lytic E isen stein series Here, we introduce the real analytic Eisenstein series E{s, z) and study their Fourier expansion. The results can be interpreted in re lation to the theory of quadratic forms and that of Laplace operators (in the case of 2-dimensional tori), and more. 3(a)). As another application we describe a method of analytic continuation of C function called the Rankin-Selberg method. 5. (a) Fundam ental p ro p e rtie s o f E{s,z), For z in the upper half plane and a complex number s satisfying Re(s) > 1, consider the series ImjzY E{s,z) = l = 2" E \CZ■■d\- 2 s (c,d) = l This series converges absolutely and determines a real analytic func tion of 2;.