Dixon-Selberg summation over local number fields. (English) Zbl 0780.11062
The authors present and prove a character sum analog of Selberg’s multidimensional beta function integral that is valid over any local completion field \(\mathbb{K}\) of \(\mathbb{Q}\) (\(\mathbb{K}\) is \(\mathbb{R}\) or a \(p\)-adic field \(\mathbb{Q}_ p\)) although if \(p=2\), it has only been proved for characters of rank less than 4:
\[
\int {{dx} \over {| x(1-x)|}} \int {{dy} \over {| y(1-y)|}} \pi_ 1(xy)\pi_ 2((1-x)(1- y))\pi^ 2_ 3(x-y)= {\textstyle {1\over 2}} \sum_ \tau S(\pi_ 1,\pi_ 2,\pi_ 3,\pi_ \tau)
\]
where
\[
S(\pi_ 1,\pi_ 2,\pi_ 3)= {{\Gamma(\pi_ 0 \pi_ 3^ 2) \Gamma(\pi_ 1)\Gamma(\pi_ 1,\pi_ 3)\Gamma(\pi_ 2)\Gamma (\pi_ 2\pi_ 3)} \over {\Gamma(\pi_ 0\pi_ 3) \Gamma(\pi_ 1\pi_ 2\pi_ 3) \Gamma(\pi_ 1\pi_ 2\pi_ 3^ 2)}}
\]
and
\[
\Gamma(\pi)=\int_ \mathbb{K} {{dx} \over {| x|}}\chi(x)\pi(x).
\]
Here \(\chi(x)\) is an additive character, \(\pi_ 0(x)= | x|\), \(\pi_ i(x)\) are arbitrary multiplicative characters. The sum is over the quadratic classes of the number field with \(\pi_ \tau\) the corresponding local quadratic character.
Reviewer: D.M.Bressoud (University Park)
MSC:
11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
11L05 | Gauss and Kloosterman sums; generalizations |
33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
Keywords:
character sum analog of Selberg’s multidimensional beta function integral; local completion field; \(p\)-adic fieldReferences:
[1] | Selberg A., Nor. Math. Tidsskr. 26 pp 71– (1944) |
[2] | Dixon A. C., Proc. London Math. Soc. 2 pp 8– (1904) |
[3] | DOI: 10.1137/0513070 · Zbl 0498.17006 · doi:10.1137/0513070 |
[4] | DOI: 10.1063/1.1704009 · Zbl 0133.45202 · doi:10.1063/1.1704009 |
[5] | DOI: 10.1016/S0550-3213(85)80004-3 · doi:10.1016/S0550-3213(85)80004-3 |
[6] | Evans R. J., Enseign. Math. 27 pp 197– (1981) |
[7] | DOI: 10.1090/S0273-0979-1981-14930-2 · Zbl 0471.10028 · doi:10.1090/S0273-0979-1981-14930-2 |
[8] | DOI: 10.1016/0370-2693(87)91357-8 · doi:10.1016/0370-2693(87)91357-8 |
[9] | DOI: 10.1142/S0217732389001970 · doi:10.1142/S0217732389001970 |
[10] | DOI: 10.1142/S0217732389001970 · doi:10.1142/S0217732389001970 |
[11] | DOI: 10.1142/S0217732389001970 · doi:10.1142/S0217732389001970 |
[12] | DOI: 10.1142/S0217732389001970 · doi:10.1142/S0217732389001970 |
[13] | DOI: 10.1142/S0217732389001970 · doi:10.1142/S0217732389001970 |
[14] | DOI: 10.1063/1.528468 · Zbl 0709.22011 · doi:10.1063/1.528468 |
[15] | Davenport H., J. Reine Angew. Math. 172 pp 151– (1935) |
[16] | DOI: 10.1112/blms/5.3.325 · Zbl 0269.10020 · doi:10.1112/blms/5.3.325 |
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