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p-adic methods in number theory

For a description of the course, click here.

Schedule:
Mondays 10.00 - 12.00 hrs. in MV:L15 (Chalmers Johanneberg)
Wednesdays 10.00 - 12.00 hrs. in MV:L14 (Chalmers Johanneberg)

Bibliography:
N. Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions.
J.-P. Serre: A Course in Arithmetic (Part I).
J. Silverman, J. Tate: Rational Points on Elliptic Curves.
J. Silverman: The arithmetic of elliptic curves.

Homeworks:
Homework 1 (click here). To be handed in on April 18.
Homework 2 (click here). To be handed in on May 8.
Homework 3 (click here). To be handed in on May 23.
Homework 4 (click here). To be handed in on June 7.

Oral exams: June, 14 to 21.
To book a time slot for the oral exam, use the following doodle.

Lecture 1 (March 26):
- Normed fields
- Completions
- The p-adic norm in Q
- p-adic series and Q_p
Reference: Koblitz, Chapter 1. For the lecture notes, click here.

Lecture 2 (March 28):
- p-adic series and Q_p
- Non-Archimedean fields, definition and properties
- The ring of integers of a non-Archimedean field
Reference: Koblitz, Chapter 1. For the lecture notes, click here.

Lecture 3 (April 4):
- Reduction maps
- Z_p as a pro-finite group and compactness.
- Hensel's Lemma
Reference: Koblitz, Chapter 1 and Serre, Chapter 2. For the lecture notes, click here.

Lecture 4 (April 9):
- Applications of Hensel´s Lemma: squares and roots of unity in Qp.
- Local fields: definition and classification.
Reference: Koblitz, Chapter 1 and Serre, Chapter 2. For the lecture notes, click here.

Lecture 5 (April 11):
- General concepts on quadratic spaces.
- Orthogonal basis for quadratic spaces.
- Forms representing zero.
- Hasse-Minkowski's Theorem (case n=2).
- The Hilbert symbol.
Reference: Serre, Chapter 4. For the lecture notes, click here.

Lecture 6 (April 16):
- Hasse-Minkowski's Theorem (case n=3).
- Equivalence of quadratic forms.
- Hyperbolic planes.
- Sum of forms.
- Forms representing a number.
Reference: Serre, Chapter 4. For the lecture notes, click here.

Lecture 7 (April 18):
- Forms representing a number.
- Basic properties of the Hilbert symbol.
- Computation of the Hilbert symbol over R and Qp.
- Existence of rational numbers with given Hilbert symbols.
Reference: Serre, Chapters 3 and 4. For the lecture notes, click here.

Lecture 8 (April 23):
- Invariants associated to quadratic forms.
- Forms over Qp representing a number.
- Hasse-Minkowski's Theorem (the remaining cases).
Reference: Serre, Chapters 3 and 4. For the lecture notes, click here.

Lecture 9 (April 25):
- p-adic distributions and measures.
- p-adic integration.
Reference: Koblitz, Chapter 2. For the lecture notes, click here.

Lecture 10 (May 2):
- A crash course on the Riemann zeta function.
- The Bernoulli numbers.
- Kummer's Theorem.
- Bernoulli distributions and regularized Bernoulli distributions.
Reference: Koblitz, Chapter 2. For the lecture notes, click here.

Lecture 11 (May 4):
- Regularized Bernoulli distributions are measures.
- Generalities on p-adic interpolation.
- p-adic interpolation of f(s)=n^s for n=1 mod p.
Reference: Koblitz, Chapter 2. For the lecture notes, click here.

Lecture 12 (May 7):
- p-adic interpolation of f(s)=n^s for n coprime to p.
- Formal (termwise) interpolation of the Riemann zeta function.
- p-adic zeta functions and p-adic Mellin transforms.
Reference: Koblitz, Chapter 2. For the lecture notes, click here.

Lecture 13 (May 9):
- Proof of Kummer's Theorem.
- Clausen and von Staudt Theorems.
Reference: Koblitz, Chapter 2. For the lecture notes, click here.

Lecture 14 (May 14):
- Geometry of elliptic curves, group law. 
- Lattices and Weierstrass equations. 
- The Weil pairing. 
Reference: Silverman, Chapters 3 and 4. For the lecture notes, click here.

Lecture 15 (May 16):
- Weierstrass equations over Qp and Fp and reduction modulo p. 
- Torsion points and the structure of the p-adic points.
Reference: Silverman. For the lecture notes, click here.

Lecture 16 (May 21):
- Lutz-Nagell Theorem.
- Minimal Weierstrass models.
- Tate modules.
Reference: Silverman. For the lecture notes, click here.

Lecture 17 (May 23):
- Tate modules, degree of an isogeny, dual isogenies.
- Hasse-Weil bound for elliptic curves over finite fields.
- Hasse-Weil zeta functions.
Reference: Silverman. For the lecture notes, click here.
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HW1.pdf
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