Dipartim. Mat. & Fis. Universit` a Roma Tre Introduction to Galois Representations Applications Plan for today Serre’s Cyclicity NATO ASI, Ohrid 2014 Conjecture Lang Trotter Conjecture Arithmetic of Hyperelliptic Curves for trace of Frobenius state of the Art August 25 - September 5, 2014 Serre’s upperbound Ohrid, the former Yugoslav Republic of Macedonia , Average Lang Trotter Conjecture Some ideas on Average results proofs Lang Trotter Conjecture for Primitive points Artin Conjecture for primitive roots Artin vs Lang Trotter Further reading Francesco Pappalardi Dipartimento di Matematica e Fisica Universit` a Roma Tre 1
Dipartim. Mat. & Fis. Plan for today Universit` a Roma Tre Plan for today Serre’s Cyclicity Topics Conjecture Lang Trotter Conjecture • Short summery of Tuesday’s Lecture for trace of Frobenius state of the Art • Facts about Elliptic curves over finite fields Serre’s upperbound Average Lang Trotter Conjecture • Serre’s Cyclicity Conjecture Some ideas on Average results proofs • Lang–Trotter Conjecture for fixed traces Lang Trotter Conjecture for Primitive points • Lang–Trotter Conjecture for primitive points Artin Conjecture for primitive roots • Artin primitive roots Conjecture Artin vs Lang Trotter Further reading 2
Dipartim. Mat. & Fis. Elliptic curves Universit` a Roma Tre E : Y 2 = X 3 + aX + b , a, b ∈ Z ; W EIERSTRASS E QUATION : ∆ E = 4 a 3 − 27 b 2 Plan for today D ISCRIMINANT OF E : Serre’s Cyclicity Conjecture • ∆ E = ( α 1 − α 2 ) 2 ( α 3 − α 2 ) 2 ( α 3 − α 1 ) 2 ( α 1 , α 2 , α 3 roots of X 3 + aX + b ); Lang Trotter Conjecture for trace of Frobenius state of the Art ⇒ X 3 + aX + b has a double root! • ∆ E = 0 ⇐ Serre’s upperbound Average Lang Trotter Conjecture Some ideas on Average results proofs Definition Lang Trotter Conjecture for Primitive points if ∆ E � = 0 = ⇒ E is called E LLIPTIC C URVE Artin Conjecture for primitive roots Group of Rational Points Artin vs Lang Trotter Further reading If K/ Q is an extension. Then E ( K ) = { ( x, y ) ∈ K 2 : y 2 = x 3 + ax + b } ∪ {∞} 3
Dipartim. Mat. & Fis. The n -torsion subgroups Universit` a Roma Tre Plan for today If n ∈ N E [ n ] := { P ∈ E ( Q ) | nP = ∞} Serre’s Cyclicity Conjecture • E [ n ] ⊂ E ( Q ) ∼ = Q / Z × Q / Z is a subgroup Lang Trotter Conjecture for trace of Frobenius • E [ n ] ∼ = C n ⊕ C n state of the Art Serre’s upperbound • E [2] = { ( α 1 , 0) , ( α 2 , 0) , ( α 3 , 0) , ∞} Average Lang Trotter Conjecture ( α 1 , α 2 , α 3 roots of x 3 + ax + b ) Some ideas on Average results proofs • E [3] is the set of inflection points Lang Trotter Conjecture for Primitive points • If n is odd, P = ( α, β ) ∈ E [ n ] = ⇒ ψ n ( α ) = 0 , Artin Conjecture for ψ n is n –division polynomials ( ∂ψ n = ( n 2 − 1) / 2 if n odd) primitive roots √ √ Artin vs Lang Trotter • E : y 3 = x 3 − 2 x = ⇒ E [2] = { (0 , 0) , ( 2 , 0) , ( − 2 , 0) , ∞} Further reading 4
Dipartim. Mat. & Fis. Representation on n -torsion points Universit` a Roma Tre � Q ( E [ n ]) = The n –torsion field: K K 2 ⊃ E [ n ] \{∞} Plan for today • Q ( E [ n ]) is Galois over Q Serre’s Cyclicity Conjecture • Gal( Q ( E [ n ]) / Q ) ⊆ Aut( E [ n ]) ∼ = GL 2 ( Z /n Z ) Lang Trotter Conjecture for trace of Frobenius state of the Art Gal( Q ( E [ n ]) / Q ) ֒ → GL 2 ( Z /n Z ) Serre’s upperbound Average Lang Trotter Conjecture Some ideas on Average results σ �→ { ( x, y ) �→ ( σ ( x ) , σ ( y )) } proofs Lang Trotter Conjecture Injective representation for Primitive points Artin Conjecture for primitive roots Artin vs Lang Trotter Theorem (Serre) Further reading If E/ Q is not CM. Then Gal( Q ( E [ ℓ ]) / Q ) � = GL 2 ( F ℓ ) only for finitely many ℓ . Conjecture ( ℓ ≤ 37 ) 5
