Kloosterman sums for prime powers in P-adic fields

Stanley J. Gurak[1]

  • [1] University of San Diego Alcala Park San Diego, CA 92110, USA

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 175-201
  • ISSN: 1246-7405

Abstract

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Let K be a field of degree n over Q p , the field of rational p -adic numbers, say with residue degree f , ramification index e and differential exponent d . Let O be the ring of integers of K and P its unique prime ideal. The trace and norm maps for K / Q p are denoted T r and N , respectively. Fix q = p r , a power of a prime p , and let η be a numerical character defined modulo q and of order o ( η ) . The character η extends to the ring of p -adic integers ℤ p in the natural way; namely η ( u ) = η ( u ˜ ) , where u ˜ denotes the residue class of u modulo q , and similarly for the root of unity ζ q u = e x p ( 2 π i u ˜ / q ) . Fix a positive integer γ ≥ r e - d for which N ( 1 + P γ ) ⊆ 1 + q ℤ p so that the (twisted) Kloosterman sums R ( η , z ) = ∑ α ∈ ( O / P γ ) * η ( N α ) ζ q T r α + z / N α ( z ∈ Z / q Z * ) are well-defined.Saliè explicitly determined R ( η , z ) in the classical case n = 1 (so K = Q p ) for q = p r with r > 1 and o ( η ) = 1 or 2 . Here I generalize Saliè’s result for the general case n > 1 for characters η with o ( η ) | p - 1 (also o ( η ) = 2 when p = 2 ), and for all γ ≥ r e - d > 1 but for a few small exceptional values r . My evaluation relies on the author’s recent explicit determination of Gauss sums for prime powers in p -adic fields and exponential sums of the form ∑ χ ( x ) a x ζ q b x .

How to cite

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Gurak, Stanley J.. "Kloosterman sums for prime powers in P-adic fields." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 175-201. <http://eudml.org/doc/10870>.

@article{Gurak2009,
abstract = {Let $K$ be a field of degree $n$ over $\{\bf Q\}_\{p\}$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\{\it P\}$ its unique prime ideal. The trace and norm maps for $K/\{\bf Q\}_\{p\}$ are denoted $Tr$ and $N$, respectively. Fix $q=p^\{r\}$, a power of a prime $p$, and let $\eta $ be a numerical character defined modulo $q$ and of order $o(\eta )$. The character $\eta $ extends to the ring of $p$-adic integers $\mathbb\{Z\}_p$ in the natural way; namely $\eta (u)=\eta (\tilde\{u\})$, where $\tilde\{u\}$ denotes the residue class of $u$ modulo $q$, and similarly for the root of unity $\zeta _\{q\}^\{u\}=exp(2\pi i \tilde\{u\}/q)$. Fix a positive integer $\gamma \ge re-d$ for which $N(1+\{\it P\}^\{\gamma \}) \subseteq 1 + q\{\mathbb\{Z\}_p\}$ so that the (twisted) Kloosterman sums\[ R(\eta ,z)=\sum \_\{\alpha \in (O/\{\it P\}^\{\gamma \})^\{*\}\} \eta (N\alpha ) \zeta \_\{q\}^\{Tr \; \alpha + z/N\alpha \} \;\;\;\;\;(z \in \{\bf Z\}/q\{\bf Z\}^\{*\} )\]are well-defined.Saliè explicitly determined $R(\eta ,z)$ in the classical case $n=1$ (so $K=\{\bf Q\}_\{p\}$) for $q=p^\{r\}$ with $r&gt;1$ and $o(\eta )=1$ or $2$. Here I generalize Saliè’s result for the general case $n &gt; 1$ for characters $\eta $ with $o(\eta )|p-1$ (also $o(\eta )=2$ when $p=2$), and for all $\gamma \ge re-d &gt;1$ but for a few small exceptional values $r$. My evaluation relies on the author’s recent explicit determination of Gauss sums for prime powers in $p$-adic fields and exponential sums of the form $\sum \chi (x)^\{ax\}\zeta _\{q\}^\{bx\}$.},
affiliation = {University of San Diego Alcala Park San Diego, CA 92110, USA},
author = {Gurak, Stanley J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Kloosterman sums; -adic fields},
language = {eng},
number = {1},
pages = {175-201},
publisher = {Université Bordeaux 1},
title = {Kloosterman sums for prime powers in P-adic fields},
url = {http://eudml.org/doc/10870},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Gurak, Stanley J.
TI - Kloosterman sums for prime powers in P-adic fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 175
EP - 201
AB - Let $K$ be a field of degree $n$ over ${\bf Q}_{p}$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and ${\it P}$ its unique prime ideal. The trace and norm maps for $K/{\bf Q}_{p}$ are denoted $Tr$ and $N$, respectively. Fix $q=p^{r}$, a power of a prime $p$, and let $\eta $ be a numerical character defined modulo $q$ and of order $o(\eta )$. The character $\eta $ extends to the ring of $p$-adic integers $\mathbb{Z}_p$ in the natural way; namely $\eta (u)=\eta (\tilde{u})$, where $\tilde{u}$ denotes the residue class of $u$ modulo $q$, and similarly for the root of unity $\zeta _{q}^{u}=exp(2\pi i \tilde{u}/q)$. Fix a positive integer $\gamma \ge re-d$ for which $N(1+{\it P}^{\gamma }) \subseteq 1 + q{\mathbb{Z}_p}$ so that the (twisted) Kloosterman sums\[ R(\eta ,z)=\sum _{\alpha \in (O/{\it P}^{\gamma })^{*}} \eta (N\alpha ) \zeta _{q}^{Tr \; \alpha + z/N\alpha } \;\;\;\;\;(z \in {\bf Z}/q{\bf Z}^{*} )\]are well-defined.Saliè explicitly determined $R(\eta ,z)$ in the classical case $n=1$ (so $K={\bf Q}_{p}$) for $q=p^{r}$ with $r&gt;1$ and $o(\eta )=1$ or $2$. Here I generalize Saliè’s result for the general case $n &gt; 1$ for characters $\eta $ with $o(\eta )|p-1$ (also $o(\eta )=2$ when $p=2$), and for all $\gamma \ge re-d &gt;1$ but for a few small exceptional values $r$. My evaluation relies on the author’s recent explicit determination of Gauss sums for prime powers in $p$-adic fields and exponential sums of the form $\sum \chi (x)^{ax}\zeta _{q}^{bx}$.
LA - eng
KW - Kloosterman sums; -adic fields
UR - http://eudml.org/doc/10870
ER -

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