论文标题
通过创意$ Q $ - 微镜
Dwork-type supercongruences through a creative $q$-microscope
论文作者
论文摘要
我们开发了一种分析方法来证明类型$$ \ sum_ {k = 0}^{（p^r-1）/d} a_kz^k \equivΩ（z）\ sum_ {k = 0}^{（p^{p^{r-1} -1} -1） z_p [[z]]} \ quad \ text {for} \; r = 1,2，\ dots，Primes $ p> 2 $和固定整数$ m，d \ ge1 $，其中$ f（z）= \ sum_ {k = 0}^\ infty a_kz^k $是“ arithmetic”超几何系列。 1969年，DWork在1969年引入了此类的$ M = D = 1 $，作为$ f（z）$的工具。我们证明了几种与$ M \ ge2 $（换句话说，超级方面）相对应的几个DWork类型的一致性，是基于构建和证明其合适的$ Q $ -Analogues，这反过来又有其自身的生存和潜力，并具有$ Q $ $ Q $ $ $ q $的质量形式和同类型algebraic colelgebraic Valieties的群体。我们的方法遵循我们引入的创意显微镜的原理，以解决$ r = 1 $ $ $ $的实例；这是第一种能够为一般$ r $建立这种类型的超级企业的方法。
We develop an analytical method to prove congruences of the type $$ \sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv ω(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]} \quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers $m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic" hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in 1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of several Dwork-type congruences corresponding to $m\ge2$ (in other words, supercongruences) are based on constructing and proving their suitable $q$-analogues, which in turn have their own right for existence and potential for a $q$-deformation of modular forms and of cohomology groups of algebraic varieties. Our method follows the principles of creative microscoping introduced by us to tackle $r=1$ instances of such congruences; it is the first method capable of establishing the supercongruences of this type for general $r$.
