Dwork-type ($q$-)(super)congruences

Abstract: The "microscope" principle is this: If a rational function A(q) of variable q vanishes at every p-th root of unity (for p prime), then A(q) == 0 modulo Φ_p(q), the p-th cyclotomic polynomial; assuming that A(1) is a well-defined rational number with a p-free denominator and specialising the congruence at q=1 we conclude with A(1) == 0 modulo p. In other words, behaviour of rational functions at p-th roots of unity may be instructive for gaining information about their values at 1 modulo p. With some "creative" extras, we can further consider divisibility by higher powers of primes (and we can even deal with not necessarily primes). In my talk, partly based on recent joint work with Victor Guo, I plan to highlight some novel outcomes of this "creative microscope" methodology -- examples of Dwork-type supercongruences for truncated hypergeometric sums.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar