Trace functions over finite fields: a study in sums of products E. - PowerPoint PPT Presentation

Trace functions over finite fields: a study in sums of products E. Kowalski ETH Z¨ urich May 29, 2014

Trace functions over finite fields: a study in sums of products E. Kowalski ETH Z¨ urich May 29, 2014 [Joint works with ´ E. Fouvry, Ph. Michel (and in part S. Ganguly, G. Ricotta); arXiv:1405.2293 ]

Trace functions The trace functions modulo a prime p are functions K : F p − → C which are “special” functions of algebraic nature.

Trace functions The trace functions modulo a prime p are functions K : F p − → C which are “special” functions of algebraic nature. ◮ Precisely, we consider trace functions of middle-extension ℓ -adic sheaves F on the affine line, pointwise pure of weight 0, brought to C by a fixed ι : ¯ Q ℓ − → C .

Trace functions The trace functions modulo a prime p are functions K : F p − → C which are “special” functions of algebraic nature. ◮ Precisely, we consider trace functions of middle-extension ℓ -adic sheaves F on the affine line, pointwise pure of weight 0, brought to C by a fixed ι : ¯ Q ℓ − → C . Such a trace function has a “complexity” c ( K ).

Trace functions The trace functions modulo a prime p are functions K : F p − → C which are “special” functions of algebraic nature. ◮ Precisely, we consider trace functions of middle-extension ℓ -adic sheaves F on the affine line, pointwise pure of weight 0, brought to C by a fixed ι : ¯ Q ℓ − → C . Such a trace function has a “complexity” c ( K ). ◮ We define c ( K ) as the minimum of c ( F ) over sheaves as above with trace function K , where � c ( F ) = rank ( F ) + | (sing. points) | + Swan x ( F ) ≥ 1 . x sing.

Trace functions The trace functions modulo a prime p are functions K : F p − → C which are “special” functions of algebraic nature. ◮ Precisely, we consider trace functions of middle-extension ℓ -adic sheaves F on the affine line, pointwise pure of weight 0, brought to C by a fixed ι : ¯ Q ℓ − → C . Such a trace function has a “complexity” c ( K ). ◮ We define c ( K ) as the minimum of c ( F ) over sheaves as above with trace function K , where � c ( F ) = rank ( F ) + | (sing. points) | + Swan x ( F ) ≥ 1 . x sing. We typically let p vary, and consider K p modulo p with bounded conductor: c ( K p ) ≤ C for all p .

Examples

Examples ◮ (Characters) K ( x ) = e ( f ( x ) / p ) or K ( x ) = χ ( f ( x )), where χ � = 1 is a multiplicative character, and f ∈ F p [ X ] is non-constant; the conductor is bounded in terms of deg( f ) only;

Examples ◮ (Characters) K ( x ) = e ( f ( x ) / p ) or K ( x ) = χ ( f ( x )), where χ � = 1 is a multiplicative character, and f ∈ F p [ X ] is non-constant; the conductor is bounded in terms of deg( f ) only; ◮ (Hyper)-Kloosterman sums: for r ≥ 1 integer 1 � y 1 + · · · + y r � � K ( x ) = Kl r ( x ) = e ; p ( r − 1) / 2 p y 1 ··· y r = x y i ∈ F p the conductor is bounded in terms of r only;

Examples ◮ (Characters) K ( x ) = e ( f ( x ) / p ) or K ( x ) = χ ( f ( x )), where χ � = 1 is a multiplicative character, and f ∈ F p [ X ] is non-constant; the conductor is bounded in terms of deg( f ) only; ◮ (Hyper)-Kloosterman sums: for r ≥ 1 integer 1 � y 1 + · · · + y r � � K ( x ) = Kl r ( x ) = e ; p ( r − 1) / 2 p y 1 ··· y r = x y i ∈ F p the conductor is bounded in terms of r only; ◮ (Point-counting) � K ( x ) = 1 − 1 , f ∈ F p [ X ] non-constant . y ∈ F p f ( y )= x the conductor is bounded in terms of deg( f ) only.

These functions occur in many applications in analytic number theory.

These functions occur in many applications in analytic number theory. Most often, one needs estimates for the “generalized exponential sums” of the type � K ( x ) , x ∈ F p

These functions occur in many applications in analytic number theory. Most often, one needs estimates for the “generalized exponential sums” of the type � K ( x ) , x ∈ F p or more naturally for inner products � K 1 ( x ) K 2 ( x ) . x ∈ F p of trace functions K 1 and K 2 .

Goals ◮ Square-root cancellation: � ≤ C √ p , � � � K 1 ( x ) K 2 ( x ) � � � x ∈ F p where C is under control (depends only on the complexity of K 1 and K 2 );

Goals ◮ Square-root cancellation: � ≤ C √ p , � � � K 1 ( x ) K 2 ( x ) � � � x ∈ F p where C is under control (depends only on the complexity of K 1 and K 2 ); ◮ Or understanding when this does not hold (“diagonal situations”), e.g., K 1 ( x ) = K 2 ( x ).

