Computing central endomorphisms of an abelian variety via reductions - PowerPoint PPT Presentation

Computing central endomorphisms of an abelian variety via reductions modulo p Edgar Costa (MIT) January 18, 2020 Joint Mathematics Meetings Joint work with Davide Lombardo and John Voight Slides available at edgarcosta.org under Research

Elliptic curves Elliptic curves can be split in two classes: E is ordinary, i.e., E , E has CM, i.e., E d for some d 0 Warmup problem How would you distinguish between these two classes? E : y 2 = x 3 + ax + b , a , b ∈ Z

Elliptic curves Elliptic curves can be split in two classes: Warmup problem How would you distinguish between these two classes? E : y 2 = x 3 + ax + b , a , b ∈ Z E is ordinary, i.e., End Q E al = Q , √ E has CM, i.e., End Q E al = Q ( − d ) for some d > 0

Elliptic curves Elliptic curves can be split in two classes: Warmup problem How would you distinguish between these two classes? E : y 2 = x 3 + ax + b , a , b ∈ Z E is ordinary, i.e., End Q E al = Q , √ E has CM, i.e., End Q E al = Q ( − d ) for some d > 0

E has CM iff w 1 w 2 c p T where c p T E p T Approaches E 1 p 1 otherwise Quaternion alg. p 1 E p T E p E p Warmup problem Counting points on E p 0 d for some d w 2 w 1 E Embedding it in 4 a 3 j -invariant How would you distinguish between these two classes? pT 2 j ( E ) = 1728 4 a 3 + 27 b 2 E has CM iff j ( E ) is in a finite set

c p T where c p T E p T Approaches E 1 p 1 otherwise Quaternion alg. p 1 E p T E p p Warmup problem E Counting points on E p 4 a 3 j -invariant How would you distinguish between these two classes? pT 2 j ( E ) = 1728 4 a 3 + 27 b 2 E has CM iff j ( E ) is in a finite set Embedding it in C E C ≃ C / Λ , Λ = w 1 Z + w 2 Z √ E has CM iff w 1 / w 2 ∈ Q ( − d ) for some d > 0

Approaches otherwise How would you distinguish between these two classes? j -invariant 4 a 3 Warmup problem j ( E ) = 1728 4 a 3 + 27 b 2 E has CM iff j ( E ) is in a finite set Embedding it in C E C ≃ C / Λ , Λ = w 1 Z + w 2 Z √ E has CM iff w 1 / w 2 ∈ Q ( − d ) for some d > 0 Counting points on E p := E mod p  Q ( T ) / c p ( T ) , # E p ̸≡ 1 mod p End Q E al ֒  → End Q E al p = Quaternion alg. ,  where c p ( T ) = 1 − ( p + 1 − # E p ) T + pT 2

Examples p y x 3 7 (27.a2) E p Quaternion algebra 2 E 3 3 p 1 3 E y 2 3 (11.a2) otherwise  Q ( T ) / c p ( T ) , # E p ̸≡ 1 mod p End Q E al ֒  → End Q E al p = Quaternion alg. ,  E : y 2 + y = x 3 − x 2 − 10 x − 20 2 ≃ Q ( √− 1 ) End Q E al 3 ≃ Q ( √− 11 ) End Q E al ⇒ End Q E al = Q

Examples (11.a2) otherwise (27.a2)  Q ( T ) / c p ( T ) , # E p ̸≡ 1 mod p End Q E al ֒  → End Q E al p = Quaternion alg. ,  E : y 2 + y = x 3 − x 2 − 10 x − 20 2 ≃ Q ( √− 1 ) End Q E al 3 ≃ Q ( √− 11 ) End Q E al ⇒ End Q E al = Q E : y 2 + y = x 3 − 7  Quaternion algebra , p = 2 mod 3  End Q E al p = √ Q ( − 3 ) , p = 1 mod 3  √ ⇝ End Q E al = Q ( − 3 )

One can compute L p T by counting points on A If L 5 T 25 is isogenous to a square of an elliptic curve; Endomorphism algebras over finite fields M 2 A A 25 ; all endomorphisms are defined over 25 T 4 , then 2 T 2 1 Example Theorem (Tate) up to isomorphism pr A Honda–Tate theory gives us we may compute 6 Let A be an abelian variety over F p , given L p ( T ) := det( 1 − t Frob | H 1 ( A )) , rk End( A F pr ) , ∀ r ≥ 1 .

