Topics in Algebraic Number Theory
This is the webpage for the course Topics in Algebraic Number Theory (4373TOANT), taught at Leiden in the spring of 2021 by Peter Stevenhagen, Mike Daas, and Jan Vonk.
The course consist roughly of two parts:
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The foundations of the theory of local fields, following notes by Stevenhagen.
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An introduction to the theory of p-adic L-functions, following notes by Vonk.
We will meet weekly on Thursdays, 9.30am - 1pm. Everything will take place on Zoom. Registered students will receive the login details via email,
and can also access the meetings through Brightspace. If you are having problems registering, or if you would like to audit the course without registering officially,
please contact the lecturers via email to be added to the mailing list.
Lectures
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4 February 2021: (Start at 10am)
Motivation and overview / Valued fields (Jan Vonk) Notes
Homework: (Due 10 Feb) Two problems chosen from chapter 1, exercises 1-10.
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11 February 2021: (Start at 10am)
Valued fields / Complete fields (Peter Stevenhagen)
Homework: (Due 17 Feb) Two problems chosen from chapter 1, exercises 10-20.
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18 February 2021:
Complete fields (Jan Vonk) Notes
Homework: (Due 25 Feb) Two problems chosen from chapter 2, exercises 10-24.
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25 February 2021: (Start at 10am)
Extending valuations (Peter Stevenhagen)
Homework: (Due 4 March) Two problems chosen from chapter 3, exercises 1-10.
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4 March 2021:
Extensions of local fields (Jan Vonk) Notes
Homework: (Due 11 March) Two problems from chapter 3 (ex. 11-18) / chapter 4 (ex. 1-14).
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11 March 2021: (Start at 10am)
Galois theory of valued fields (Peter Stevenhagen)
Homework: (Due 18 March) Two problems chosen from chapter 5, exercises 1-15.
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18 March 2021:(Start at 10am)
The Kronecker-Weber theorem (Peter Stevenhagen)
Homework: (Due 25 March) Two problems chosen from all chapters.
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25 March 2021:
Lecture time devoted to exercises only.
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1 April 2021:
Easter Break, no lecture.
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8 April 2021:
Foundations of p-adic analysis (Jan Vonk) Notes
Homework: (Due 15 April) Two problems chosen from chapter 2.
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15 April 2021:
Foundations of p-adic analysis (Jan Vonk) Notes
Homework: (Due 22 April) Two problems chosen from chapter 2.
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22 April 2021:
Measures and Distributions (Jan Vonk) Notes
Homework: (Due 29 April) Two problems chosen from chapters 2-3.
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29 April 2021:
p-Adic L-functions (Jan Vonk) Notes
Homework: (Due 6 May) Two problems chosen from chapters 3-4.
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6 May 2021:
p-Adic L-functions (Jan Vonk)
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20 May 2021:
Class numbers of cyclotomic fields (Jan Vonk)
There will be weekly homework assignments for this course. Every Thursday after the lecture, a short list of problems will be announced on this webpage,
two of which are due to be handed in before the next lecture starts. You have free choice which two problems from the list you decide to hand in for grading each week.
Handing in your solutions can be done by sending an email to Mike Daas with your typeset solutions attached.
Recordings:
Lecture 1: (Missing)
Lecture 2: https://video.leidenuniv.nl/media/t/1_09p0jtrf
Lecture 3: https://video.leidenuniv.nl/media/t/1_x7bmaeul
Lecture 4: https://video.leidenuniv.nl/media/t/1_bot5vhau
Lecture 5: https://video.leidenuniv.nl/media/t/1_93lk3dqn
Lecture 6: https://video.leidenuniv.nl/media/t/1_o4qrfltw
Lecture 7: https://video.leidenuniv.nl/media/t/1_46pnkjnm
Lecture 8: https://video.leidenuniv.nl/media/t/1_71xvl053
Lecture 9: https://video.leidenuniv.nl/media/t/1_lv4frqlu
Lecture 10: https://video.leidenuniv.nl/media/t/1_n7exkfvb
Lecture 11: https://video.leidenuniv.nl/media/t/1_d1cmdokc
Lecture 12: https://video.leidenuniv.nl/media/t/1_7v03tbcg
Lecture 13: https://video.leidenuniv.nl/media/t/1_8ruj4glw