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Filippo Alberto Edoardo
Filippo Alberto Edoardo
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I do not know if you are willing to restrict to abelian $K/\mathbb{Q}$: if not, please throw away my whole answer. But let me assume that your base field is any number field $F$, as it does not change anything: and, accordingly, that $K/F$ is abelian.

Suppose first that $K/F$ is a ray class field for some conductor $\mathfrak{f}$ (for $F=\mathbb{Q}$ it means $K$ is either $\mathbb{Q}(\zeta_f)$ or $\mathbb{Q}(\zeta_f)^+$ for some $f$). Then you are simply considering the partial zeta function $\zeta({\bf{1}},s)$ where $\bf{1}$ is the trivial class in the ray class group modulo $\mathfrak{f}$. Those are well-studied and can be interpreted as suitable Mellin transform, pretty much as the usual Dedekind zeta function and admits the usual analytic continuation (see J. Neukirch Algebraic Number Theory, Chapter VII, Section 5, theorem 5.5 for the Mellin transform statement, Theorem 5.9 for analytic continuation caveat: I only have the German version and I ignore if numbering has been changed in the translation). This settles also the case in which you consider some different splitting behaviour which would be controlled by class field theory pretty much the same way – i. e. your Euler product would be that of $\zeta,{\bf{c}},s$$\zeta({\bf{c}},s)$ for the class $\bf{c}$ in the ray class field containing precisely the integral ideals whose splitting time is your favorite one.

If now your abelian extension is only contained in a ray class field, you need to follow more closely Tate's thesis (see the last chapter of Cassels and Fröhlich's Algebraic Number Theory) and chose the right measure to put on the unit ball inside $\mathbb{Q}_p$ for the primes you want to throw away – I imagine that you should chose the measure giving volume $1$ to some suitable subgroup depending on your local extension, but I am not checking. I guess that this would not affect anything at all, because Tate's strategy is somehow ''formal'' in the sense that it just depends on some very nice property of adelic Fourier transform. As for a reference, I suggest D. Ramakrishan and R. Valenza Fourier Analysis on Number Fields.

Let me also observe that what you are asking for is ''natural'' in the sense that when looking at zeta functions as being suitable measures on Galois group, your Euler product at negative integers $k$ gives the special values of the measure agains the function $\chi_S|\cdot|^k$ where $\chi_S$ is the characteristic function of the set of primes $S$ seen as elements of the Galois group via class field theory – this is discussed very well in the introduction to Deligne and Ribet's paper about special values of $L$-functions at negative integers.

If your extension is not abelian, I think the situation is much more involved because it is not enough to play the same game simply on adeles group: and, as said before, you'd rather forget everything I have written...

I do not know if you are willing to restrict to abelian $K/\mathbb{Q}$: if not, please throw away my whole answer. But let me assume that your base field is any number field $F$, as it does not change anything: and, accordingly, that $K/F$ is abelian.

Suppose first that $K/F$ is a ray class field for some conductor $\mathfrak{f}$ (for $F=\mathbb{Q}$ it means $K$ is either $\mathbb{Q}(\zeta_f)$ or $\mathbb{Q}(\zeta_f)^+$ for some $f$). Then you are simply considering the partial zeta function $\zeta({\bf{1}},s)$ where $\bf{1}$ is the trivial class in the ray class group modulo $\mathfrak{f}$. Those are well-studied and can be interpreted as suitable Mellin transform, pretty much as the usual Dedekind zeta function and admits the usual analytic continuation (see J. Neukirch Algebraic Number Theory, Chapter VII, Section 5, theorem 5.5 for the Mellin transform statement, Theorem 5.9 for analytic continuation caveat: I only have the German version and I ignore if numbering has been changed in the translation). This settles also the case in which you consider some different splitting behaviour which would be controlled by class field theory pretty much the same way – i. e. your Euler product would be that of $\zeta,{\bf{c}},s$ for the class $\bf{c}$ in the ray class field containing precisely the integral ideals whose splitting time is your favorite one.

If now your abelian extension is only contained in a ray class field, you need to follow more closely Tate's thesis (see the last chapter of Cassels and Fröhlich's Algebraic Number Theory) and chose the right measure to put on the unit ball inside $\mathbb{Q}_p$ for the primes you want to throw away – I imagine that you should chose the measure giving volume $1$ to some suitable subgroup depending on your local extension, but I am not checking. I guess that this would not affect anything at all, because Tate's strategy is somehow ''formal'' in the sense that it just depends on some very nice property of adelic Fourier transform. As for a reference, I suggest D. Ramakrishan and R. Valenza Fourier Analysis on Number Fields.

Let me also observe that what you are asking for is ''natural'' in the sense that when looking at zeta functions as being suitable measures on Galois group, your Euler product at negative integers $k$ gives the special values of the measure agains the function $\chi_S|\cdot|^k$ where $\chi_S$ is the characteristic function of the set of primes $S$ seen as elements of the Galois group via class field theory – this is discussed very well in the introduction to Deligne and Ribet's paper about special values of $L$-functions at negative integers.

If your extension is not abelian, I think the situation is much more involved because it is not enough to play the same game simply on adeles group: and, as said before, you'd rather forget everything I have written...

