Pries: Algebraic Number Theory Projects.
There are a wide range of possible projects to choose. Here are a couple topics that are
interesting. I’m happy to help you choose a topic which matches your interests.
Quality is more important than quantity but you could aim for a 15 minute talk and an 810 page paper. Make sure to have an introduction, theory/proofs, examples, pictures/graphs
as appropriate, and bibliography.
1. Reciprocity laws and Gauss sums
(a) Gauss and Jacobi sums, proof of quadratic reciprocity, number of solutions to
x21 + x22 + . . . x2n = 1 or xn + y n = 1 in Fp .
(b) Cubic or biquadratic reciprocity.
(c) Eisenstein reciprocity law (proof uses Gauss sums and Stickelberger relation)
2. Lattices
(a) Dirichlet’s unit theorem. Units in cyclotomic fields.
(b) LLL algorithm
3. Topics with applications outside number theory.
(a) Bernouli numbers, regular primes, Fermat’s last theorem
(b) Quadratic forms, local global principal, Hasse principal
4. Analogy between number fields and function fields.
(a) Cyclotomic function fields, Carlitz modules, Drinfeld modules.
(b) ABC theorem (number field version and function field version).
(c) Artin’s conjecture (number field version and function field version).
(d) Class number formulas (number field version and function field version).
5. Zeta functions - generating series counting the number of prime ideals.
(a) Dedekind zeta functions (including the Riemann zeta function!) characterizing
primes in algebraic number fields.
(b) Zeta functions of equations defined over finite fields. Weil conjectures - number
of points of curve over finite field.
(c) Dirichlet L-functions. Dirichlet’s theorem of arithmetic progressions: (if gcd(a, n) =
1, then there are infinitely many primes p s.t. p ≡ a mod n).
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