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. - 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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