If L 5 T 25 is isogenous to a square of an elliptic curve; Endomorphism algebras over finite fields M 2 A A 25 ; all endomorphisms are defined over 25 T 4 , then 2 T 2 1 Example Theorem (Tate) up to isomorphism pr A Honda–Tate theory gives us we may compute 6 Let A be an abelian variety over F p , given L p ( T ) := det( 1 − t Frob | H 1 ( A )) , rk End( A F pr ) , ∀ r ≥ 1 . One can compute L p ( T ) by counting points on A

If L 5 T 25 is isogenous to a square of an elliptic curve; 1 M 2 A A 25 ; all endomorphisms are defined over 25 T 4 , then 2 T 2 Endomorphism algebras over finite fields Theorem (Tate) Example we may compute 6 Let A be an abelian variety over F p , given L p ( T ) := det( 1 − t Frob | H 1 ( A )) , rk End( A F pr ) , ∀ r ≥ 1 . One can compute L p ( T ) by counting points on A Honda–Tate theory gives us End Q ( A F pr ) up to isomorphism

Endomorphism algebras over finite fields Theorem (Tate) we may compute Example Let A be an abelian variety over F p , given L p ( T ) := det( 1 − t Frob | H 1 ( A )) , rk End( A F pr ) , ∀ r ≥ 1 . One can compute L p ( T ) by counting points on A Honda–Tate theory gives us End Q ( A F pr ) up to isomorphism If L 5 ( T ) = 1 − 2 T 2 + 25 T 4 , then all endomorphisms are defined over F 25 ; A F 25 is isogenous to a square of an elliptic curve; √ End Q A al ≃ M 2 ( Q ( − 6 ))

7, L 7 T all endomorphisms of A 7 are defined over 7 H 1 A A 7 over 49 is isogenous to a square of an elliptic curve Example continued 1 T 2 49 T 2 2 1 6 T A 7 M 2 10 A 49 49 T 4 , and: 6 T 2 1 For p (262144.d.524288.1) M 2 A = Jac( y 2 = x 5 − x 4 + 4 x 3 − 8 x 2 + 5 x − 1 ) For p = 5, L 5 ( T ) = 1 − 2 T 2 + 25 T 4 , and: all endomorphisms of A 5 are defined over F 25 det( 1 − T Frob 2 5 | H 1 ( A )) = ( 1 − 2 T + 25 T 2 ) 2 A 5 over F 25 is isogenous to a square of an elliptic curve √ End Q A al 5 ≃ M 2 ( Q ( − 6 ))

Example continued A M 2 (262144.d.524288.1) A = Jac( y 2 = x 5 − x 4 + 4 x 3 − 8 x 2 + 5 x − 1 ) For p = 5, L 5 ( T ) = 1 − 2 T 2 + 25 T 4 , and: all endomorphisms of A 5 are defined over F 25 det( 1 − T Frob 2 5 | H 1 ( A )) = ( 1 − 2 T + 25 T 2 ) 2 A 5 over F 25 is isogenous to a square of an elliptic curve √ End Q A al 5 ≃ M 2 ( Q ( − 6 )) For p = 7, L 7 ( T ) = 1 + 6 T 2 + 49 T 4 , and: all endomorphisms of A 7 are defined over F 49 det( 1 − T Frob 2 7 | H 1 ( A )) = ( 1 + 6 T + 49 T 2 ) 2 A 7 over F 49 is isogenous to a square of an elliptic curve √ End Q A al 7 ≃ M 2 ( Q ( − 10 ))

