Solutions to arithmetic differential equations in algebraically closed fields

Buium, Alexandru; Miller, Lance Edward

doi:10.1016/j.aim.2020.107342

Cited by 4 publications

(12 citation statements)

References 8 publications

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“…Corollaries 5.14, 3.9 and Proposition 3.13). This can be viewed as a simultaneous generalization of the main results in [Bu95] and [BM20].…”

Section: Resultsmentioning

confidence: 58%

“…The homomorphisms ψ alg are not given by algebraic (or even by analytic) functions in the coordinates but rather by analytic (in fact rigid analytic) functions in the coordinates and their various "δ π ′ -derivatives" for various π ′ 's dividing π. The recipe for defining these maps involves the notion of total δ-overconvergence which is analogous to the one in [BM20] and will be explained in the body of the paper. Note that since E is projective over R π we have E(K alg ) = E(R alg ); however, it is an important feature of the theory that the images of the maps ψ alg are not contained in R alg .…”

Section: Resultsmentioning

confidence: 99%

“…In the framework of [Bu95,Bu05] the only solutions of arithmetic ODEs that were defined were "unramified solutions" i.e., solutions (with coordinates) in the completion R of the maximum unramified extension of Z p . Subsequently the δoverconvergence machinery in [BS11,BM20] allowed one to define "ramified solutions" to the main arithmetic ODEs of the theory, i.e., solutions in the ring of integers R alg of the algebraic closure K alg of K := R[1/p] and sometimes even in the ring of integers C • p of the complex p-adic field C p .…”

Section: Introductionmentioning

confidence: 99%

“…, φ n of Frobenius automorphisms of K alg . Remarkably this approach, combined with the δ-overconvergence technique in [BS11,BM20], allows one to define solutions to our equations in R alg . As an application we will again construct arithmetic Manin maps which (as in [BS10a] but unlike in [BS09b]) have order 1; so the various arithmetic differentiation operators at p conspire, again, to create lower order arithmetic Manin maps.…”

Section: Introductionmentioning

confidence: 99%

“…reference N pr N ari N geo ramified solutions defined [Bu95] 1 1 0 NO [BM20] 1 1 0 YES [BS10a] 1 1 1 NO [BS09b] n n 0 NO This paper 1 n 0 YES 1.3. Terminology.…”

Section: Introductionmentioning

confidence: 99%

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Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

Buium,

Miller

2021

Preprint

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A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. We also prove structure results for the spaces of "PDE differential modular forms defined on the ordinary locus." We also produce a system of differential equations satisfied by our PDE modular forms based on Serre and Euler operators. ALEXANDRU BUIUM AND LANCE EDWARD MILLER 7. Partial δ-modular forms 40 7.1. Basic definitions 40 7.2. Jet construction 41 7.3. The case n = 2, π = p 44 7.4. δ-Serre-Tate expansions 46 7.5. δ-period maps 51 7.6. Theorem of the Kernel and Reciprocity Theorem 56 7.7. Crystalline construction 59 7.8. Forms on the ordinary locus 60 7.9. Finite covers defined by δ-modular forms 64 7.10. Application to Modular parameterizations 66 7.11. δ-Serre operators 68 References 70

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“…Corollaries 5.14, 3.9 and Proposition 3.13). This can be viewed as a simultaneous generalization of the main results in [Bu95] and [BM20].…”

Section: Resultsmentioning

confidence: 58%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…reference N pr N ari N geo ramified solutions defined [Bu95] 1 1 0 NO [BM20] 1 1 0 YES [BS10a] 1 1 1 NO [BS09b] n n 0 NO This paper 1 n 0 YES 1.3. Terminology.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

Buium,

Miller

2021

Preprint

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Canonical Witt formal scheme extensions and p-torsion groups

Bertapelle

Mazzari

Saha

2023

Annali di Matematica

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Reversible Thinking of Fifth Graders: Focus on Linier Equations

Maf’ulah

2022

j. ilm. sekol. Dasar

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Reversible thinking is the mental ability to return the way of thinking to the original point. This study explores the reversible thinking of fifth graders on linear equations. This descriptive-exploratory research was conducted on 96 fifth-grade elementary school students. Data collection is done by providing reversible thinking and interviews. The technique used for data analysis is qualitative and quantitative description analysis. The assignment contains a linear equation (called the initial equation). Students are asked to make an equation equivalent to the initial equation. All equations that students successfully created were then analyzed and classified based on the strategies used. Next, one student with the most diverse approach was selected to be explored more deeply in reversible thinking. The results of the study, there are ten categories of students' reversible thinking strategies, among others, first, moving the elements of the initial equation builder. Second, determine the unknown factors. Third, operate both sides of the initial equation with the same number. Fourth, operate both as equations of equivalent expressions. Fifth, change the known building elements in the created equation. Sixth refers to the value of the unknown component then converts that value into a presentation. Seventh, transforming the building elements that are known in the initial equation. Eighth refers to the value of the initial equation variable. Ninth, using the definition of subtraction. Tenth, using the commutative property of the operation of the number.

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Solutions to arithmetic differential equations in algebraically closed fields

Cited by 4 publications

References 8 publications

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

Canonical Witt formal scheme extensions and p-torsion groups

Reversible Thinking of Fifth Graders: Focus on Linier Equations

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