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Gauss sums (cf. [[Gauss sum|Gauss sum]]) were first introduced by J.L. Lagrange in his work on algebraic equations. They were subsequently named for and developed by C.F. Gauss in the "Disquisitiones" as part of his work on the division of the circle. Since then, the study of special exponential sums has been synonymous with the study of cyclotomy. In the 19th century other exponential sums were studied in connection with the arithmetic of cyclotomic fields (cf. [[Cyclotomic field|Cyclotomic field]]), most notably Jacobi sums, which are intimately connected with the multiplicative structure of finite fields (cf. also [[Finite field|Finite field]]). Exponential sums capture the relation between additive and multiplicative properties of finite fields and, together with the language of [[Harmonic analysis|harmonic analysis]], provide a very powerful tool in many applications, particularly communication science and coding theory.

Other types of sums that have arisen in a natural way from the study of additive problems are the so-called Kloosterman sums, which appear in the study of partitions of integers and in the more general Hardy–Littlewood–Ramanujan methods. From the point of view of harmonic analysis, Kloosterman sums can be viewed as Fourier transforms of Gauss sums.

In the 20th century, the most significant breakthrough in the study of exponential sums came from the application of algebraic-geometric methods. First, H. Hasse discovered a close connection between bounds for certain cubic exponential sums that generalize quadratic Gauss sums and the number of rational points on certain elliptic curves (cf. also [[Elliptic curve|Elliptic curve]]). The formulation of this relation given by E. Artin establishes a correspondence between bounds for exponential sums and the so-called Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) for function fields in the hyper-elliptic case. The subsequent developments in [[Algebraic geometry|algebraic geometry]], leading to Weil's proof of the Riemann hypothesis for algebraic curves over finite fields, resulted in renewed interest in the study of exponential sums.

A. Weil was the first to realize that many sums, including Gauss and Kloosterman sums, could be viewed as constructions over the projective line. For example, if  $  f ( x ) $
is a rational function on the projective line  $  \mathbf P $, 
i.e., the quotient of two polynomials, then the sum

$$ 
S ( f ) = \sum _ {\chi \in \mathbf F _ {q} } e ^ {2 \pi i \cdot { {( { \mathop{\rm tr} } f ( q ) ) } / p } } ,
$$

where the poles of  $  f $
have been excluded, has a representation as a trace, namely as a sum of the Frobenius characteristic roots:

$$ 
S ( f ) = \alpha _ {1} + \dots + \alpha _ {D} ,
$$

where  $  \{ \alpha _ {i} \} $
is a subset of all the roots of the [[Zeta-function|zeta-function]] of the Artin–Schreier covering (cf. also [[Artin–Schreier code|Artin–Schreier code]]; [[Artin–Schreier theory|Artin–Schreier theory]])

$$ 
{\mathcal C}: Y  ^ {p} - Y = f ( x ) .
$$

This includes the special case where  $  f $
is a polynomial of degree  $  n $(
and where the characteristic of the finite field does not divide  $  n $) 
and the integer  $  D $
can then be calculated exactly as  $  n - 1 $. 
Using the bound  $  | {\alpha  ^ {i} } | = q ^ {1/2 } $(
Weil's Riemann hypothesis), one obtains

$$ 
\left | {S ( f ) } \right | \leq  ( n - 1 ) \sqrt q .
$$

==Exponential sums of Weil–Bombieri type.==
The methods of Weil were later studied by E. Bombieri, who not only saw how to replace the projective line by more general algebraic curves, but began the systematic study of exponential sums in several variables using Artin–Schreier coverings (see [[#References|[a1]]] and [[Artin–Schreier theory|Artin–Schreier theory]]). Bombieri's estimates were strengthened and applied to the solution of several significant problems in coding theory (see [[#References|[a2]]]).

