Creative microscoping

Abstract: Let $A_k=2^{-6k}{\binom{2k}k}^3$ for $k=0,1,\dots$,. Though traditional techniques of establishing the hypergeometric evaluation

$\sum \limits_{k=0}^\infty(-1)^k(4k+1)A_k =\frac2\pi$

and (super)congruences

$\sum \limits_{k=0}^{p-1}(-1)^k(4k+1)A_k \equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{for primes}\; p>2$

share certain similarities, they do not display intrinsic reasons for the two to be related. In my talk I will outline basic ingredients of a method developed in joint works with Victor Guo, which does the missing part, also for many other instances of such arithmetic duality. The main idea is constructing suitable $q$-deformations of the infinite sum (and many such sums are already recorded in the $q$-literature), and then look at the asymptotics of that at roots of unity. Interestingly enough, the $q$-deformations may offer more. For example, the $q$-deformation of the above infinite sum also implies $$ \sum_{k=0}^\infty A_k =\frac{\Gamma(1/4)^4}{4\pi^3} =\frac{8L(f,1)}{\pi} \quad\text{and}\quad \sum_{k=0}^{p-1}A_k\equiv a(p)\pmod{p^2} $$ (in fact, the latter congruences in their stronger modulo $p^3$ form proven by Long and Ramakrishna), where $a(p)$ is the $p$-th Fourier coefficient of (the weight 3 modular form) $f=q\prod_{m=1}^\infty(1-q^{4m})^6$.

($NB:$ The variable $q$ in the last definition is related to the modular parameter $\tau$ through $q=e^{2\pi i\tau}$ and has nothing to do with the $q$ in the $q$-deformation.)

number theory

Audience: researchers in the discipline

Upstate New York Online Number Theory Colloquium