On the state of Wans Conjecture Newton Polygon of f HP () - PowerPoint PPT Presentation

Wan’s Conjecture Phong Le ∆ L On the state of Wan’s Conjecture Newton Polygon of f HP (∆) Ordinarity Phong Le Decomposition Theorems Department of Mathematics Niagara University April 2012 Upstate Number Theory Conference Rochester, NY

Wan’s Conjecture Laurent Polynomials Phong Le Let q = p a where p is a prime and a is a positive integer. Let ∆ F q denote the field of q elements. L For a Laurent polynomial f ∈ F q [ x ± 1 1 , . . . , x ± 1 Newton n ] we may Polygon of f represent f as: HP (∆) J Ordinarity � a j x V j , a j � = 0 , f = Decomposition Theorems j = 1 where each exponent V j = ( v 1 j , . . . , v nj ) is a lattice point in v 1 j v nj Z n and the power x V j is the product x · . . . · x n . 1 Example 2 10 x 1 x 2 f ( x 1 , x 2 ) = + + 82 2 x 1 = { ( − 1 , 0 ) , ( 1 , 2 ) , ( 0 , 0 ) } lattice points

Wan’s Conjecture F p (∆) Phong Le ∆ L Let ∆( f ) denote Newton polyhedron of f , that is, the convex Newton closure of the origin and { V 1 , . . . , V J } , the integral Polygon of f exponents of f . HP (∆) Ordinarity Definition Decomposition Given a convex integral polytope ∆ which contains the Theorems origin, let F q (∆) be the space of functions generated by the monomials in ∆ with coefficients in the algebraic closure of F q , a field of q elements. In other words, F q (∆) = { f ∈ F q [ x ± 1 1 , . . . , x ± 1 n ] | ∆( f ) ⊆ ∆ } .

Wan’s Conjecture The polytope ∆ Phong Le ∆ L Example Newton Polygon of f HP (∆) • Ordinarity Decomposition Theorems Let ∆ be the polytope generated by f ( x , y , z ) = 1 / z + x 5 z + y 5 z . • ◦ •

Wan’s Conjecture The polytope ∆ Phong Le ∆ L Example Newton Polygon of f HP (∆) ( 1 , 0 , 5 ) • Ordinarity ( 1 , 1 , 4 ) • Decomposition Theorems ( 1 , 2 , 3 ) It is also the convex • closure of the lattice ( 1 , 3 , 2 ) • points (including interior ( 1 , 4 , 1 ) • points). ( 1 , 5 , 0 ) • ( 0 , 0 , 0 ) ◦ • ( − 1 , 0 , 0 )

Wan’s Conjecture The polytope ∆ Phong Le ∆ L Example Newton Polygon of f HP (∆) x 0 x 5 2 • Ordinarity x 0 x 1 x 4 2 • Decomposition We can correspond Theorems x 0 x 2 1 x 3 2 • each lattice point to a x 0 x 3 1 x 2 2 • monomial in n variables x 0 x 4 1 x 2 (including interior • x 0 x 5 points). 1 • c ◦ • x − 1 0

Wan’s Conjecture The polytope ∆ Phong Le ∆ L Example Newton Polygon of f HP (∆) x 0 x 5 2 • Ordinarity x 0 x 1 x 4 2 • Decomposition F p (∆) is space of Theorems x 0 x 2 1 x 3 2 • functions the generated x 0 x 3 1 x 2 2 • by these monomials x 0 x 4 1 x 2 • (including interior x 0 x 5 points). 1 • c ◦ • x − 1 0

Wan’s Conjecture M q (∆) Phong Le ∆ L Definition Newton The Laurent polynomial f is called non-degenerate if for Polygon of f each closed face δ of ∆( f ) of arbitrary dimension which HP (∆) does not contain the origin, the n partial derivatives Ordinarity Decomposition Theorems { ∂ f δ , . . . , ∂ f δ } ∂ x 1 ∂ x n have no common zeros with x 1 · · · x n � = 0 over the algebraic closure of F q . Definition Let M q (∆) be the functions in F q (∆) that are non-degenerate.

