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I am not getting the definition of Conductor of a ray class field. I know the following definition Let $K$ be a number field. The theorems of class field theory tell us that given any modulus $\mathfrak{m}$ for $K$, there is a unique Abelian extension $K_{\mathfrak{m}}$ such that the kernel of the Artin map of $K_{\mathfrak{m}}/K$ with respect to $\mathfrak{m}$ is precisely the subgroup of principal fractional ideals congruent to $1 \pmod{\mathfrak{m}}$. This is the Ray class field.

  1. Could any one give an idea of conductor of ray class field?
  2. How the conductor of ray class field are generated?
  3. or any material explaining this?

Thank you very much in advance

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    $\begingroup$ The conductor is attached to a general finite abelian extension, not specifically ray class fiels, and tells you the smallest ray class field containing the original abelian extension. $\endgroup$
    – KCd
    May 2, 2014 at 10:19
  • $\begingroup$ @KCd do you know any book or material explaining about this $\endgroup$
    – Math123
    May 4, 2014 at 13:47
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    $\begingroup$ Almost any book on class field theory would discuss this. You may need to read the text carefully to find the precise statement you are seeking. $\endgroup$
    – KCd
    May 4, 2014 at 15:17

2 Answers 2

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Ray class field modulo $\mathfrak{m}$ is the maximal abelian extension of conductor $\mathfrak{m}$. The ray class field of conductor $m=1$ is the Hilbert Class field. In particular Hilbert Class field is the ring class field of maximal order in general ring class field of an order of conductor $\mathfrak{m}$ is the intermediate between the Hilbert Class field and the ray class field of conductor $\mathfrak{m}$.

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    $\begingroup$ Books by Cox: Primes of the form $x^2+ny^2$, Stevenhagen's article which have come in MSRI vol 44 are very good references $\endgroup$
    – user152394
    May 22, 2014 at 5:45
  • $\begingroup$ Thank you very much. Its Really a good reference on these $\endgroup$
    – Math123
    May 22, 2014 at 5:59
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Maybe a bit late but "History of Class Field Theory" was quite enlightening to me. http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/cfthistory.pdf

Nancy Childress, "Class Field Theory" book is very clear and should help you getting throught this "conductor" business more easily.

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  • $\begingroup$ Have you really gone through nancy childress book and Conrad's lecture notes for the above doubt? $\endgroup$
    – Math123
    May 22, 2014 at 5:49
  • $\begingroup$ Conrad's lecture page 9-10 gives you the definition of conductors- I mentioned this paper for its very nice and short treatment of CFT, you might want to get more details elsewhere. Then, I remembered that Nancy Childress made a particular emphasis on conductors (for characters and so on). $\endgroup$
    – saradi
    May 22, 2014 at 19:40

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