Spectral theory and analytic geometry over non-Archimedean by Vladimir G. Berkovich

By Vladimir G. Berkovich

The aim of this booklet is to introduce a brand new thought of analytic house over a non-Archimedean box. regardless of the entire disconnectedness of the floor box, those analytic areas have the standard topological homes of a posh analytic area, corresponding to neighborhood compactness and native arcwise connectedness. This makes it attainable to use the standard notions of homotopy and singular homology. The booklet incorporates a homotopic characterization of the analytic areas linked to sure periods of algebraic types and an interpretation of Bruhat-Tits constructions by way of those analytic areas. the writer additionally reports the relationship with the sooner concept of a inflexible analytic area. Geometrical issues are used to acquire a few purposes, and the analytic areas are used to build the principles of a non-Archimedean spectral conception of bounded linear operators. This ebook calls for a history on the point of uncomplicated graduate classes in algebra and topology, in addition to a few familiarity with algebraic geometry. it'd be of curiosity to investigate mathematicians and graduate scholars operating in algebraic geometry, quantity concept, and -adic research

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It is perhaps worth mentioning that, as Hecke also proved, the converse of Proposition 13 holds as well: if an (n ≥ 1) are complex numbers of polynomial growth and the function L∗ (f, s) defined by (45) continues analytically to the whole complex plane and satisfies the functional equation (46), then f (z) = ∞ 2πinz is a cusp form of weight k on Γ1 . 3 Modular Forms and Algebraic Number Theory In §3, we used the theta series θ(z)2 to determine the number of representations of any integer n as a sum of two squares.

No non-modular proof of the asymptotic formula (38) is known. Before continuing with the theory, we look at some examples, starting in rank 8. Define the lattice Λ8 ⊂ R8 to be the set of vectors belonging to either Z8 or (Z+ 12 )8 for which the sum of the coordinates is even. This is unimodular because the lattice Z8 ∪ (Z + 12 )8 contains both it and Z8 with the same index 2, and is even because x2i ≡ xi (mod 2) for xi ∈ Z and x2i ≡ 14 (mod 2) for xi ∈ Z + 12 . The lattice Λ8 is sometimes denoted E8 because, if we choose the Z-basis ui = ei − ei+1 (1 ≤ i ≤ 6), u7 = e6 + e7 , u8 = − 21 (e1 + · · · + e8 ) of Λ8 , then every ui has length 2 and (ui , uj ) for i = j equals −1 or 0 according whether the ith and jth vertices (in a standard numbering) of the “E8 ” Dynkin diagram in the theory of Lie algebras are adjacent or not.

Corollary (Theorem of Lagrange). Every positive integer is a sum of four squares. ♥ For another simple application of the q-expansion principle, we introduce two variants θM (z) and θF (z) (“M” and “F” for “male” and “female” or “minus sign” and “fermionic”) of the function θ(z) by inserting signs or by shifting the indices by 1/2 in its definition: θM (z) = 2 (−1)n q n = 1 − 2q + 2q 4 − 2q 9 + · · · , n∈Z 2 qn θF (z) = n∈Z+1/2 = 2q 1/4 + 2q 9/4 + 2q 25/4 + · · · . 28 D. Zagier These are again modular forms of weight 1/2 on Γ0 (4).

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