I do not know if you are willing to restrict to abelian $K/\mathbb{Q}$: if not, please throw away my whole answer. But let me assume that your base field is any number field $F$, as it does not change anything: and, accordingly, that $K/F$ is abelian.

Suppose first that $K/F$ is a ray class field for some conductor $\mathfrak{f}$ (for $F=\mathbb{Q}$ it means $K$ is either $\mathbb{Q}(\zeta_f)$ or $\mathbb{Q}(\zeta_f)^+$ for some $f$). Then you are simply considering the partial zeta function $\zeta({\bf{1}},s)$ where $\bf{1}$ is the trivial class in the ray class group modulo $\mathfrak{f}$. Those are well-studied and can be interpreted as suitable Mellin transform, pretty much as the usual Dedekind zeta function and admits the usual analytic continuation (see J. Neukirch Algebraic Number Theory, Chapter VII, Section 5, theorem 5.5 for the Mellin transform statement, Theorem 5.9 for analytic continuation caveat: I only have the German version and I ignore if numbering has been changed in the translation). This settles also the case in which you consider some different splitting behaviour which would be controlled by class field theory pretty much the same way – i. e. your Euler product would be that of $\zeta({\bf{c}},s)$ for the class $\bf{c}$ in the ray class field containing precisely the integral ideals whose splitting time is your favorite one.

If now your abelian extension is only contained in a ray class field, you need to follow more closely Tate's thesis (see the last chapter of Cassels and Fröhlich's Algebraic Number Theory) and chose the right measure to put on the unit ball inside $\mathbb{Q}_p$ for the primes you want to throw away – I imagine that you should chose the measure giving volume $1$ to some suitable subgroup depending on your local extension, but I am not checking. I guess that this would not affect anything at all, because Tate's strategy is somehow ''formal'' in the sense that it just depends on some very nice property of adelic Fourier transform. As for a reference, I suggest D. Ramakrishan and R. Valenza Fourier Analysis on Number Fields.

Let me also observe that what you are asking for is ''natural'' in the sense that when looking at zeta functions as being suitable measures on Galois group, your Euler product at negative integers $k$ gives the special values of the measure agains the function $\chi_S|\cdot|^k$ where $\chi_S$ is the characteristic function of the set of primes $S$ seen as elements of the Galois group via class field theory – this is discussed very well in the introduction to Deligne and Ribet's paper about special values of $L$-functions at negative integers.

If your extension is not abelian, I think the situation is much more involved because it is not enough to play the same game simply on adeles group: and, as said before, you'd rather forget everything I have written...

Source Link
Filippo Alberto Edoardo
Filippo Alberto Edoardo
  • 5.7k
  • 2
  • 36
  • 53

I do not know if you are willing to restrict to abelian $K/\mathbb{Q}$: if not, please throw away my whole answer. But let me assume that your base field is any number field $F$, as it does not change anything: and, accordingly, that $K/F$ is abelian.

Suppose first that $K/F$ is a ray class field for some conductor $\mathfrak{f}$ (for $F=\mathbb{Q}$ it means $K$ is either $\mathbb{Q}(\zeta_f)$ or $\mathbb{Q}(\zeta_f)^+$ for some $f$). Then you are simply considering the partial zeta function $\zeta({\bf{1}},s)$ where $\bf{1}$ is the trivial class in the ray class group modulo $\mathfrak{f}$. Those are well-studied and can be interpreted as suitable Mellin transform, pretty much as the usual Dedekind zeta function and admits the usual analytic continuation (see J. Neukirch Algebraic Number Theory, Chapter VII, Section 5, theorem 5.5 for the Mellin transform statement, Theorem 5.9 for analytic continuation caveat: I only have the German version and I ignore if numbering has been changed in the translation). This settles also the case in which you consider some different splitting behaviour which would be controlled by class field theory pretty much the same way – i. e. your Euler product would be that of $\zeta,{\bf{c}},s$ for the class $\bf{c}$ in the ray class field containing precisely the integral ideals whose splitting time is your favorite one.

If now your abelian extension is only contained in a ray class field, you need to follow more closely Tate's thesis (see the last chapter of Cassels and Fröhlich's Algebraic Number Theory) and chose the right measure to put on the unit ball inside $\mathbb{Q}_p$ for the primes you want to throw away – I imagine that you should chose the measure giving volume $1$ to some suitable subgroup depending on your local extension, but I am not checking. I guess that this would not affect anything at all, because Tate's strategy is somehow ''formal'' in the sense that it just depends on some very nice property of adelic Fourier transform. As for a reference, I suggest D. Ramakrishan and R. Valenza Fourier Analysis on Number Fields.

Let me also observe that what you are asking for is ''natural'' in the sense that when looking at zeta functions as being suitable measures on Galois group, your Euler product at negative integers $k$ gives the special values of the measure agains the function $\chi_S|\cdot|^k$ where $\chi_S$ is the characteristic function of the set of primes $S$ seen as elements of the Galois group via class field theory – this is discussed very well in the introduction to Deligne and Ribet's paper about special values of $L$-functions at negative integers.

If your extension is not abelian, I think the situation is much more involved because it is not enough to play the same game simply on adeles group: and, as said before, you'd rather forget everything I have written...