By Shai M. J. Haran

In this quantity the writer extra develops his philosophy of quantum interpolation among the genuine numbers and the p-adic numbers. The *p*-adic numbers include the *p*-adic integers *Z _{p}*which are the inverse restrict of the finite jewelry

*Z/p*. this offers upward push to a tree, and chance measures w on

^{n}*Z*correspond to Markov chains in this tree. From the tree constitution one obtains specified foundation for the Hilbert area

_{p}*L*(

_{2}*Z*). the true analogue of the

_{p},w*p*-adic integers is the period [-1,1], and a chance degree w on it offers upward thrust to a different foundation for

*L*([-1,1],

_{2}*w*) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For particular (gamma and beta) measures there's a "quantum" or "

*q*-analogue" Markov chain, and a unique foundation, that inside of yes limits yield the true and the p-adic theories. this concept might be generalized variously. In illustration thought, it's the quantum basic linear team

*GL*(

_{n}*q*)that interpolates among the p-adic staff

*GL*(

_{n}*Z*), and among its genuine (and complicated) analogue -the orthogonal

_{p}*O*(and unitary

_{n}*U*)groups. there's a comparable quantum interpolation among the genuine and p-adic Fourier remodel and among the true and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.

_{n}

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**Sample text**

6) It is easy to see that the p-adic substitution p and real substitution η yield the (non-symmetric) p-adic and real β-chains, respectively; p N q-β-chain −→ the p-adic β-chain (N → ∞), 56 3 Real Beta Chain and q-Interpolation η N q-β-chain −→ the real β-chain (N → ∞). In this sense, the q-β-chain interpolates between the p-adic and the real βchain. Notice that if we take the limit N → ∞ in the p-adic substitution p N , the probability on the walk from (i, j) to (i + 1, j) vanishes unless j = 0.

Measure on Zp /Z∗p and an absolute value on V (Qp ) by ⎧ ⎨max{|x|p , |y|p } |x, y|p = |(x, y)|p := ⎩ |x|2 + |y|2 η η if p = η, if p = η (x, y) ∈ V (Qp ). For p = η, put V (Zp ) : = Zp × Zp , V ∗ (Zp ) : = (x, y) ∈ Zp |x, y|p = 1 . Then the projective line P1 (Qp ) is expressed as P1 (Qp ) = V ∗ (Qp )/Q∗p = V ∗ (Zp )/Z∗p . For all p ≥ η, there is a canonical distance function ρp : P1 (Qp ) × P1 (Qp ) → [0, 1] deﬁned by ρp (x1 : y1 ), (x2 : y2 ) := |x1 y2 − y1 x2 |p |x1 , y1 |p · |x2 , y2 |p The function ρp is well-deﬁned.

Cd ((x : y), Then it is easy to see that the stabilizer group Stab(0) of 0 = (1 : 0) is given by Stab(0) = 10 cd ∈ P GL(Zp ) c ∈ Zp , d ∈ Z∗p . This shows that Stab(0) is isomorphic to the semi-direct product Z∗p the isomorphic map Stab(0) 10 cd −→ (d, c) ∈ Z∗p Zp by Zp and, hence, we obtain a smaller quotient P1 (Zp )/Z∗p Zp of P1 (Zp ) than P1 (Zp )/Z∗p . From the tree of P1 (Zp )/Z∗p , we obtain the following tree corresponding to P1 (Zp )/Z∗p Zp since two elements (pn : 1) and (1 : 1) are equivalent under the action of Z∗p Zp for all n ∈ N (Figs.