论文标题
与Q-差异操作员相关的强Frobenius结构
Strong Frobenius structures associated with q-difference operators
论文作者
论文摘要
强大的Frobenius结构的概念在$ p $ - 亚种差分运算符的理论中经常研究。在目前的工作中，我们介绍了针对$ q $ - 差异运算符的强Frobenius结构概念的新定义。该定义的相关性得到了两个主要结果的支持。第一个处理\ emph {confluence}。我们表明，如果$ q $ -difference运算符$ l_q $具有较强的Frobenius结构，用于$ h $的prime $ p $，并且如果$ h $ $ h $，则是$ p $ - ad-adic差异操作员从$ l_q $获得的$ q $ tytting to to to tte 1 tte 1，那么$ l $ to $ l $ a $ p $ a $ p $ a $ p $ agient $ h $ h $ h $ h $ h $ h $。第二个涉及一致性模量循环多项式。我们表明，如果$ f（q，z）\ in \ mathbb {z} [q] [q] [[z]] $是$ q $ - 差异操作员的解决方案，具有强大的frobenius结构，则适用于$ p $ p $ p $ f（q，z）$满足$ p $ p $ p $ - th cyclotomic polynomial的一些一致性。 André和Di Vizio引入了与$ Q $ - 差异操作员相关的强frobenius结构的另一个定义，我们还指出了为什么它们的定义不适合我们的应用：Confluence and Allocuence Modulo Cylotomic colotomic colletomic tolynomials。最后，我们表明，订单1的一些$ Q $ - 流行地几个地面运算符具有强大的Frobenius强度，对于无限的许多素数。
The notion of strong Frobenius structure is classically studied in the theory of $p$-adic differential operators. In the present work, we introduce a new definition of the notion of strong Frobenius structure for $q$-difference operators. The relevance of this definition is supported by two main results. The first one deals with \emph{confluence}. We show that if the $q$-difference operator $L_q$ has a strong Frobenius structure for a prime $p$ with period $h$ and if $L$ is the $p$-adic differential operator obtained from $L_q$ by letting $q$ tend to 1, then $L$ has a strong Frobenius structure for $p$ with period $h$. The second one deals with congruence modulo cyclotomic polynomials. We show that if $f(q,z)\in\mathbb{Z}[q][[z]]$ is a solution of a $q$-difference operator having strong Frobenius structure for $p$ then $f(q,z)$ satisfies some congruences modulo the $p$-th cyclotomic polynomial. Another definition of strong Frobenius structures associated with $q$-difference operators has been introduced by André and Di Vizio and we also point out why their definition is not suitable for our applications: confluence and congruence modulo cyclotomic polynomials. Finally, we show that some $q$-hypergeometric operators of order 1 have a strong Frobenius strong for infinitely many primes numbers.
