Math 702 schedule

Monday Wednesday Friday
Sept. 3

In Lecture:
  • No Class!
Sept. 5

In Lecture:
  • No Class!
Sept. 7

In Lecture:
  • Algebraic numbers, algebraic integers and transcendental numbers

Associated Reading:
  • Lang, Chapter I, sections 1.1 and 1.2
  • Hardy and Wright, section 11.7
  • Samuel, chapter 2, sections 2.1 to 2.5

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Sept. 10

In Lecture:
  • Norms and Traces
  • Integral closures
  • Prime ideals and maximal ideals

Associated Reading:
  • Lang, Chapter I, sections 1.2, 1.3, 1.4, 1.5.
  • Samuel, chapter 2.

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Sept. 12

In Lecture:
  • Integral closures, continued
  • Traces and norms, finiteness of integral closures, continued
  • Dual bases

Associated Reading:
  • Lang, chapter I, sections 1.2, 1.3, 1.4, 1.5
  • Lang, chapter 3, section 3.1.
  • Samuel, chapter 2.

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Sept. 14

In Lecture:

No class - next class will be Monday, Sept. 17.
Sept. 17

In Lecture:
  • Discriminants

Associated Reading:
  • Lang, Chapter I, sections 1.2, 1.3, 1.4, 1.5.
  • Samuel, chapter 2.

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Sept. 19

In Lecture:
  • Discriminants, continued
  • Unramified extensions of rings

Associated Reading:
  • Lang, chapter I, sections 1.2, 1.3, 1.4, 1.5
  • Lang, chapter 3, section 3.1.
  • Samuel, chapter 2.

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Sept. 21

In Lecture:

No class - next class will be Friday, Sept. 28
Sept. 24

In Lecture:

No class - next class will be Friday, Sept. 28
Sept. 26

In Lecture:

No class - next class will be Friday, Sept. 28
Sept. 28

In Lecture:
  • Rings of cyclotomic integers
  • Statement of Kronecker Weber Theorem, Hilbert's 12th problem

Associated Reading:
  • Samuel, chapter 2, section 2.9
  • Samuel, chapter 3.
  • Lang, chapter 4, section 4.1.
  • Lang, chapter 1, section 1.6

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Oct. 1

In Lecture:
  • Drinfeld modules

Associated Reading:
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Oct. 3

In Lecture:
  • Constructing abelian extensions of imaginary quadratic fields via elliptic functions.
  • Definition of Dedekind rings
  • Integral closures of Dedekind rings in finite separable extensions of their fraction fields

Associated Reading:
  • The book Elliptic functions and rings of integers by Ph. Cassou-Nogues and M. J. Taylor.
  • Washington, chapters 1 and 2.
  • Samuel, chapter 2 section 2.9 and Chapter 3.
  • Samuel, chapter 5, sections 5.1 and 5.2.
  • Lang, chapter 1, section 1.6.
  • Hartshorne, chapters 1.6 and 2.6.

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Oct. 5

In Lecture:
  • Fractional ideals of Dedekind rings form a group
  • Ideal class groups, Dedekind subrings of function fields of curves.
  • Decomposition of primes

Associated Reading:
  • Washington, chapters 1 and 2.
  • Hartshorne, chapters 1.6 and 2.6
  • Samuel, chapter 5, sections 5.1 and 5.2.
  • Lang, chapter 1, section 1.6.

