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UID:20147b3e996a63ebcfccfc14b354ae77
CATEGORIES:Number Theory Seminar
CREATED:20250116T200936
SUMMARY:The shifted convolution of divisor functions in function fields
LOCATION:HLL 525
DESCRIPTION:Abstract: We will discuss ongoing work with Alexandra Florea, Matilde Lalín
 , and Amita Malik on the shifted convolution problem for divisor functions 
 in function fields. This involves studying the average value of $d(f) d(f+h
 )$ where $h$ is a fixed polynomial (having possibly large degree $m$) in $m
 athbb{F}_q[T]$ and $f$ runs over all monic polynomials in $mathbb{F}_q[T]$ 
 of degree $n$, where $n$ goes to infinity. Our techniques mirror the classi
 cal approach of Estermann in the integer setting. The main new ingredient i
 s a functional equation for the Estermann function (equivalently, a Voronoi
  summation formula for the divisor function) that seems be to novel for thi
 s setting. If time permits, we will discuss a related result involving the 
 shifted convolution of the norm-counting functions of quadratic extensions.
  The talk should be accessible to those unfamiliar with function fields.\n
X-ALT-DESC;FMTTYPE=text/html:<p><span data-olk-copy-source="MessageBody" style="direction: ltr; margin: 
 0px; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, 
 Helvetica, sans-serif; font-size: 12pt; color: #000000;">Abstract: We will 
 discuss ongoing work with Alexandra Florea, Matilde Lalín, and Amita Malik 
 on the shifted convolution&nbsp;problem for divisor functions in function f
 ields. This involves studying the average value of $d(f) d(f+h)$ where $h$ 
 is a fixed polynomial (having possibly large degree $m$) in&nbsp;$mathbb{F}
 _q[T]$ and $f$ runs over all monic polynomials in&nbsp;$mathbb{F}_q[T]$ of 
 degree $n$, where $n$ goes to infinity. Our techniques mirror the classical
  approach of Estermann in&nbsp;the integer setting. The main new ingredient
  is a functional equation for the Estermann function&nbsp;(equivalently, a 
 Voronoi summation formula for the divisor function) that seems be to novel 
 for this setting. If time permits, we will discuss a related result involvi
 ng the shifted convolution&nbsp;of the norm-counting functions of quadratic
  extensions. The talk should be accessible to those unfamiliar with functio
 n fields.</span></p>
CONTACT:Anurag Sahay
DTSTAMP:20260826T145752
DTSTART;TZID=America/New_York:20250128T140000
DTEND;TZID=America/New_York:20250128T150000
SEQUENCE:0
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