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On a conjecture of Furusho over function fields
Inventiones mathematicae ( IF 2.6 ) Pub Date : 2020-07-14 , DOI: 10.1007/s00222-020-00988-1
Chieh-Yu Chang , Yoshinori Mishiba

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p -adic MZV’s satisfy the same $${\mathbb {Q}}$$ Q -linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers $${\mathfrak {s}}=(s_1, \ldots , s_r)$$ s = ( s 1 , … , s r ) , we construct a corresponding t -module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the $$\infty $$ ∞ -adic MZV and v -adic MZV at $${\mathfrak {s}}$$ s , completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of $$r=1$$ r = 1 . Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any $$\infty $$ ∞ -adic linear relation on MZV’s with a sub- t -module of a corresponding giant t -module. This makes it possible to arrive at the same linear relation for v -adic MZV’s.

中文翻译:

关于Furuso关于函数域的猜想

在多 zeta 值 (MZV's) 的经典理论中,Furusho 提出了一个猜想,断言 p-adic MZV's 满足相同的 $${\mathbb {Q}}$$ Q -线性关系,它们对应的实值 MZV 对应物满足. 在本文中,我们验证了 Furusho 猜想的函数域类比的更强版本,因为我们能够处理给定有理函数域的代数闭包上的所有线性关系,而不仅仅是有理线性关系。对于每个正整数元组 $${\mathfrak {s}}=(s_1, \ldots , s_r)$$ s = ( s 1 , … , sr ) ,我们构造一个相应的 t 模和一个特定的有理点. 这种结构的精细分辨率(通过纤维联积)实际上允许我们在 $${\mathfrak {s}}$$ s 处获得 $$\infty $$ ∞ -adic MZV 和 v -adic MZV 的很好的对数解释,在 $$r=1$$ r = 1 的情况下完全概括了 Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) 的工作。此外,它使我们能够应用 Yu 的子 t 模定理(Yu in Ann Math (2) 145(2):215–233, 1997),将 MZV 上的任何 $$\infty $$ ∞ -adic 线性关系与相应巨型 t 模块的子 t 模块。这使得可以得到 v -adic MZV 的相同线性关系。215–233, 1997),将 MZV 上的任何 $$\infty $$ ∞ -adic 线性关系与相应巨型 t 模的子 t 模联系起来。这使得可以得到 v -adic MZV 的相同线性关系。215–233, 1997),将 MZV 上的任何 $$\infty $$ ∞ -adic 线性关系与相应巨型 t 模的子 t 模联系起来。这使得可以得到 v -adic MZV 的相同线性关系。
更新日期:2020-07-14
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