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SUMMARY:Matija Kazalicki (University of Zagreb)
DTSTART:20201116T121500Z
DTEND:20201116T131500Z
DTSTAMP:20260423T023054Z
UID:WarsawNT/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WarsawNT/12/
 ">Congruences for sporadic sequences\, three fold covers of the elliptic m
 odular surfaces and modular forms for non-congruence subgroups</a>\nby Mat
 ija Kazalicki (University of Zagreb) as part of Warsaw Number Theory Semin
 ar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 1979\, in the cou
 rse of the proof of the irrationality of $\\zeta(2)$\n Ap\\'ery introduced
  numbers $b_n = \\sum_{k=0}^n {n \\choose k}^2{n+k\n\\choose k}$ that are\
 , surprisingly\, integral solutions of recursive\nrelations  $$(n+1)^2 u_{
 n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0.$$\nZagier performed a computer sea
 rch on first 100 million triples\n$(A\,B\,C)\\in \\mathbb{Z}^3$ and found 
 that the recursive relation\ngeneralizing $b_n$\n$$(n+1)u_{n+1} - (An^2+An
 +B)u_n + C n ^2 u_{n-1}=0\,$$\nwith the initial conditions $u_{-1}=0$ and 
 $u_0=1$ has (non-degenerate\ni.e. $C(A^2-4C)\\ne 0$) integral solution for
  only six more triples\n(whose solutions are so called sporadic sequences)
  .\n\nStienstra and Beukers showed that the for prime $p\\ge 5$\n\\begin{e
 quation*}\nb_{(p-1)/2} \\equiv \\begin{cases} 4a^2-2p \\pmod{p} \\textrm{ 
 if } p =\na^2+b^2\,\\textrm{ a odd}\\\\ 0 \\pmod{p} \\textrm{ if } p\\equi
 v 3\n\\pmod{4}.\\end{cases}\n\\end{equation*}\n\nRecently\, Osburn and Str
 aub proved similar congruences for all but one\nof the six Zagier's sporad
 ic sequences (three cases were already known\nto be true by the work of St
 ienstra and Beukers) and we proved the\ncongruence for the sixth sequence.
 \n\nIn this talk we describe congruences for the Ap\\'ery numbers\n$b_{(p-
 1)/3}$ (and also for the other sporadic sequences).\nFor that we study Atk
 in and Swinnerton-Dyer type of congruences\nbetween Fourier coefficients o
 f cusp forms for non-congruence\nsubgroups\, $L$-functions of three covers
  of elliptic modular surfaces\nand Galois representations attached to thes
 e covers.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/12/
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