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Oct. 8

In Lecture:
  • Completion of the proof that the fractional ideals of a Dedekind ring form a group

Associated Reading:
  • Samuel, chapter 5, sections 5.1 and 5.2.
  • Lang, chapter 1, section 1.6, 1.7, 1.8

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Oct. 10

In Lecture:
  • Decomposition of prime ideals in finite separable extensions

Associated Reading:
  • Samuel, chapter 5, sections 5.1 and 5.2.
  • Lang, chapter 1, section 1.6, 1.7, 1.8

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Oct. 12

In Lecture:
  • Decomposition of prime ideals in Galois extensions
  • Decomposition and Inertia groups

Associated Reading:
  • Samuel, chapter 5, sections 5.1 and 5.2.
  • Lang, chapter 1, section 1.6, 1.7, 1.8

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Oct. 15

In Lecture:
  • Decomposition of primes in extensions generated by a single element
  • Valuations and discrete valuations.
  • Examples of discrete valuation rings
  • Discrete valuations of fraction fields of Dedekind rings

Associated Reading:
  • Lang, chapter 2

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Oct. 17

In Lecture:
  • Discrete valuations on higher dimensional function fields, connection with blow-ups
  • Absolute values on fields and completions of fields
  • The canonical absolute values of number fields and function fields
  • The product formula
  • Formal power series expressions for elements of completions.

Associated Reading:
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Oct. 19

In Lecture:
  • More on formal power series for elements of completions
  • Pictures of Z_p
  • When I-adic completions of rings are compact and totally disconnected.

Associated Reading:
  • Lang, Chapter 2

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Oct. 22

In Lecture:
  • No class - fall break!
Oct. 24

In Lecture:
  • Adeles of global fields
  • Weak and Strong approximation theorems

Associated Reading:
  • Lang, Chapter 2

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Oct. 26

In Lecture:
  • No class
Oct. 29

In Lecture:
  • No class - Hurricaine Sandy!
Oct. 31

In Lecture:
  • Minkowski's Lemma
  • End of the proof of the strong approximation theorem for rings of integers

Associated Reading:
  • Lang, Chapter 2

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Nov. 2

In Lecture:
  • No class
Nov. 5

In Lecture:
  • No class
Nov. 7

In Lecture:
  • Quasi-crystals and the strong approximation theorem
  • Weak and strong approximation theorem for algebraic groups.

Associated Reading:
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Nov. 9

In Lecture:
  • The Riemann Roch theorem on curves, and strong approximation

Associated Reading:
  • Hartshorne, "Algebraic Geometry", Chapter IV.
  • Lang, chapter 2

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Nov. 12

In Lecture:
  • The Riemann Roch theorem on curves, continued
  • Finiteness of Pic^0 for curves over finite fields.

Associated Reading:
  • Hartshorne, "Algebraic Geometry", Chapter IV.
  • Lang, chapter 2

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Nov. 14

In Lecture:
  • Riemann Roch and Goppa codes
  • Geometry of numbers
  • Finiteness of class numbers, finite generation of unit groups

Associated Reading:
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Nov. 16

No Class!
Nov. 19

No Class!
Nov. 21

No Class!
Nov. 23

No Class! - Thanksgiving!
Nov. 26

In Lecture:
  • Finiteness of ideal class groups (classical proof)

Associated Reading:
  • Lang, chapters 2 and 5

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Nov. 28

In Lecture:
  • Finiteness of class numbers (classical proof, completed)
  • Dirichlet unit theorem (classical proof)

Associated Reading:
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Video of class (downloadable)
Nov. 30

In Lecture:
  • Dirichlet unit theorem (classical proof, completed)

Associated Reading:
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Dec. 3

In Lecture:
  • Lenstra's approach to the unit theorem.
  • Examples of computing units
  • Continued fractions and real quadratic units
  • Arakelov theory in dimension one

Associated Reading:
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Dec. 5

In Lecture:
  • Arakelov theory in dimension 1, continued

Associated Reading:
  • Lang, chapters 2 and 5
  • The paper "Presentation de la Theorie d'Arakelov", by L. Szpiro, A.M.S. Contemporary Mathematics series no. 67.

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Dec. 7

In Lecture:
  • End of Arakelov theory in dimension 1.

Associated Reading:
  • Lang, chapters 2 and 5
  • The paper "Presentation de la Theorie d'Arakelov", by L. Szpiro, A.M.S. Contemporary Mathematics series no. 67.

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Last updated: 11/5/12
Send e-mail comments to: ted@math.upenn.edu