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SUMMARY:Chris Williams (Nottingham)
DTSTART:20241001T130000Z
DTEND:20241001T140000Z
DTSTAMP:20260421T154017Z
UID:UEAPS/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UEAPS/38/">C
 ongruences between eigensystems for GL(n)</a>\nby Chris Williams (Nottingh
 am) as part of ANTLR seminar\n\nLecture held in NEWSCI 0.03.\n\nAbstract\n
 In this talk\, I will discuss congruences between modular forms\, and -- m
 ore generally -- between systems of eigenvalues on GL(n)\, modular forms b
 eing the special case $n=2$. In a broad sense\, this seeks to answer the f
 ollowing type of question: Let $p$ be prime and $f$ be a modular form of l
 evel $\\Gamma_0(M)$ where $p$ divides $M$. For a given integer $m$\, does 
 there exist another eigenform $g$ congruent to $f$ modulo $p^m$?\n\nFor mo
 dular forms\, the answer is yes. Even better\, such congruences can be cap
 tured geometrically via 1-dimensional families' of eigenforms (via the eig
 encurve'). This geometric object has had profound consequences in Iwasawa 
 theory and the Langlands program. In this talk\, I will attempt to give a 
 gentle introduction to  $p$-adic families\, and describe joint work with D
 aniel Barrera and Andy Graham\, where we consider some of the problems in 
 generalising them to higher dimension. Here the picture becomes more subtl
 e -- whilst all families for GL(2) are $1$-dimensional\, for GL(4)\, there
  can be classical families of dimension $0\, 1$\, or $2$ attached to the s
 ame automorphic representation.\n
LOCATION:https://researchseminars.org/talk/UEAPS/38/
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