Wan’s Conjecture Definition of the L -function Phong Le ∆ L Newton Polygon of f Let f ∈ F q [ x ± 1 1 , . . . , x ± 1 n ] . Let ζ p be a p -th root of unity and HP (∆) q = p a . For each positive integer k , consider the Ordinarity exponential sum: Decomposition Theorems ζ Tr k f ( x 1 ,..., x n ) � S ∗ k ( f ) = . p ( x 1 ,..., x n ) ∈ F ∗ qk The behavior of S ∗ k ( f ) as k increases is difficult to understand.

Wan’s Conjecture L -function Phong Le ∆ L To better understand S ∗ k ( f ) we define the L -function as Newton Polygon of f follows: HP (∆) F q , F q 2 , . . . F q k , . . . Ordinarity S ∗ S ∗ S ∗ Decomposition 1 ( f ) , 2 ( f ) , . . . k ( f ) , . . . Theorems 2 ( f ) T 2 k ( f ) T k S ∗ S ∗ S ∗ 1 ( f ) T + 2 + . . . + k + . . . � ∞ � k ( f ) T k L ∗ ( f , T ) = exp � S ∗ . k k = 1 By a theorem of Dwork-Bombieri-Grothendieck L ( f , T ) is a rational function.

Wan’s Conjecture NP ( f ) Phong Le ∆ L Adolphson and Sperber showed that if f is non-degenerate Newton Polygon of f ∞ L ∗ ( f , T ) ( − 1 ) n − 1 = HP (∆) � A i ( f ) T i , A i ( f ) ∈ Z [ ζ p ] Ordinarity i = 0 Decomposition Theorems is a polynomial of degree n ! Vol (∆) . Definition Define the Newton polygon of f , denoted NP ( f ) to be the lower convex closure in R 2 of the points ( k , ord q A k ( f )) , k = 0 , 1 , . . . , n ! Vol (∆) .

Wan’s Conjecture Phong Le Example ∆ • L Newton Polygon of f For p = q = 3 and ◦ HP (∆) f = 1 x 1 + x 1 x 2 2 + x 1 x 2 3 . Ordinarity One can computed directly: Decomposition L ( f , T ) − 1 = Theorems • • − 27 T 4 0 T 3 18 T 2 + + + + 8 T 1 ↓ ( 4 , 3 ) ( 3 , ∞ ) ( 2 , 2 ) ( 1 , 0 ) ( 0 , 0 )

Wan’s Conjecture The Hodge Polygon Phong Le There exists a combinatorial lower bound to the Newton ∆ polygon called the Hodge polygon HP (∆) . This is L constructed using the cone generated by ∆ consisting of all Newton Polygon of f rays passing through nonzero points of ∆ emanating from HP (∆) the origin. Ordinarity Decomposition Example Theorems •

Wan’s Conjecture Main Question Phong Le ∆ L Newton Polygon of f HP (∆) Definition Ordinarity When NP ( f ) = HP (∆) we say f is ordinary . Decomposition Theorems Generic Newton Polygon Let GNP (∆ , p ) = inf f ∈ M p (∆) NP ( f ) . Adophson and Sperber showed that GNP (∆ , p ) ≥ HP (∆) for every p .

