We develop an analytical method to prove congruences of the type$\sum _{k=0}^{(p^r-1)/d}{A}_k{z}^k\equiv \omega (z)\sum _{k=0}^{(p^{r-1}-1)/d}{A}_k{z}^{pk}(\mathrm{mod}{p}^{mr}{\mathbb{Z}}_p[[z]])\text{for}r=1,2,\dots ,$ for primes $p>2$ and fixed integers $m,d⩾1$, where $f(z)={\sum }_{k=0}^{\infty }{A}_k{z}^k$ 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 $m⩾2$ (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.

我们开发一种分析方法来证明该类型的全等$\sum _{ķ=0}^{（p^{\mathrm{[R}}-\mathrm{1个}）/d}{\mathrm{一种}}_{ķ}{ž}^{ķ}\equiv \omega （ž）\sum _{ķ=0}^{（p^{\mathrm{[R}-\mathrm{1个}}-\mathrm{1个}）/d}{\mathrm{一种}}_{ķ}{ž}^{pķ}（\mathrm{模}{p}^{米\mathrm{[R}}{\mathbb{ž}}_p[[ž]]）\text{对于}\mathrm{[R}=\mathrm{1个}，2，\dots ，$ 素数 $p>2$ 和固定整数 $米，d⩾\mathrm{1个}$，在哪里 $F（ž）={\sum }_{ķ=0}^{\infty }{\mathrm{一种}}_{ķ}{ž}^{ķ}$是“算术”超几何级数。这样的全等$米=d=\mathrm{1个}$是由Dwork在1969年引入的，它是p -adic分析连续性的工具$F（ž）$。我们证明了与Dwork型同余对应的$米⩾2$（换句话说，超同余）是基于构造和证明其合适的q类比的，它们反过来也具有存在的权利和潜力，可以模块化形式和代数变体的同调群进行q形变。我们的方法遵循我们引入的解决方案的创新性微观范围原理$\mathrm{[R}=\mathrm{1个}$这种一致性的实例；它是第一个能够为一般r建立这种超同余的方法。