Algorithmics of Function Fields 3 Geometry Lecture 3 Weierstrass Places Mathematical Algorithmic Geometry Background Computation of Weierstrass Placs for Function Fields Isomorphisms and Automor- phisms Mathematical Background Summer School UNCG 2016 Computation of Isomorphisms Applications Florian Hess 1 / 24
Algorithmics of Function Fields 3 Geometry Weierstrass Places Mathematical Background Computation of Weierstrass Weierstrass Places Placs Isomorphisms and Automor- phisms Mathematical First Part Background Computation of Isomorphisms Applications 2 / 24
Weierstrass Places Algorithmics of Function Fields 3 Geometry Assume K perfect and let P be a place of degree one of F / K . Weierstrass The Weierstrass semigroup for P is the additive semisubgroup Places of Z ≥ 0 defined by Mathematical Background Computation of W ( P ) = {− v P ( f ) | f ∈ F × with v Q ( f ) ≥ 0 for all Q � = P } Weierstrass Placs Isomorphisms and Automor- Theorem. There is a semisubgroup W of Z ≥ 0 such that phisms Mathematical Background W = W ( P ) Computation of Isomorphisms Applications for almost all P . Moreover, #( Z ≥ 0 \ W ( P )) = g in general and Z ≥ 0 \ W ( P ) = { 1 , . . . , g } if char( F ) = 0. If W ( P ) � = W then P is called Weierstrass place of F / K . Theorem. There exist Weierstrass places if and only if g ≥ 2. Their number is between 2 g + 2 and ( g − 1) g ( g + 1) for char( F ) = 0 and in O ( g 3 ) in general. 3 / 24
Sketch Algorithmics of Function Fields 3 Geometry Let W denote a canonical divisor. The first observation is Weierstrass Places Mathematical L ( nP ) � = L (( n − 1) P ) iff L ( W − nP ) = L ( W − ( n − 1) P ) . Background Computation of Weierstrass Placs Isomorphisms Thus can/need to study zero and poles of function in L ( W ) for and Automor- all P . This can be done using the following tools and objects: phisms Mathematical Background Computation of ◮ Higher Derivatives of algebraic functions, Isomorphisms Applications ◮ Wronskian Determinant associated to L ( W ), ◮ Invariant divisor. The Weierstrass places are then the places in the support of this invariant divisor. 4 / 24
Sketch - Essential Idea Algorithmics of Function Fields 3 Geometry Roughly speaking, if f ∈ F has a zero of order n � = 0 at a place P of degree one, then its i -th derivative D ( i ) ( f ) with i ≤ n has Weierstrass Places a zero of order n − i at P . Mathematical Background Computation of Weierstrass Let f 1 , . . . , f g be a basis of L ( W ) and suppose P �∈ supp( W ). Placs Isomorphisms and Automor- The existence or non-existence of functions in L ( W ) with phisms Mathematical prescribed zero orders ε i at a P can be cast as the linear Background Computation of independece of the vectors Isomorphisms Applications ( D ( ε i ) ( f 1 )( P ) , . . . , D ( ε i ) ( f g )( P )) . Places P where linear independence does not hold are precisely the zeros of the Wronskian determinant �� � D ( ε i ) ( f j ) � det . i , j 5 / 24
Higher Derivatives - Example ∗ Algorithmics of Function Fields 3 Geometry We begin by way of example. Weierstrass Places Suppose f ∈ C [ x ]. Then also f ∈ C [ t ][ x ] and we can write Mathematical Background Computation of Weierstrass deg( f ) Placs � λ i ( t )( x − t ) i f = Isomorphisms and Automor- phisms i =0 Mathematical Background with λ i ∈ C [ t ]. The i -th derivative f ( i ) of f then satisfies Computation of Isomorphisms Applications f ( i ) ( t ) = i ! · λ i ( t ) . We wish to generalise this to arbitrary function fields and characteristic. Note that if p = char( F ) > 0 then uninterestingly f ( p ) ( t ) = 0, so we will take the λ i as higher derivatives of f . 6 / 24
Local Expansions ∗ Algorithmics of Function Fields 3 Geometry Let P be a place of degree one and π a local uniformizer of P , Weierstrass Places so v p ( π ) = 1. Mathematical Background Computation of Weierstrass For every f ∈ F and n ∈ Z there are uniquely determined Placs Isomorphisms m ∈ Z and λ i ∈ K such that and Automor- phisms Mathematical � n � Background � λ i π i f − ≥ n + 1 . Computation of v P Isomorphisms Applications i = m This leads to a K -algebra monomorphism F → K (( t )) into the ring of Laurent series over K which maps π to t . 7 / 24