Example continued (262144.d.524288.1) A = Jac( y 2 = x 5 − x 4 + 4 x 3 − 8 x 2 + 5 x − 1 ) For p = 5, L 5 ( T ) = 1 − 2 T 2 + 25 T 4 , and: all endomorphisms of A 5 are defined over F 25 det( 1 − T Frob 2 5 | H 1 ( A )) = ( 1 − 2 T + 25 T 2 ) 2 A 5 over F 25 is isogenous to a square of an elliptic curve √ End Q A al 5 ≃ M 2 ( Q ( − 6 )) For p = 7, L 7 ( T ) = 1 + 6 T 2 + 49 T 4 , and: all endomorphisms of A 7 are defined over F 49 det( 1 − T Frob 2 7 | H 1 ( A )) = ( 1 + 6 T + 49 T 2 ) 2 A 7 over F 49 is isogenous to a square of an elliptic curve √ End Q A al 7 ≃ M 2 ( Q ( − 10 )) ⇒ End R A al ̸ = M 2 ( C )

L i B i , then i n 2 i L i Higher genus if the Mumford–Tate conjecture holds for A. t 1 L i i and t 1 e i n i i t We can effectively compute Theorem (C–Mascot–Sijsling–Voight, C–Lombardo–Voight) 1 e 2 i t A K i Set e 2 with center L i . t This is done by just counting points. We may factor End A al uniquely as End A al ≃ ∏ M n i ( B i ) , i = 1 where B i are division algebras

Higher genus Theorem (C–Mascot–Sijsling–Voight, C–Lombardo–Voight) if the Mumford–Tate conjecture holds for A. t 1 L i i and t 1 e i n i i t We can effectively compute e 2 Set e 2 t t This is done by just counting points. We may factor End A al uniquely as End A al ≃ ∏ M n i ( B i ) , i = 1 where B i are division algebras with center L i . i := dim L i B i , then ∑ rk End( A K ) = i [ L i : Q ] . i n 2 i = 1

Higher genus t if the Mumford–Tate conjecture holds for A. and We can effectively compute Theorem (C–Mascot–Sijsling–Voight, C–Lombardo–Voight) e 2 This is done by just counting points. Set e 2 t We may factor End A al uniquely as End A al ≃ ∏ M n i ( B i ) , i = 1 where B i are division algebras with center L i . i := dim L i B i , then ∑ rk End( A K ) = i [ L i : Q ] . i n 2 i = 1 t , { e i n i } i = 1 ,..., t , { L i } i = 1 ,..., t ,

Higher genus t if the Mumford–Tate conjecture holds for A. and We can effectively compute Theorem (C–Mascot–Sijsling–Voight, C–Lombardo–Voight) e 2 This is done by just counting points. Set e 2 t We may factor End A al uniquely as End A al ≃ ∏ M n i ( B i ) , i = 1 where B i are division algebras with center L i . i := dim L i B i , then ∑ rk End( A K ) = i [ L i : Q ] . i n 2 i = 1 t , { e i n i } i = 1 ,..., t , { L i } i = 1 ,..., t ,

Y m over One factor at a time A an abelian variety over numberfield F Mumford–Tate conjecture holds for A Let S be the set of primes of F such that: (i) A has good reduction at (ii) A with Y geometrically simple (iii) L Theorem (Zywina, C-Lombardo-Voight) The set S has positive density. If m is unknown, sharp upper bound for m may be obtained. A al ∼ Y n , with Y simple, i.e., End A al ≃ M n ( B ) L := Z (End Q A al ) m 2 := dim L End Q A al

One factor at a time A an abelian variety over numberfield F Mumford–Tate conjecture holds for A Theorem (Zywina, C-Lombardo-Voight) The set S has positive density. If m is unknown, sharp upper bound for m may be obtained. A al ∼ Y n , with Y simple, i.e., End A al ≃ M n ( B ) L := Z (End Q A al ) m 2 := dim L End Q A al Let S be the set of primes p of F such that: (i) A has good reduction at p (ii) A p ∼ Y m over F p with Y geometrically simple (iii) L ֒ → Q (Frob p )