Number Theory

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

Posted On March 23, 2017 at 10:17 am by / Comments Off on 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.

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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;.

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