Wan’s Conjecture Generic Ordinarity Phong Le ∆ L Main Question Newton When is GNP (∆ , p ) = HP (∆) ? Polygon of f HP (∆) If GNP (∆ , p ) = HP (∆) we say ∆ is generically ordinary at Ordinarity p . Decomposition Adolphson and Sperber conjectured that if p ≡ 1 ( mod D (∆)) the Theorems M p (∆) is generically ordinary. Wan showed that this is not quite true, but if we replace D (∆) with an effectively computable D ∗ (∆) this is true. Wan’s Conjecture p →∞ GNP (∆ , p ) = HP (∆) lim

Wan’s Conjecture Example of Ordinarity Phong Le ∆ L Newton Polygon of f • HP (∆) Ordinarity Decomposition Theorems • • Recall for p = q = 3 and f = 1 x 1 + x 1 x 2 2 + x 1 x 2 3 , the Newton polygon of L ( f , T ) ( − 1 ) ( n − 1 ) = − 27 T 4 + 18 T 2 + 8 T + 1.

Wan’s Conjecture Example Phong Le ∆ L Newton Polygon of f HP (∆) Ordinarity Decomposition Theorems • The Newton polygon ∆( f ) the polytope spanned by the origin, ( − 1 , 0 , 0 ) , ( 1 , 2 , 0 ) and ( 1 , 0 , 2 ) . • HP (∆( f )) is the lower convex hull of the points ( 0 , 0 ) , ( 1 , 0 ) and ( 4 , 3 ) which is identical to NP ( f ) . • From this we see that the Newton Polygon is equal to the Hodge polygon. Hence f is ordinary.

Wan’s Conjecture • In 2002 Zhu showed that Wan’s Conjecture holds for Phong Le the one variable case. ∆ • This was done by considering a specific family x d + ax . L • Through direct computation she found the Generic Newton Polygon of f Newton Polygon to be the lower convex hull of the HP (∆) points Ordinarity ( n , n ( n + 1 ) Decomposition + ǫ n ) Theorems 2 d Where p →∞ ǫ n = 0 lim • The Hodge polygon can be shown to be the lower convex hull of the points: ( n , n ( n + 1 ) ) 2 d

Wan’s Conjecture • In 2004 Regis Blache showed that Wan’s Conjecture Phong Le holds for families of the form: ∆ a d 1 1 x d 1 1 + a d 1 − 11 x d 1 − 1 + . . . + a 01 L 1 Newton Polygon of f + a d 2 2 x d 2 2 + a d 2 − 12 x d 2 − 1 + . . . + a 02 2 HP (∆) . Ordinarity . . Decomposition Theorems + a d n n x d n n + a d n − 1 n x d n − 1 + . . . + a 0 n n • These are families of polynomials with no cross terms like x 1 x 2 . • This was accomplished primarily by ‘factoring’ the Newton Polygon by variable. That is, he reduced this special multivariable case into the single variable case. • He also addressed ’rectangular’ families such as those generated by the polytope ( 0 , 0 ) , ( d 1 , 0 ) , ( 0 , d 2 ) , ( d 1 , d 2 ) .

Wan’s Conjecture • Last year Liu tackled these two specific families: Phong Le a ( 3 , 0 ) x 3 1 + a ( 0 , 3 ) x 3 2 + a ( 1 , 2 ) x 1 x 2 2 + a ( 2 , 1 ) x 2 x 1 2 + a ( 1 , 1 ) x 1 x 1 ∆ 2 L + a ( 2 , 0 ) x 2 1 + + a ( 0 , 2 ) x 2 2 + a ( 1 , 0 ) x 1 + a ( 0 , 1 ) x 2 + a ( 0 , 0 ) Newton Polygon of f and HP (∆) Ordinarity a ( 3 , 0 ) x 3 1 + a ( 1 , 1 ) x 1 x 1 2 + a ( 2 , 0 ) x 2 1 + + a ( 0 , 2 ) x 2 2 + a ( 1 , 0 ) x 1 Decomposition Theorems + a ( 0 , 1 ) x 2 + a ( 0 , 0 ) • This is an isosceles right triangle with leg length 3, and a leg length 2 isosceles right triangle with an additional point at ( 3 , 0 ) . • This was done in an entirely brute force method, computing the Newton Polygon speficially for these two families and showing that they tend toward the Hodge Polygon as p tends to infinity: