Imaginaries in pseudo- p -adically closed fields Joint with Samaria - PowerPoint PPT Presentation

Imaginaries in pseudo- p -adically closed fields Joint with Samaria Montenegro Silvain Rideau UC Berkeley July    / 

L A field extension K B L is totally p-adic if every p -adic valuation of K L A field K is pseudo-p-adically closed if it is existentially closed (as a L A field is bounded if it has finitely many extensions of any given Bounded pseudo- p -adically closed fields Definition positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. degree. Proposition (Montenegro) Let K be a bounded pseudo- p -adically closed field. There are finitely many p -adic valuations on K and they are definable in the ring language.  /  L A valuation is p-adic if the residue field is F p and p has minimal

L A field K is pseudo-p-adically closed if it is existentially closed (as a L A field is bounded if it has finitely many extensions of any given Bounded pseudo- p -adically closed fields Definition positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. degree. Proposition (Montenegro) Let K be a bounded pseudo- p -adically closed field. There are finitely many p -adic valuations on K and they are definable in the ring language.  /  L A valuation is p-adic if the residue field is F p and p has minimal L A field extension K B L is totally p-adic if every p -adic valuation of K

L A field is bounded if it has finitely many extensions of any given Bounded pseudo- p -adically closed fields Definition positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. degree. Proposition (Montenegro) Let K be a bounded pseudo- p -adically closed field. There are finitely many p -adic valuations on K and they are definable in the ring language.  /  L A valuation is p-adic if the residue field is F p and p has minimal L A field extension K B L is totally p-adic if every p -adic valuation of K L A field K is pseudo-p-adically closed if it is existentially closed (as a

Bounded pseudo- p -adically closed fields Definition positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. degree. Proposition (Montenegro) Let K be a bounded pseudo- p -adically closed field. There are finitely many p -adic valuations on K and they are definable in the ring language.  /  L A valuation is p-adic if the residue field is F p and p has minimal L A field extension K B L is totally p-adic if every p -adic valuation of K L A field K is pseudo-p-adically closed if it is existentially closed (as a L A field is bounded if it has finitely many extensions of any given

Bounded pseudo- p -adically closed fields Definition positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. degree. Proposition (Montenegro) Let K be a bounded pseudo- p -adically closed field. There are finitely many p -adic valuations on K and they are definable in the ring language.  /  L A valuation is p-adic if the residue field is F p and p has minimal L A field extension K B L is totally p-adic if every p -adic valuation of K L A field K is pseudo-p-adically closed if it is existentially closed (as a L A field is bounded if it has finitely many extensions of any given

L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . p eliminates imaginaries in L G (HMR). L L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n n � O � . S  � and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field.

p eliminates imaginaries in L G (HMR). L L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . S  � and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � .

p eliminates imaginaries in L G (HMR). L S  � and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � . L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n .

p eliminates imaginaries in L G (HMR). L L Algebraically closed valued fields eliminate imaginaries in L G The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � . L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . S  � Γ and s  can be identified with the valuation.

p eliminates imaginaries in L G (HMR). L The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � . L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . S  � Γ and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G

The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � . L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . S  � Γ and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G L Q p eliminates imaginaries in L G (HMR).

L Let L i denote n copies of L G , with sorts G i , sharing the sort F . L Let K  l K , L � � i L i 8 K  and T � Th L � K � . L Let M � T , M i be the algebraic closure of M with an extension of v i and M i be the p -adic closure of M inside M i Let U i x g be v i -open, then � i U i x g . Let K  b A b F � M � and s i t i > S i n � M � . If � i s i � M i L i � A � t i � s i � i B n � M L � A � � t i � i B n then An orthogonality result Proposition Remark  /  L Let K be a bounded pseudo- p -adically closed fields with n p -adic valuations � v i � i B n .

L Let K  l K , L � � i L i 8 K  and T � Th L � K � . L Let M � T , M i be the algebraic closure of M with an extension of v i and M i be the p -adic closure of M inside M i Let U i x g be v i -open, then � i U i x g . Let K  b A b F � M � and s i t i > S i n � M � . If � i s i � M i L i � A � t i � s i � i B n � M L � A � � t i � i B n then An orthogonality result Proposition Remark  /  L Let K be a bounded pseudo- p -adically closed fields with n p -adic valuations � v i � i B n . L Let L i denote n copies of L G , with sorts G i , sharing the sort F .

L Let M � T , M i be the algebraic closure of M with an extension of v i and M i be the p -adic closure of M inside M i Let U i x g be v i -open, then � i U i x g . Let K  b A b F � M � and s i t i > S i n � M � . If � i s i � M i L i � A � t i � s i � i B n � M L � A � � t i � i B n then An orthogonality result Proposition Remark  /  L Let K be a bounded pseudo- p -adically closed fields with n p -adic valuations � v i � i B n . L Let L i denote n copies of L G , with sorts G i , sharing the sort F . L Let K  l K , L � � i L i 8 K  and T � Th L � K � .

Let U i x g be v i -open, then � i U i x g . Let K  b A b F � M � and s i t i > S i n � M � . If � i s i � M i L i � A � t i � s i � i B n � M L � A � � t i � i B n Proposition then An orthogonality result Remark  /  L Let K be a bounded pseudo- p -adically closed fields with n p -adic valuations � v i � i B n . L Let L i denote n copies of L G , with sorts G i , sharing the sort F . L Let K  l K , L � � i L i 8 K  and T � Th L � K � . L Let M � T , M i be the algebraic closure of M with an extension of v i and M i be the p -adic closure of M inside M i