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PMATH 940

Topics in Number Theory

Graduate

Winter 2026 · Fermat’s Last Theorem

Topics

  • Moduli spaces of elliptic curves
  • Galois representations of elliptic curves and modular forms
  • Deformation theory of Galois representations
  • Modularity lifting
Winter 2023 · Analytic Methods on Diophantine Problems

Topics

  • Circle method
  • Waring’s problem
  • Vinogradov’s mean value theorem
  • Connections to Bourgain’s discrete restriction conjecture
  • Possible multidimensional and function-field extensions
Fall 2022 · Diophantine Approximation

Topics

  • Rational approximation of real numbers
  • Dirichlet and Hurwitz theorems
  • Continued fractions and Pell’s equation
  • Metric Diophantine approximation
  • Approximation of algebraic numbers
  • Roth’s theorem
  • Schmidt’s simultaneous approximation theorem
Winter 2020 · Second Course in Analytic Number Theory

Topics

  • Gauss sums
  • Dirichlet L-functions and class number formula
  • Zeros, moments and pair correlations of zeta
  • Explicit formula and Voronoi summation
  • Partition problems
  • Modular forms and Hecke operators
  • Selberg trace formula
Fall 2020 · The Geometry of Numbers

Topics

  • Lattices and lattice points in convex bodies
  • Minkowski’s theorem
  • Covering minima
  • Lenstra–Lenstra–Lovász algorithm
  • Reduction of quadratic and cubic forms
  • Leech lattice and sphere packing
  • Kissing numbers
  • Heights of subspaces and Siegel’s lemma
Winter 2019 · Elliptic Curves

Topics

  • in Number Theory: Elliptic Curves J. X. H. Wang
  • List of Topics
  • Weierstrass form
  • Isogenies and dual isogenies
  • Differential forms
  • Elliptic curves over C and F_p
  • Formal groups
  • Elliptic curves over local fields
  • Elliptic curves over global fields (Mordell-Weil)
  • Arithmetic statistics
Sources