Abstract
We develop an analytical method to prove congruences of the type∑k=0(pr−1)/dAkzk≡ω(z)∑k=0(pr−1−1)/dAkzpk(modpmrZp[[z]])forr=1,2,…, for primes p>2 and fixed integers m,d⩾1, where f(z)=∑k=0∞Akzk 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.
Highlights
Extending his work on the rationality of the zeta function of an algebraic variety defined over a finite field, Dwork [2] considered a question of continuing analytical solutions f (z) =of linear differential verify that the truncated sums fr(z) =equations pr −1 k=0Ak p-adically. zk, whereA general strategy was to r = 0, 1, 2, . . . , satisfy the so-called Dwork congruences [33]fr+1(z) fr (z p ) ≡ fr (z ) fr−1(zp)for r = 1, 2, . . . (1.1)
The principal goal of this paper is to extend the approach of [17] and establish general techniques for proving Dwork-type supercongruences using the method of creative microscoping, which we initiated in [23] for proving r = 1 instances of such supercongruences
(1.7) are congruent to 0 modulo [nr], while letting q → qn and n → nr−1 in the above congruences, we see that the right-hand sides of them are congruent to 0 modulo
Summary
Notice that the argument extends to the cases when f1(zp) ≡ 0 (mod pZp[[z]]) but f1(zp) ≡ 0 (mod pmZp[[z]]) for some m 2, provided the congruences (1.1) hold modulo a higher power of p, for example, fr+1(z) fr (z p ) It is this type of congruences that we refer to as Dwork-type supercongruences; other truncations of the initial power series are possible as well, usually of the type fr(z) =. The principal goal of this paper is to extend the approach of [17] and establish general techniques for proving Dwork-type supercongruences using the method of creative microscoping, which we initiated in [23] for proving r = 1 instances of such supercongruences.
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