Dipartim. Mat. & Fis. Reducing modulo primes Universit` a Roma Tre Facts about elliptic curves over finite fields • p prime, p ∤ ∆ E p | Y 2 = X 3 + aX + b }∪ {∞} ( X, Y ) ∈ F 2 Plan for today • E ( F p ) = { Serre’s Cyclicity • E ( F p ) ∼ = C k ⊕ C nk for some k | p − 1 Conjecture Lang Trotter Conjecture • k = 1 above iff E ( F p ) is cyclic for trace of Frobenius state of the Art • # E ( F p ) = p + 1 − a p ( a p is the T RACE OF F ROBENIUS ) Serre’s upperbound | a p | ≤ 2 √ p ; Average Lang Trotter • H ASSE BOUND : Conjecture Some ideas on Average results proofs • Ψ p : E ( F p ) → E ( F p ) , ( x, y ) �→ ( x p , y p ) Lang Trotter Conjecture for Primitive points it is an endomorphism of E/ F p Artin Conjecture for • Ψ p ∈ End( E ) satisfies T 2 − a p T + p primitive roots Artin vs Lang Trotter • Z [Ψ p ] ⊂ End( E ) Further reading • If the equality hold above, we say that E is ordinary at p . Otherwise we say that it is supersingular • E/ F p is supersingular ⇐ ⇒ E [ p ] = {∞} ⇐ ⇒ a p = 0 6
Dipartim. Mat. & Fis. Serre’s Cyclicity Conjecture Universit` a Roma Tre Let E/ Q and set π cyclic ( x ) = # { p ≤ x : E ( F p ) is cyclic } . E Plan for today Conjecture (Serre) Serre’s Cyclicity Conjecture The following asymptotic formula holds Lang Trotter Conjecture for trace of Frobenius x state of the Art π cyclic ( x ) ∼ δ cyclic x → ∞ Serre’s upperbound E E log x Average Lang Trotter Conjecture Some ideas on Average results proofs where ∞ µ ( n ) Lang Trotter Conjecture � δ cyclic = for Primitive points E # Gal( Q ( E [ n ]) / Q ) Artin Conjecture for n =1 primitive roots Artin vs Lang Trotter • Since E ( F p ) ∼ = C k ⊕ C kn Further reading and E [ ℓ ] ∼ = C ℓ ⊕ C ℓ for all ℓ � = p E ( F p ) is cyclic iff E [ ℓ ] � E ( F p ) ∀ ℓ prime ℓ � = p • So we may rewrite π cyclic ( x ) = # { p ≤ x : E [ ℓ ] � E ( F p ) ∀ ℓ prime , ℓ � = p } . E 7
Dipartim. Mat. & Fis. Serre’s Cyclicity Conjecture Universit` a Roma Tre We can apply inclusion exclusion principle: Plan for today π cyclic ( x ) = # { p ≤ x : E [ ℓ ] � E ( F p ) ∀ ℓ prime , ℓ � = p } E Serre’s Cyclicity Conjecture � � = π ( x ) − π E,ℓ ( x ) + π E,ℓ 1 ℓ 2 ( x ) − · · · Lang Trotter Conjecture for trace of Frobenius ℓ prime ℓ 1 ,ℓ 2 primes state of the Art Serre’s upperbound Average Lang Trotter where π ( x ) := # { p ≤ x } and if k ∈ N , Conjecture Some ideas on Average results proofs π E,k ( x ) := # { p ≤ x : E [ k ] ⊆ E ( F p ) } Lang Trotter Conjecture for Primitive points Artin Conjecture for primitive roots Hence, if µ is the M¨ obius function, then Artin vs Lang Trotter Further reading � π cyclic ( x ) = µ ( k ) π E,k ( x ) E k ∈ N We will study π E,k ( x ) by mean of the Chebotarev density Theorem. 8
Dipartim. Mat. & Fis. Chebotarev Density Theorem (from tuesday) Universit` a Roma Tre If K/ Q be Galois and p is prime unramified in K , the Artin Symbol � K/ Q � � � σ ∈ Gal( K/ Q ) : ∃ p prime of K above p s.t. := σα ≡ α N p mod p ∀ α ∈ O p Plan for today Serre’s Cyclicity � � K/ Q Conjecture = { id } then p splits completely in K/ Q Note that p Lang Trotter Conjecture (i.e p O ⊂ O is the product of [ K : Q ] prime ideals) for trace of Frobenius state of the Art Serre’s upperbound Theorem (Chebotarev Density Theorem) Average Lang Trotter Conjecture Let K/ Q be finite and Galois, and let C ⊂ Gal( K/ Q ) be a union of Some ideas on Average results proofs conjugation classes. Then the density of the primes p such that Lang Trotter Conjecture for Primitive points � � K/ Q # C ⊂ C equals # Gal( K/ Q ) . Artin Conjecture for p primitive roots In particular, if C = { id } , then the density of the primes p such that Artin vs Lang Trotter � � K/ Q 1 = { id } equals # Gal( K/ Q ) . Further reading p If K = Q ( E [ n ]) , then � Q ( E [ n ]) / Q � E [ n ] ⊂ E ( F p ) ⇐ ⇒ = { id } p 9
Dipartim. Mat. & Fis. Chebotarev Density Theorem and Serre’s Cyclicity Conj. Universit` a Roma Tre If K = Q ( E [ n ]) , then � Q ( E [ n ]) / Q � E [ n ] ⊂ E ( F p ) ⇐ ⇒ = { id } Plan for today p Serre’s Cyclicity Conjecture Also recall that π E,k ( x ) := # { p ≤ x : E [ k ] ⊆ E ( F p ) } Lang Trotter Conjecture for trace of Frobenius state of the Art π cyclic � Serre’s upperbound ( x ) = µ ( k ) π E,k ( x ) E Average Lang Trotter Conjecture k ∈ N Some ideas on Average results proofs � � Q ( E [ n ]) / Q � � � = µ ( k )# p ≤ x : = { id } Lang Trotter Conjecture p for Primitive points k ∈ N Artin Conjecture for primitive roots Artin vs Lang Trotter To proceed we need a quantitative versions of the Chebotarev Density Further reading Theorem. Let � � K/ Q � � π C / G ( x ) := # p ≤ x : ⊂ C . p 10
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