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The average Lang Trotter Conjecture for imaginary quadratic fields Francesco Pappalardi Chennai - January, 2002 0-0

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 1 ✞ ☎ ✝ Notations. ✆ E : Y 2 = X 3 + aX + b • Elliptic curve: − ∆ E = 4 a 3 + 27 b 2 � = 0); ( a, b ∈ Z , p | Y 2 = X 3 + aX + b } ( X, Y ) ∈ F 2 • E ( F p ) = { ; • Trace of Frobenius: a p ( E ) = p − # E ( F p ); | a p ( E ) | ≤ 2 √ p ; • Hasse bound: • Lang Trotter function: r ∈ Z π r E ( x ) = # { p ≤ x | a p ( E ) = r } . Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 2 ✞ ☎ The Lang Trotter Conjecture ✝ ✆ If r � = 0 or E not CM, ✞ ☎ √ x π r E ( x ) ∼ C E,r log x , C E,r ≥ 0 . ✝ ✆ √ x 1 π r 1 E ( x ) ≈ � Prob( a p ( E ) = r ) ≈ = = = = > 2 √ p ∼ log x . 2 √ p p ≤ x Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 3 ✞ ☎ ✝ State of the Art. ✆ • M. Deuring (1941): If E has CM π E, 0 ( x ) ∼ 1 x log x ; 2 • J. P. Serre (1981) , Elkies, Kaneko, K. Murty, R. Murty, N. Saradha, Wan (1988):  x (log log x ) 2 if r � = 0  log 2 x π E,r ( x ) ≪ if r = 0 and x 3 / 4 E not CM  • N. Elkies, E. Fouvry, R. Murty (1996) π E, 0 ( x ) ≫ log log log x/ (log log log log x ) 1+ ǫ (Stronger results on GRH) Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 4 ✞ ☎ Average Lang Trotter Conjecture ✝ ✆ E. Fouvry, R. Murty (1996), C. David, F. P. (1997) C x = { E : Y 2 = X 3 + aX + b || a | , | b | ≤ x log x, } Then √ x 1 � π E,r ( x ) ∼ c r as x → ∞ . |C x | log x E ∈C x where l ( l 2 − l − 1) � − 1 � l | GL 2 ( F l ) Tr= r | � c r = 2 1 − 1 ( l − 1)( l 2 − 1) = 2 � � . l 2 π π | GL 2 ( F l ) | l l | r l ∤ r Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 5 ✞ ☎ Representation on n -torsion points. ✝ ✆ For n ∈ N • E [ n ] = { P ∈ E ( C ) | nP = O} ⊂ E ( C ) ( n -torsion subgroup); • E [ n ] ∼ = Z /n Z × Z /n Z ; � • Q ( E [ n ]) = K ; ( Q ( E [ n ]) Galois over Q ); K 2 ⊃ E [ n ] \{O} • Aut( E [ n ]) ∼ = GL 2 ( Z /n Z ) ; Gal( Q ( E [ n ]) / Q ) − → GL 2 ( Z /n Z ) . σ �→ { ( x 1 , x 2 ) �→ ( σ ( x 1 ) , σ ( x 2 )) } . injective representation. Theorem.(Serre) If E not CM , Gal( Q ( E [ l ]) / Q ) = GL 2 ( F l ) except finitely many l . Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 6 ✞ ☎ Chebotarev Density Thm. & Lang–Trotter Conj. ✝ ✆ • p ramifies in Q ( E [ l ]) < = = = > p | l ∆ E ; • p ∤ l ∆ E , σ p ⊂ Gal( Q ( E [ l ]) / Q ) (Frobenius conjugacy class); • Gal( Q ( E [ l ]) / Q ) ⊆ GL 2 ( F l ), σ p has characteristic polynomial T 2 − a p ( E ) T + p . • a p ( E ) ≡ Tr( σ p ) mod l ; • π E,r ( x ) ≤ # { p ≤ x | a p ( E ) ≡ r (mod l ) } ; • Chebotarev Density Theorem, l ≫ 0, Prob( a p ( E ) ≡ r mod l ) ∼ | GL 2 ( F l ) Tr= r | . | GL 2 ( F l ) | Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 7 ✞ ☎ Lang–Trotter Constant ✝ ✆ π r E ( x ) C E,r = lim √ x x →∞ log x ∃ m E,r ∈ N s.t. m E,r | Gal( Q ( E [ m E,r ]) / Q ) Tr= r | l | GL 2 ( F l ) Tr= r | C E,r = 2 � . π | Gal( Q ( E [ m E,r ]) / Q ) | | GL 2 ( F l ) | l ∤ m E,r Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 8 ✞ ☎ ✝ More Notations. ✆ • K finite Galois / Q ; • E elliptic curve defined over O K ; • ∆ E discriminant ideal of E/ O K ; • p ∈ Z unramified in K / Q , p ∤ N (∆ E ); • p ⊂ O K , p | p ; • E p reduction of E over O K / ( p ); • E p ( O K / ( p )) = N ( p ) + 1 − a E ( p ); � • Hasse bound | a E ( p ) | ≤ 2 N ( p ); • degree of p : N ( p ) = p deg K ( p ) . Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 9 ✞ ☎ A Variation of Lang–Trotter Conjecture ✝ ✆ f | [ K : Q ]. General Lang–Trotter function: π r,f E ( x ) = # { p ≤ x | deg K ( p ) = f, a E ( p ) = r } . Conjecture: ∃ c E,r,f ∈ R ≥ 0 such that  x if E has CM and r = 0  log x  √ x   if f = 1   π r,f log x E ( x ) ∼ c E,r,f log log x if f = 2      1 otherwise.  Example. K = Q ( i ): π r, 1 ↔ split primes ≡ 1 mod 4; π r, 2 ↔ inert primes ≡ 3 mod 4 Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 10 ✞ ☎ Statement of Today’s Result ✝ ✆ Theorem. (C. David & F. Pappalardi) K = Q ( i ) , r ∈ Z , r � = 0  �  α = a 1 + a 2 i, β = b 1 + b 2 i ∈ Z [ i ] , �    �    E : Y 2 = X 3 + αX + β 4 α 3 − 27 β 2 � = 0 � C x = � �    �  max {| a 1 | , | a 2 | , | b 1 | , | b 2 |} < x log x   � Then ✓ ✏ 1 � π r, 2 E ( x ) ∼ c r log log x. |C x | ✒ ✑ E ∈C x � � − r 2 l ( l − 1 − ) c r = 1 l � ) . � − 1 � 3 π ( l − 1)( l − l l> 2 Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 11 ✞ ☎ Sketch of proof. 1/8 ✝ ✆ Deuring’s Thm. q = p n , r odd (simplicity), s.t. r 2 − 4 q > 0 . � F q − isomorphism classes of E/ F q � = H ( r 2 − 4 q ) . with a q ( E ) = r h ( r 2 − 4 p 2 ) Kronecker class numbers : H ( r 2 − 4 p 2 ) = 2 f 2 � . w ( r 2 − 4 p 2 ) f 2 f 2 | r 2 − 4 p 2 √ � r 2 − 4 p 2). h ( D ) = class number, w ( D ) = #units in Z [ D + D ] ⊂ Q ( ✓ ✏ H ( r 2 − 4 p 2 ) 1 E ( x ) = 1 � π r, 2 � Step 1: + O (1) . p 2 |C x | 2 E ∈C x p ≤ x ✒ ✑ p ≡ 3 mod 4 Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 12 ✞ ☎ Sketch of proof. 