Iwasawa Theory and Motivic L-functions

Abstract

Abstract:We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.

Values of Zeta Functions of Arithmetic Surfaces At

“…One obtains a prediction for the special value of at by combining for the motives for all j . For more details, see [2, 16, 23, 24, 15, 10, 11, 12].…”

“…To predict L * (M,0) ∈ R × , they introduced similar exact sequences for each prime p; this is the conjecture C BK (M ) [15] for p. One obtains a prediction for the special value of ζ HW (V 0 ,s) at s = r by combining C BK (M ) for the motives h j (V 0 )(r) for all j. For more details, see [2,16,23,24,15,10,11,12].…”

“…The torsion in C(1) of(22) satisfiesχ(C(1) tor ) = w • [Br(X)] [Pic(X) tor ] • [Pic(X) tor ] . (50)We can rewrite C(1) C as0 → O × F ⊗ C degree 0 Coker(γ M0 ) degree Hom(Pic(X),C) degree Pic(X) ⊗ C degree Ker(γ M2 )degree Hom(O × F ,C) the sequence with determinant δ 0 → Hom(Z,C) → Hom(Pic(X),C) → Hom(Pic 0 (X),C) → 0;• and the sequence with determinant det(ψ) h : Hom(Pic 0 (X),C)degree Pic 0 (X) ⊗ C degree 3where h is the Arakelov intersection pairing[27, Conjecture 2.3.7] as in(11);• the sequence with determinant δ 0 → Pic 0 (X) ⊗ C → Pic(X) ⊗ C d − → C → 0;• the sequence(12) with determinant R 0 → C → Ker(γ M2 ) degree Hom…”

Values of zeta functions of arithmetic surfaces at $s=1$

“…To predict L * (M, 0) ∈ R × , they introduce similar exact sequences for each prime p; this is the conjecture C BK (M ) [15] for p. One obtains a prediction for the special value 2 ζ * (V, r) by combining C BK (M ) for the motives h j (V 0 )(r) for all j. For more details, see [2,16,23,24,15,11,12,13].…”

On the Weil-étale topos of regular arithmetic schemes

“…If X → Spec(Z) is a (proper, flat, regular) arithmetic scheme with a section then RΓ(Spec(Z) W , Z) is a direct summand of RΓ(X W , Z). Hence by [13] H 4 (X W , Z) will not be a finitely generated abelian group and d) does not hold. Even if one could find a definition of Spec(Z) W with the expected Z-cohomology the definition of X W as a fibre product (Definition 9) will not be the right one.…”