Generic Place ∗ Algorithmics of Function Fields 3 Geometry Let x be a separating element of F / K and y ∈ F such that Weierstrass Places F = K ( x , y ). Mathematical Background Computation of Denote F ′ = K ( x ′ , y ′ ) an isomorphic copy of F and let FF ′ / F ′ Weierstrass Placs be the constant field extension. Isomorphisms and Automor- phisms There is place P of degree one of FF ′ / F ′ which is the unique Mathematical Background common zero of x − x ′ and y − y ′ . Moreover, x − x ′ is a local Computation of Isomorphisms Applications uniformizer of P . This place P is called generic place of F / K . The generic place is independently of the choice of x and y generated by the set of f − f ′ for f ∈ F . 8 / 24
Higher Derivatives ∗ Algorithmics of Function Fields 3 Geometry Weierstrass For every f ∈ F it holds that v P ( f ) ≥ 0. Via local expansions Places Mathematical we obtain the monomorphism Background Computation of Weierstrass Placs φ : F → F ′ [[ t ]] , Isomorphisms and Automor- phisms and we define the D ( i ) x ( f ) by Mathematical Background Computation of Isomorphisms ∞ Applications D ( i ) � x ( f )( x − x ′ ) i . φ ( f ) = i =0 Then D ( i ) x ( f ) is called i -th derivative of f with respect to x . 9 / 24
Higher Derivatives and Algorithmics of Function Fields Local Expansions at Places ∗ 3 Geometry Weierstrass Places Mathematical Background A local uniformizer π is also a separating element of F / K . Computation of Weierstrass Placs If v P ( f ) ≥ 0 then D ( i ) Isomorphisms π ( f )( P ) is the i -th coefficient of the and Automor- phisms power series expansion of f at P in π . Mathematical Background Computation of The element π − π ′ ∈ FF ′ is also a local uniformizer of the Isomorphisms Applications generic place of F / K . Thus the D ( i ) π ( f ) can be expressed in terms of the D ( i ) x ( f ) and vice versa. This is used to define the invariant divisor (under change of x ) mentioned above. 10 / 24
Algorithmics of Function Fields 3 Geometry Weierstrass Places Mathematical Background Computation of Weierstrass Isomorphisms and Automorphisms Placs Isomorphisms and Automor- phisms Mathematical Second Part Background Computation of Isomorphisms Applications 11 / 24
Isomorphisms Algorithmics of Function Fields 3 Geometry Weierstrass Let F (1) / K and F (2) / K be two function fields over K . Places Mathematical Background Computation of A homomorphism φ from F (1) / K to F (2) / K is a K -algebra Weierstrass Placs homomorphism F (1) → F (2) , which is necessarily injective. Isomorphisms and Automor- phisms Mathematical If φ is surjective it is called an isomorphism. Background Computation of Isomorphisms Applications A homomorphism φ is defined by its images in F (2) on generators of F (1) over K . Suppose F (2) /φ ( F (1) ) is separable and g (1) ≥ 2. Theorem. Then φ is an isomorphism if and only if g (1) = g (2) . 12 / 24
Automorphisms Algorithmics of Function Fields 3 Geometry Weierstrass Places Mathematical Background An isomorphism φ of F / K with itself is called an automorphism Computation of Weierstrass of F / K . They form a group which is denoted by Aut( F / K ). Placs Isomorphisms and Automor- phisms Theorem. The automorphism group Aut( F / K ) is finite. If in Mathematical Background particular char( F ) = 0 then Computation of Isomorphisms Applications #Aut( F / K ) ≤ 84( g − 1) . In general, #Aut( F / K ) is roughly bounded by 16 g 4 . 13 / 24
Computation of Isomorphisms Algorithmics of Function Fields 3 Geometry Weierstrass We assume that g (1) = g (2) ≥ 2 and K is the exact constant Places Mathematical field of F (1) / K and F (2) / K , for otherwise they are not Background Computation of Weierstrass isomorphic. All this can be checked beforehand. Placs Isomorphisms and Automor- There are different (better) techniques for g = 0 or g = 1 and phisms Mathematical for hyperelliptic function fields. Background Computation of Isomorphisms Applications We compute isomorphisms of complete regular curves C with a distinguished point by computing defining equations for C that are almost uniquely determined. We assume that K is perfect. 14 / 24
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