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Buium A. — Arithmetic Differential Equations
Buium A. — Arithmetic Differential Equations



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Название: Arithmetic Differential Equations

Автор: Buium A.

Аннотация:

This work develops an arithmetic analogue of the theory of ordinary differential equations, in which functions are replaced by integer numbers, the derivative operator is replaced by a Fermat quotient operator, and differential equations are replaced by arithmetic differential equations. The book partly follows a series of papers written by the author; however, a substantial part of the material presented here has never been published before. For most of the book, the only prerequisites are the basic facts of algebraic geometry and number theory. Author information is not given.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 310

Добавлена в каталог: 18.11.2009

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Предметный указатель
$a(\Sigma)$, definition      4.1
$ad_r$, proposition      3.39
$Aut_{\delta}(B/A)$, definition      5.7
$A^*_{\langle s\rangle}$, equation      2.29
$A_{((p))}$, equation      2.24
$B(X_\delta)$, definition      2.14
$B(\boldsymbol{X,L})$, $BS(\boldsymbol{X,L})$, definition      2.46
$Cheb_d$, definition      1.29
$EL_r$, equation      3.36
$E_{Weier}$, equation      8.99
$f^r$, equation      6.16
$f^r_{crys}$, equation      8.114
$f^r_{jet}$, equation      8.105
$f^r_{jet}(E,\omega)$, equation      7.13
$f^\partial$, equation      8.118
$f^{1,2}_{jet}$, equation      8.109
$f^{ab}$, equation      6.15
$f_\partial$, corollary      8.58
$f_{lift}(E,\omega)$, equation      7.18
$f_{lift}(\boldsymbol{G}_m,dx/x)$, equation      7.19
$H^0(\boldsymbol{A})$, $H^1(\boldsymbol{A})$, definition      1.18
$H^0(\boldsymbol{X, L}; B)$, equation      2.51
$H^0(\boldsymbol{X,L})$, $H^1(\boldsymbol{X,L})$, equations      1.22 2.44
$H^0(\Sigma, \Lambda)$, $H^1(\Sigma, \Lambda)$, equation      4.12
$H^0_K(\Sigma, \Lambda)$, equation      4.36
$H^1$, $H^{10}$, $H^{01}$, equation      8.15
$I^r(w)$, definitions      8.44 8.45
$k\langle u\rangle$, equation      5.18
$k\langle X, L\rangle$, equation      2.42
$k\langle x\rangle$, equation      5.15
$k\langle x\rangle_*$, $k\langle x\rangle^*$, equation      5.14
$k\langle X_\delta\rangle$, equation      2.17
$k\langle \boldsymbol{X}, \boldsymbol{L}\rangle$, equation      2.49
$K_{X/Y}$, $K_X$, equation      1.11
$Lat_d$, definition      1.29
$L^r$, equation      4.50
$L^w$, equation      2.34
$L^{[s]}$, equation      2.35
$M^r$, $M^r(w)$, definitions      8.44 8.45
$M^r_X$, $M^r_X(w)$, $\overline{M}^r_X$, $\overline{M}^r_X(w)$, equations      3.48 8.80
$M_{Weier}$, equation      8.98
$Norm_\lambda$, remark      4.30
$N^r$, equation      7.6
$ord(\overline{w})$, equation      2.9
$O^-_{\sigma_2}(\widetilde{P})$, $O^+_{\sigma_2}(\widetilde{P})$, equations      1.38 1.39
$Proj_{\delta}A^*$, definition      2.21
$Proj_{\delta}R(X,L)^*$, equation      2.63
$Proj_{\delta}R(\boldsymbol{X},\boldsymbol{L})^*$, equation      2.71
$P^{(e)}$, definition      8.36
$P_\sigma$, equation      1.34
$R(X,L)^*_{\langle P\rangle}$, equation      2.67
$R(\boldsymbol{X,L})^*$, equation      2.48
$R(\boldsymbol{X,L})_*$, equation      2.60
$R[x^{(\leqslant r)}]$, equation      5.3
$R\langle X,L\rangle$, equation      2.42
$R\langle X_\delta\rangle$, equation      2.16
$R\langle \boldsymbol{X,L}\rangle$, equation      2.49
$R\{x\}_*$, equation      5.7
$R\{x_*,y\}^*$, equation      5.11
$R^1(\boldsymbol{X,L})^*$, equation      2.50
$s(\Sigma)$, definition      4.1
$T^{w/d}$, equation      3.13
$T_p(A_0)$, equation      8.3
$T_PX$, equation      1.13
$T_PX_\delta$, definition      2.14
$v(\Sigma)$, equation      4.2
$W = \boldsymbol{Z}[\phi]$, definition      2.7
$W_+$, equation      2.11
$W_{deg \ne 0}$, equation      4.63
$X/\sigma$, equation      1.3
$X^D(U)$, $X^D(U,\Gamma_0(l))$, example      8.24
$X_\delta$, equation      2.63
$X_{ord}$, equation      8.43
$Y_1(N)$, equation      2.87
$Y_1(N,l)$, equation      2.88
$Z_p$-accumulating multipliers (resp. Schwarzians), definition      4.59
$[F] \circ [g_0,\ldots,g_r]$,      equation 4.43
$[f] \in H^1(\boldsymbol{X,L})$, equation      2.45
$\boldsymbol{A}^N$, equation      1.8
$\boldsymbol{F}(r, \lambda)$, equation      2.40
$\boldsymbol{F}(\lambda)$, equation      1.17
$\boldsymbol{G}_a$, $\boldsymbol{G}_m$, equation      1.9
$\boldsymbol{K = K_X}$, equation      1.18
$\boldsymbol{K}^w$, equation      4.15
$\boldsymbol{K}^w_r$, equation      4.14
$\boldsymbol{L}$, equation      2.37
$\boldsymbol{L}^w$, $\boldsymbol{L}^{[s]}$, $\boldsymbol{L} \otimes \boldsymbol{M}$, equation      2.38
$\boldsymbol{L}_s$, equation      7.2
$\boldsymbol{L}_w$, equations      6.12 6.49 7.78 8.61
$\boldsymbol{O}$, equation      2.39
$\boldsymbol{V}(M)$, equation      1.7
$\boldsymbol{X}$, equation      1.1
$\boldsymbol{X}^{on}$, definition      1.4
$\boldsymbol{X}^{\Gamma, \tau}$, equation      6.38
$\boldsymbol{X}_F$, $\boldsymbol{X}_C$, $\boldsymbol{X}_\wp$, equation      2.86
$\boldsymbol{X}_\delta$, equation      2.71
$\boldsymbol{X}_\tau$, equation      6.14
$\boldsymbol{X}_{G,N,\epsilon}$, equation      7.1
$\boldsymbol{X}_{Q_1,Q_2}$, equation      6.48
$\boldsymbol{X}_{set}$, equation      2.72
$\boldsymbol{X}_{t^2+c,g,p}$, equation      7.77
$\Delta$ (discriminant), equation      8.97
$\delta$-base locus, definitions      2.14 2.46 2.53
$\delta$-base point, definition      2.14 2.46
$\delta$-character, definition      7.9
$\delta$-cohomology, definition      2.31
$\delta$-degree, definition      5.11
$\delta$-derivation, definition      2.11
$\delta$-dominant, definition      2.15
$\delta$-function, definition      2.22
$\delta$-Galois group, definition      5.11
$\delta$-graded ring, definition      2.18
$\delta$-independent, definition      2.10
$\delta$-invariants, definitions      2.31 4.14 5.11
$\delta$-line bundle, definitions      2.24 2.25
$\delta$-localized, definition      2.14
$\delta$-operator, definition      2.66
$\delta$-polynomial function, definition      2.4
$\delta$-polynomials, definition      3.1
$\delta$-rational, definitions      2.14 2.40 2.45
$\delta$-ring, definition      2.2
$\delta$-ringed set, definition      2.14
$\delta$-rule, definition      2.66
$\delta$-section, definition      2.24
$\delta$-slope zero, definition      5.11
$\delta$-tangent map, equation      2.20
$\delta$-tangent space, definition      2.14
$\delta$-trivial, definition      2.14 2.40 2.45
$\Delta=\Delta^L$, equations      1.21 2.43
$\Delta=\Delta^\Lambda=\Delta^B$, equation      4.9
$\Delta_{12}$, equation      6.47
$\Delta_{mn}$, equation      7.48
$\hat{A}$, $A^{for}$, equation      1.6
$\hat{X}$, $X^{for}$, equation      1.10
$\Lambda(r,B)$, $\Lambda_r$, $\Lambda_\gamma$, definition      4.13
$\Lambda^n_s$, equation      7.46
$\lambda^w$, equation      2.12
$\Lambda_{sj}$, lemma      3.34
$\mathcal{C}_R$, equation      1.14
$\mathcal{C}_\delta$, equation      2.19
$\mathcal{C}_{set}$, equation      1.23
$\mathcal{D}$, equation      3.54
$\mathcal{F}(X)$, equation      2.3
$\mathcal{O}^r(T(X))^+$, equation      3.21
$\mathcal{O}^r(X)$, equation      2.32
$\mathcal{O}^{\infty} = \mathcal{O}^{\infty}_X$, equation      2.33
$\mathcal{O}_{X_\delta,P}$, equation      2.18
$\mathcal{R}$, equation      1.26
$\mathcal{R}^*_\delta$, equation      2.27
$\mathcal{R}_\delta$, equation      2.2
$\nabla: H^1 \to H^1 \otimes \Omega$, equation      8.30
$\nabla: X(R) \to J^r(X)(R)$, equations      3.8
$\nu$-canonical line bundle, definition      1.7
$\Omega_\sigma$, equation      1.34
$\omega_{Weier}$, equation      8.100
$\overline{f}^1_{jet}$, equation      8.110
$\overline{f}_{def}$, equation      8.112
$\overline{H}$, equation      8.42
$\partial^*_j$, equation      8.95
$\partial^{can}$, equation      4.27
$\partial_j$, $\partial_*$, $\partial_{**}$, definition      3.40
$\phi$-module, definition      2.12
$\phi$-polynomial functions, definition      2.6
$\psi_1$, definition      7.16
$\psi_1$, equation      7.5
$\psi_2$, definition      7.18
$\Psi_E$, equation      7.35
$\psi_{2,nor}$, theorem      7.22
$\psi_{can}$, definition      7.24
$\Psi_{G_a}$, equation      6.18
$\Psi_{G_m}$, equation      4.40
$\Sigma(T)$, definition      4.1
$\Sigma_a(T)$, equation      4.38
$\tau_i$, equation      7.9
$\underline{BS}$, $\overline{BS}$, $\underline{B}$, definition      2.53
$\varepsilon$, equation      8.82
$\{g,f\}_\delta$, equation      2.26
(m,n,r)-flat canonical bundle, definition      4.10
a(C), equation      4.58
Adjunction operator, proposition      3.39
Admissibile subgroup, definition      1.21
Analytic uniformization of a correspondence, definition      1.23
Backward orbit, equation      1.39
Basic $\delta$-character of order $\le 2$, definition      7.18
Basic $\delta$-character of order 1, definition      7.16
Canonical $R[[t_{ij}]]$-basis of de Rham, definition      8.7
Canonical $\delta$-character, definition      7.24
Canonical derivation, definition      4.21
Canonical line bundle of a correspondence, definition      1.7
Canonical line bundle of a formal correspondence, definition      4.2
Canonical R[[T]]-basis of de Rham, definition      8.10
Categorical quotient, definition      1.1
Chebyshev polynomial, definition      1.29
Commensurable correspondences, definition      1.10
Conjugate operators, definition      3.40
Correspondence, definition      1.1
Critically finite, definition      1.42
d, equation      3.45
d-coregular $\delta$-section, definition      4.53
d-perfect $\delta$-section, definition      4.53
d-regular $\delta$-section, definition      4.53
deg(w), equation      2.7
Dense correspondence, definition      1.9
Dimension of a correspondence, definition      1.4
Disjoint union of correspondences, definition      1.4
Dual numbers, equation      1.5
E, equation      4.52
E-arithmetic subgroup, definition      1.33
E-arithmetic uniformization, definition      1.36
Equivalent correspondences, definition      1.10
Etale correspondence, definition      1.4
Euler derivation operator, remark      3.58
Euler — Lagrange operator/equation, equation      3.36
Evaluation maps, definition      2.14
False 1-form, definition      8.1
Fixed point, definition      4.48
Flat $\delta$-line bundle, definition      2.26
Flat correspondence, definition      7.1
Flat line bundle, definition      1.6
Flat uniformization, definition      1.23
Formal $\delta$-line bundle, definition      4.13
Formal correspondence, definition      4.1
Formal line bundle, definition      4.2
Formally flat $\delta$-line bundle, definition      4.54
Forward orbit, equation      1.38
Frobenius tangent scheme (resp. bundle), definition      3.8
FT(Y/k), equation      3.5
Full subgroup, definitions      2.58 6.22
Galois cover of a correspondence, definition      1.11
Generalized root of unity, definition      4.29
Generic $\delta$-fiber, definition      2.81
Generic rule, definition      2.68
Germ of a $\delta$-line bundle, definition      4.50
Germ of a $\delta$-section, definition      4.53
Germ of a correspondence, definition      4.49
Global $\delta$-p-ring, definition      2.8
Global p-ring, definition      1.2
Group formal scheme, definition      1.3
Hamiltonian flow, definition      3.51
Hecke correspondence, definition      8.11
Hecke form $P^{(e)}$, definition      8.36
Hecke n-cycle, definition      8.17
Hyperbolic uniformization, definition      1.23
Infinitesimal symmetry, definitions      3.48 4.20
Infinitesimally universal Abelian scheme, definition      8.6
Internal Kodaira — Spencer class, definitions      3.10 8.50
Irreducible base, definition      1.4
Irreducible correspondence, definition      1.4
KS(X), equation      3.10
Lattes function, definition      1.29
Left etale correspondence, definition      1.4
Lift of Frobenius, definition      3.6
Linearization of a $\delta$-line bundle, definition      2.25
Linearizing series, definition      4.6
Local $\delta$-p-ring, definition      2.8
Local p-ring, definition      1.2
Multiplicative polynomial, definition      1.29
Multiplier, definition      4.1
n-cycle on a correspondence, definition      4.56
n-th power of a correspondence, definition      1.4
Normalized basis of de Rham, definitions      8.4 8.5
Normalized basis of Tate module, definition      8.3
Open immersion of correspondences, definition      1.4
Open subcorrespondences, definition      1.4
ord(w), equation      2.8
P, definition      8.33
p-derivation, definition      2.1
p-jet space, definition      3.5
Postcritically finite, definitions      1.38 1.42
Primitively postcritically finite, definition      1.42
Principal sets, definition      2.14
Prolongation operators, definition      3.40
Prolongation sequence, definition      3.2
Purely $\delta$-Galois, definition      5.11
Q-anticanonical $\delta$-line bundle, definition      2.27
Q-canonical $\delta$-line bundle, definition      2.27
Quadratically $\phi$-independent, definition      4.66
R(X,L)*, equation      2.41
Ramanujan form P, definition      8.33
Regular Hecke fixed point, definition      8.19
Relative generic $\delta$-fiber, definition      2.49
Representation on a set, definition      2.20
Restricted $\delta$-function, definition      2.4
Restricted $\phi$-functions, definition      2.6
R{T}, definition      3.1
R{x}*, equation      5.10
R{x}, equation      5.2
s(C), equation      4.58
Saturated set of places, equation      2.84
Schwarzian, definition      4.1
Smooth $\delta$-modular form of genus 1, definition      8.44
Smooth $\delta$-modular form on D, definition      8.45
Smooth $\delta$-modular function of genus 1, definition      8.44
Smooth $\delta$-modular function on D, definition      8.45
Smooth isogeny covariant $\delta$-modular form of genus 1, definition      8.44
Smooth isogeny covariant $\delta$-modular form on D, definition      8.45
Spherical correspondence, definition      6.1
Spherical uniformization, definition      1.23
Strict $\delta$-base point, definition      2.46
Symmetric of radius $\rho$, definition      4.62
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