These goals can often be reached, by exploiting the features of the underlying algebraic geometry:

These goals can often be reached, by exploiting the features of the underlying algebraic geometry: ◮ There is a powerful and very flexible formalism for trace functions, including: 1. Stability under algebraic operations; 2. Stability under Fourier transform, convolution(s), etc; 3. The Grothendieck-Lefschetz trace formula

These goals can often be reached, by exploiting the features of the underlying algebraic geometry: ◮ There is a powerful and very flexible formalism for trace functions, including: 1. Stability under algebraic operations; 2. Stability under Fourier transform, convolution(s), etc; 3. The Grothendieck-Lefschetz trace formula ◮ This formalism is compatible with the complexity: operations on trace functions with bounded complexity result in other trace functions with bounded complexity;

These goals can often be reached, by exploiting the features of the underlying algebraic geometry: ◮ There is a powerful and very flexible formalism for trace functions, including: 1. Stability under algebraic operations; 2. Stability under Fourier transform, convolution(s), etc; 3. The Grothendieck-Lefschetz trace formula ◮ This formalism is compatible with the complexity: operations on trace functions with bounded complexity result in other trace functions with bounded complexity; ◮ And we have the general form of Deligne’s Riemann Hypothesis over finite fields.

A version of the Riemann Hypothesis Theorem (Quasi-orthogonality) ◮ Suppose K 1 and K 2 are trace functions modulo p associated to geometrically irreducible sheaves F 1 , F 2 . Then � ≤ C √ p � � � K 1 ( x ) K 2 ( x ) � � � x ∈ F p where C depends only on c ( K 1 ) , c ( K 2 ) , unless F 1 and F 2 are geometrically isomorphic.

A version of the Riemann Hypothesis Theorem (Quasi-orthogonality) ◮ Suppose K 1 and K 2 are trace functions modulo p associated to geometrically irreducible sheaves F 1 , F 2 . Then � ≤ C √ p � � � K 1 ( x ) K 2 ( x ) � � � x ∈ F p where C depends only on c ( K 1 ) , c ( K 2 ) , unless F 1 and F 2 are geometrically isomorphic. ◮ In this “diagonal” case, there exists α with | α | = 1 such that K 1 ( x ) = α K 2 ( x ) and � ≤ C √ p . � � � K 1 ( x ) K 2 ( x ) − ¯ α p � � � x ∈ F p

Examples ◮ (Weil-Deligne bounds) � y 1 + · · · + y r | Kl r ( x ) | = p − ( r − 1) / 2 � �� � e � ≤ r . � � p � y 1 ··· y r = x

Examples ◮ (Weil-Deligne bounds) � y 1 + · · · + y r | Kl r ( x ) | = p − ( r − 1) / 2 � �� � e � ≤ r . � � p � y 1 ··· y r = x ◮ (A “non-bound”) For � xy � P 2 ( X ) = X 2 − 1 , � P 2 ( Kl 2 ( y 2 )) e K ( x ) = , p y ∈ F p we have � � � K ( x ) K ( γ · x ) � ≥ p + O (1) � � � x ∈ F p γ · x � = ∞ if γ ∈ PGL 2 ( F p ) is � − 1 � � 0 � � 4 � 0 16 − 16 γ = Id , , , (and 4 others) . 0 1 1 0 1 4

Philosophy ◮ The Riemann Hypothesis can be used as a black box in many applications, using known examples of trace functions and their properties;

Philosophy ◮ The Riemann Hypothesis can be used as a black box in many applications, using known examples of trace functions and their properties; ◮ But the more one knows, the better (for instance, to identify geometrically irreducible trace functions);

Philosophy ◮ The Riemann Hypothesis can be used as a black box in many applications, using known examples of trace functions and their properties; ◮ But the more one knows, the better (for instance, to identify geometrically irreducible trace functions); ◮ This talk will attempt to explain, in a specific context, how to make the box slightly less dark.

Sums of products We often find in applications that we need to bound sums like � K 1 ( x ) · · · K n ( x ) M ( x ) x ∈ F p where K i , 1 ≤ i ≤ n , are trace functions, as well as M , and often M ( x ) = 1 or M ( x ) = e ( hx / p ) for some h ∈ F p .

Sums of products We often find in applications that we need to bound sums like � K 1 ( x ) · · · K n ( x ) M ( x ) x ∈ F p where K i , 1 ≤ i ≤ n , are trace functions, as well as M , and often M ( x ) = 1 or M ( x ) = e ( hx / p ) for some h ∈ F p . In particular, one often has K i ( x ) = K ( a i x + b i ) for some other fixed trace function K and a i ∈ F × p , b i ∈ F p . The ( a i , b i ) are not necessarily distinct.

Examples ◮ Proof of the Burgess bound: k even, K i ( x ) = χ ( x + b i ) , M ( x ) = 1 .