2/8 ✝ ✆ Given f 2 | r 2 − 4 p 2 , • d = ( r 2 − 4 p 2 ) /f 2 ( ≡ 1 mod 4); � d � • χ d ( n ) = ; n • L ( s, χ d ) Dirichlet L –function; • h ( d ) = ω ( d ) | d | 1 / 2 L (1 , χ d ) (class number formula). 2 π ✗ ✔ Step 2. H ( r 2 − 4 p 2 ) 1 = 2 1 L (1 , χ d ) � � � + O (1) . p 2 p 2 2 π f p ≤ x f ≤ 2 x p ≤ x p ≡ 3 mod 4 p ≡ 3 mod 4 ( f, 2 r )=1 4 p 2 ≡ r 2 mod f 2 ✖ ✕ Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 13 ✞ ☎ Sketch of proof. 3/8 ✝ ✆ Lemma A. [Analytic] Let d = ( r 2 − 4 p 2 ) /f 2 , ∀ c > 0, 1 � x � � � L (1 , χ d ) log p = k r x + O . log c x f f ≤ 2 x p ≤ x ( f, 2 r )=1 p ≡ 3 mod 4 4 p 2 ≡ r 2 mod f 2 where ∞ ∞ 1 1 � a � � � b ≡ 3 mod 4 , � b ∈ ( Z / 4 nf 2 Z ) ∗ � � � � k r = # . � 4 b 2 ≡ r 2 − af 2 (4 nf 2 ) nϕ (4 nf 2 ) f n � a ∈ ( Z / 4 n Z ) ∗ f =1 n =1 Lemma B. [Euler product] With above notations, � � − r 2 l − 1 − k r = 2 l � ) . � − 1 � 3 ( l − 1)( l − l l> 2 Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 14 ✞ ☎ Sketch of proof. 4/8 ✝ ✆ Start from � e − n/U � | d | 3 / 16+ ǫ � d � 1 � d � � � L (1 , χ d ) = n = + O n n n U 1 / 2 n ∈ N n ∈ N follows from � e − n/U L ( s + 1 , χ d )Γ( s + 1) U s � d � � = L (1 , χ d ) + s ds n n ℜ ( s )= − 1 n ∈ N 2 applying Burgess, L (1 / 2 + it, χ d ) ≪ | t | 2 | d | 3 / 16+ ǫ and obtain e − n � x 11 / 8+ ǫ � d � � 1 U � � � � L (1 , χ d ) log p = log p + O U 1 / 2 f nf n f ≤ 2 x p ≤ x f ≤ 2 x, p ≤ x ( f, 2 r )=1 p ≡ 3 mod 4 n ∈ N p ≡ 3 mod 4 4 p 2 ≡ r 2 mod f 2 4 p 2 ≡ r 2 mod f 2 ( f, 2 r )=1 Universit` a Roma Tre

The average Lang Trotter Conjecture for imaginary quadratic fields Chennai, January 2002 15 ✞ ☎ Sketch of proof. 5/8 ✝ ✆ e − n � d � � � 1 x U � � � � L (1 , χ d ) log p = log p + O log c x f nf n f ≤ 2 x p ≤ x f ≤ V, p ≤ x p ≡ 3 mod 4 n ≤ U log U p ≡ 3 mod 4 ( f, 2 r )=1 4 p 2 ≡ r 2 mod f 2 4 p 2 ≡ r 2 mod f 2 ( f, 2 r )=1 where U = x 1 − ǫ . Easy to deal with f > V = (log x ) a , n > U log U . d � � Since character modulo 4 n n � d � a � � � � � log p = log p n n a ∈ ( Z / 4 n Z ) ∗ p ≤ x p ≤ x, p ≡ 3 mod 4 p ≡ 3 mod 4 ( r 2 − 4 p 2 ) /f 2 ≡ a mod 4 n 4 p 2 ≡ r 2 mod f 2 a � � � � ψ 1 ( x, 4 nf 2 , b ) = n a ∈ ( Z / 4 n Z ) ∗ b ∈ ( Z / 4 nf 2 Z ) ∗ b ≡ 3 mod 4 4 b 2 ≡ r 2 − af 2 mod 4 nf 2 � where as usual ψ 1 ( x, 4 nf 2 , b ) = log p 2 ≤ p ≤ x, p ≡ b mod 4 nf 2 Universit` a Roma Tre