Rational approximations to values of $E$-functions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913936996040704 |
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| author | Fischler, Stéphane Rivoal, Tanguy |
| author_facet | Fischler, Stéphane Rivoal, Tanguy |
| contents | We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under strong assumptions on algebraic independence of $E$-functions, satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance. We remove them using a new generalization of Shidlovskii's lemma, analogous to zero estimates on commutative algebraic groups in which obstructions come from algebraic subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12043 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational approximations to values of $E$-functions Fischler, Stéphane Rivoal, Tanguy Number Theory 11J82 (Primary), 11J91 (Secondary) We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under strong assumptions on algebraic independence of $E$-functions, satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance. We remove them using a new generalization of Shidlovskii's lemma, analogous to zero estimates on commutative algebraic groups in which obstructions come from algebraic subgroups. |
| title | Rational approximations to values of $E$-functions |
| topic | Number Theory 11J82 (Primary), 11J91 (Secondary) |
| url | https://arxiv.org/abs/2312.12043 |