{"id":16989,"date":"2023-07-25T13:57:01","date_gmt":"2023-07-25T05:57:01","guid":{"rendered":"https:\/\/virtual.math.ntnu.edu.tw\/?p=16989"},"modified":"2023-07-25T14:07:26","modified_gmt":"2023-07-25T06:07:26","slug":"20230727_speech","status":"publish","type":"post","link":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/2023\/07\/25\/20230727_speech\/","title":{"rendered":"[NTNU Number Theory Seminar] <\/span>\u301007\u670827\u65e5\u3011\u4e8e\u9756 \/ The special Gamma Values —From n! to Crystalline. Periods"},"content":{"rendered":"\t\t
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The special Gamma Values ---From n! to Crystalline. Periods<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\u6642\u3000\u9593\uff1a2023-07-27 13:30 (\u661f\u671f\u56db) \/ \u5730\u3000\u9ede\uff1aM310\u00a0<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\u4e8e\u9756<\/strong> \u9662\u58eb<\/strong><\/div>
\u53f0\u7063\u5927\u6578\u5b78\u540d\u8b7d\u6559\u6388<\/strong><\/div>
\u6e05\u83ef\u5927\u5b78\u69ae\u8b7d\u8b1b\u5ea7\u6559\u6388<\/strong><\/div>
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ABSTRACT<\/p>\n

Euler interpolated n! to the classical Gamma function with \u0393(n+1) :=n!. Since \u0393(1\/2) = \u221a<\/span>\u03c0 is transcendental, one wanders about the <\/span>Gamma values at other proper fractions. In 1976 G. V. Chudnovsky <\/span>succeeded in proving \u0393(1\/3), \u0393(1\/4) are also transcendental and alge\u0002braically independent from \u03c0. In his work the periods of specific ellip\u0002tic curves play a vital role. One actually conjectures that the Gamma <\/span>values at all proper fractions are transcendental and Lang-Rohrlich <\/span>even speculates that all algebraic relations among these transcendental <\/span>special values come from well-known functional equations of Euler\u2019s <\/span>Gamma function. <\/span><\/p>\n

Y. Morita in 1975 looked at the n! from p-adic arithmetic, p any fixed prime number. He introduced p-adic Gamma function \u0393p defined on Zp with values in the algebraic closure of Qp satisfying \u0393p(n + 1) = \u2212n\u0393p(n), if p \u0338= n. This case \u0393p(1\/2) = \u221a\u00b11 is always algebraic.
In 1979, Gross-Koblitz discovered that if n \u2261 1 (mod p), all proper fractions with denominator n turns out to be algebraic. This leads to conjecture that the values of the p-adic Gamma at the other proper fractions should be transcendental, as they are related to the crystalline
periods of certain abelian varieties by Ogus 1989.<\/p>


We are interested in factorials and Gamma functions for function fields in the positive characteristic worlds, after Carlitz, Goss, Thakur, An\u0002derson, Brownawell, and Papanikolas. Particularly I will report on recent progress of special v-adic Gamma values for finite prime v, and the ongoing work by Chieh-Yu Chang, Fu-Tsun Wei and me.<\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\u53cb\u5584\u5217\u5370<\/a><\/p>","protected":false},"excerpt":{"rendered":"

The special Gamma Values —From n! to Crystalline. […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1,124],"tags":[],"_links":{"self":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/16989"}],"collection":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=16989"}],"version-history":[{"count":11,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/16989\/revisions"}],"predecessor-version":[{"id":18496,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/16989\/revisions\/18496"}],"wp:attachment":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=16989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=16989"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=16989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}