In a typical application of the Riemann hypothesis, one starts with an expression such as

$$ 
S ( f ) = \alpha _ {1} + \dots + \alpha _ {D} .
$$

It happens in many cases, especially in those that arise in the application of codes over the binary field, that the complex numbers  $  \alpha _ {i} $
not only satisfy  $  | {\alpha _ {i} } | = q ^ {1/2 } $, 
but also appear in complex-conjugate pairs. Furthermore, as algebraic integers, they are stable under the action of the Galois group of the rational numbers. This realization, first observed by J.-P. Serre, permits a finer utilization of the inequality of arithmetic-geometric means to obtain better Archimedean bounds for exponential sums than those found by Weil and Bombieri. A typical improvement has the following form: If  $  f $
is a polynomial of odd degree  $  d $, 
in which case the genus of the Artin–Schreier covering  $  Y  ^ {2} - Y = f ( x ) $
is  $  { {( d - 1 ) } / 2 } $, 
then

$$ 
\left | {\sum _ {x \in \mathbf F _ {2  ^ {f}  } } ( - 1 ) ^ { { \mathop{\rm tr} } f ( x ) } } \right | \leq  \left ( {
\frac{d - 1 }{2}
 } \right ) [ 2 \sqrt {2  ^ {f} } ] .
$$

This is a remarkable bound. It has application to coding theory (see [[#References|[a5]]]) and improvements to it were suggested by coding theory. The exponential sum over the binary field has integer values and very often it also has divisibility properties that permit further improvements. In pursuing these ideas further, one is lead to consider, in addition to the Archimedean bounds,  $  p $-
adic bounds that reflect the divisibility properties that the sums satisfy.

To carry these ideas further, another great theme in the study of exponential sums has been incorporated, namely the Chevalley–Warning theorem. The methodology can best be described as follows.

By using improved versions of the Chevalley–Warning theorem and the Ax theorem (see [[#References|[a3]]]), sharp bounds for the divisibility of the values of exponential sums over different field extensions are obtained. In turn, these are transferred, by arguments using formulas of Newton type, to divisibility properties of the roots of the [[L-function| $  L $-
function]] associated with the exponential sum. Armed with this new information, one can pull off some further powers from the roots of the zeta-function without destroying their integral nature. Subsequently, Serre's old procedure can be applied to derive bounds.

These methods have many applications. One example is as follows. Let  $  f $
be a polynomial of degree  $  7 $. 
Then (see [[#References|[a6]]] for examples)

$$ 
\left | {\sum _ {\chi \in \mathbf F _ {2 ^ {3m }  } } ( - 1 ) ^ { { \mathop{\rm tr} } f ( x ) } } \right | \leq  6 \cdot 2 ^ {m - 1 } [ 2 ^ {1 - m } \sqrt q ] .
$$

==A  $  p $-adic Serre bound.==
Looking at the original 1949 formulation of the Weil conjectures, one sees from the structure of the zeta-function of a non-singular algebraic variety as a kind of Euler–Poincaré characteristic that an [[Algebraic variety|algebraic variety]] of dimension  $  d $
has an internal skeleton very close to projective  $  d $-
dimensional space. More precisely, the difference between the number of rational points on  $  X $
over  $  \mathbf F _ {q  ^ {m}  } $
and on the projective space  $  \mathbf P  ^ {d} $
is given by a formula of the type

$$ 
\# X ( \mathbf F _ {q  ^ {m}  } ) - \# \mathbf P  ^ {d} ( \mathbf F _ {q  ^ {m}  } ) =
$$

$$ 
= 
\sum _ { 0 \leq  i \leq  2d } ( - 1 )  ^ {i} \left ( \sum _ {j =1 } ^ { {B _ i } } \omega _ {ij }   ^ {m} \right ) ,
$$

where, for  $  i $
even,  $  B _ {i} $
is the dimension of the primitive part of the  $  i $
th [[Cohomology group|cohomology group]]  $  H  ^ {i} $
and, for  $  i $
odd,  $  B _ {i} = { \mathop{\rm dim} } H  ^ {i} $. 
The bound  $  | {\omega _ {ij }  } | = q ^ { {{d i } / 2 } } $
is, of course, the Riemann hypothesis for non-singular projective algebraic varieties.

