Congruences like Atkin’s for the partition function

Let p ( n stretchy="false">) p(n) be the ordinary partition function. In 1960s Atkin found a number of examples congruences form upper Q cubed script l plus beta right-parenthesis identical-to 0 mod Q 3 ℓ + β 0 mod width="0.333em" encoding="application/x-tex">p( Q^3 \ell n+\beta )\equiv 0\pmod \ell where alttext="script l"> encoding="application/x-tex">\ell and alttext="upper Q"> encoding="application/x-tex">Q are prime alttext="5 less-than-or-equal-to 31"> 5 ≤<!-- ≤ <mml:mn>31 encoding="application/x-tex">5\leq \leq 31 ; these lie in two natural families distinguished by square class alttext="1 minus 24 1 −<!-- − <mml:mn>24 encoding="application/x-tex">1-24\beta \pmod . recent decades much work has been done to understand Superscript m Baseline m encoding="application/x-tex">p(Q^m\ell It is now known that there many such when alttext="m greater-than-or-equal-to 4"> ≥<!-- ≥ <mml:mn>4 encoding="application/x-tex">m\geq 4 , scarce (if they exist at all) equals 1 comma 2"> = , 2 encoding="application/x-tex">m=1, 2 for 0"> encoding="application/x-tex">m=0 only 5 7 11"> 7 11 encoding="application/x-tex">\ell =5, 7, 11 For like Atkin’s (when 3"> encoding="application/x-tex">m=3 ), more have but little else seems known. Here we use theory modular Galois representations prove every 5"> \geq 5 infinitely first family which he discovered least alttext="17 slash 24"> 17 / encoding="application/x-tex">17/24 primes second family.