One of the principal aims of [[#References|[a5]]] and [[#References|[a6]]] is to describe a development of Serre's ideas called a  $  p $-
adic Serre bound. A surprising consequence of this development is that some of the inner sums in the above formula are empty, in the sense that the modified Betti numbers are zero (cf. [[Betti number|Betti number]]). More precisely (see [[#References|[a4]]], [[#References|[a3]]] and [[#References|[a6]]]), let  $  V \subset  \mathbf P  ^ {N} $
be a smooth projective variety (cf. [[Projective scheme|Projective scheme]]) defined by the vanishing of  $  r $
polynomials  $  F _ {1} \dots F _ {r} \in \mathbf F _ {q} [ x _ {0} \dots x _ {N} ] $. 
Let  $  \mu $
be the integer defined by

$$ 
\mu =  \max  \left ( f \left \lceil  {
\frac{N + 1 - \sum _ {i = 1 } ^ { r }  { \mathop{\rm deg} } ( F _ {i} ) }{ \max  _ {1 \leq  i \leq  r } \{ { \mathop{\rm deg} } ( F _ {i} ) \} }
 } \right \rceil , \right .
$$

$$ 
\left . \left [ f \left ( {
\frac{N + 1 - \sum _ {i = 1 } ^ { r }  w _ {p} ( F _ {i} ) }{ \max  _ {1 \leq  i \leq  r } \{ w _ {p} ( F _ {i} ) \} }
 } \right ) \right ] \right ) .
$$

Then, for  $  i $
odd, each eigenvalue  $  \alpha $
of  $  { \mathop{\rm Frob} } _ {q} $
acting on the [[Etale cohomology|étale cohomology]]  $  H  ^ {i} ( {\overline{V}\; } , \mathbf Q _ {l} ) $
is divisible by  $  p  ^  \mu  $. 
Consequently,  $  \alpha =0 $
whenever  $  2 \mu \geq  i f $. 
Furthermore, for the trace one has

$$ 
\left | { { \mathop{\rm tr} } ( { \mathop{\rm Frob} } _ {q} ) } \right | \leq  {
\frac{b _ {i} }{2}
 } p  ^  \mu  [ 2 \cdot q ^ {i/2 } p ^ {- \mu } ] .
$$

(Here,  $  b _ {i} = { \mathop{\rm dim} } H  ^ {i} ( {\overline{V}\; } , \mathbf Q _ {l} ) $; 
for the  $  w _ {p} ( F _ {i} ) $.) 
For  $  i $
even, each eigenvalue  $  \alpha $
of  $  { \mathop{\rm Frob} } _ {q} $
acting on the primitive cohomology  $  { \mathop{\rm Prim} }  ^ {i} ( {\overline{V}\; } , \mathbf Q _ {l} ) $
is divisible by  $  p  ^  \mu  $. 
Consequently,  $  \alpha = 0 $
whenever  $  2 \mu \geq  i f $.

Further developments of the ideas described above work very well in situations that are geometrically very similar to curves, e.g., complete intersections.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Bombieri, "On exponential sums in finite fields" ''Amer. J. Math.'' , '''88''' (1966) pp. 71–105</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.J. Moreno, O. Moreno, "An improved Bombieri–Weil bound and applications to coding theory" ''J. Number Th.'' , '''42''' : 1 (1992) pp. 32–46</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> O. Moreno, C.J. Moreno, "An elementary proof of a partial improvement to the Ax–Katz theorem" G. Cohen (ed.) T. Mora (ed.) O. Moreno (ed.) , ''Proc. of Applied Algebra, Algebraic Algorithms and Error–Correcting Codes--AAECC-10'' , ''Lecture Notes in Computer Science'' , '''673''' , Springer (1993) pp. 257–268</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Moreno, C.J. Moreno, "Improvements of the Chevalley–Warning and the Ax–Katz theorems" ''Amer. J. Math.'' , '''117''' : 1 (1995) pp. 241–244</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> O. Moreno, C.J. Moreno, "The MacWilliams–Sloane conjecture on the tightness of the Carlitz–Uchiyama bound and the weights of duals of BCH codes" ''IEEE Trans. Inform. Theory'' , '''40''' : 6 (1994) pp. 1894–1907</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> O. Moreno, C.J. Moreno, "A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110720/b11072068.png" />-adic Serre bound" ''preprint'' (??)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. Weil, "On some exponential sums" ''Proc. Nat. Acad. Sci. USA'' , '''34''' (1948) pp. 204–207 {{MR|0027006}} {{ZBL|0032.26102}} </TD></TR></table>