The Explicit Sato-Tate Conjecture in Arithmetic Progressions Trajan - PowerPoint PPT Presentation

Background Results Sketch of Proof The Explicit Sato-Tate Conjecture in Arithmetic Progressions Trajan Hammonds, Casimir Kothari, Hunter Wieman thammond@andrew.cmu.edu, ckothari@princeton.edu, hlw2@williams.edu Joint with Noah Luntzlara, Steven J. Miller, and Jesse Thorner October 7, 2018, Qu´ ebec-Maine Number Theory Conference

Background Results Sketch of Proof Motivation Theorem (Prime Number Theorem) π ( x ) := # { p ≤ x : p is prime } ∼ Li ( x ) .

Background Results Sketch of Proof Motivation Theorem (Prime Number Theorem) π ( x ) := # { p ≤ x : p is prime } ∼ Li ( x ) . Theorem Refinement to arithmetic progressions: Let a , q be such that gcd( a , q ) = 1 . Then 1 π ( x ; q , a ) := # { p ≤ x : p prime and p ≡ a mod q } ∼ ϕ ( q ) Li ( x ) .

Background Results Sketch of Proof Modular Forms Recall that a modular form of weight k on SL 2 ( Z ) is a function f : H → C with ∞ � a f ( n ) q n , q = e 2 π iz f ( z ) = n =0 and f ( γ z ) = ( cz + d ) k f ( z ) for all γ ∈ SL 2 ( Z ) .

Background Results Sketch of Proof Modular Forms Recall that a modular form of weight k on SL 2 ( Z ) is a function f : H → C with ∞ � a f ( n ) q n , q = e 2 π iz f ( z ) = n =0 and f ( γ z ) = ( cz + d ) k f ( z ) for all γ ∈ SL 2 ( Z ) . By restricting to the action of a congruence subgroup Γ ⊂ SL 2 ( Z ) of level N , we can associate that level to our modular form f ( z ).

Background Results Sketch of Proof Modular Forms Recall that a modular form of weight k on SL 2 ( Z ) is a function f : H → C with ∞ � a f ( n ) q n , q = e 2 π iz f ( z ) = n =0 and f ( γ z ) = ( cz + d ) k f ( z ) for all γ ∈ SL 2 ( Z ) . By restricting to the action of a congruence subgroup Γ ⊂ SL 2 ( Z ) of level N , we can associate that level to our modular form f ( z ). We say a modular form is a cusp form if it vanishes at the cusps of Γ; hence a f (0) = 0 for a cusp form f ( z ).

Background Results Sketch of Proof Newforms We say f is a Hecke eigenform if it is a cusp form and T n f = λ ( n ) f for n = 1 , 2 , 3 , . . . , where T n is the Hecke operator.

Background Results Sketch of Proof Newforms We say f is a Hecke eigenform if it is a cusp form and T n f = λ ( n ) f for n = 1 , 2 , 3 , . . . , where T n is the Hecke operator. A newform is a cusp form that is an eigenform for all Hecke operators.

Background Results Sketch of Proof Newforms We say f is a Hecke eigenform if it is a cusp form and T n f = λ ( n ) f for n = 1 , 2 , 3 , . . . , where T n is the Hecke operator. A newform is a cusp form that is an eigenform for all Hecke operators. For a newform, the coefficients a f ( n ) are multiplicative.

Background Results Sketch of Proof Newforms We say f is a Hecke eigenform if it is a cusp form and T n f = λ ( n ) f for n = 1 , 2 , 3 , . . . , where T n is the Hecke operator. A newform is a cusp form that is an eigenform for all Hecke operators. For a newform, the coefficients a f ( n ) are multiplicative. We consider holomorphic cuspidal newforms of even weight k ≥ 2 and squarefree level N .

Background Results Sketch of Proof The Ramanujan Tau Function Ramanujan tau function: ∞ ∞ (1 − q n ) 24 = τ ( n ) q n = q − 24 q 2 +252 q 3 + · · · . � � ∆( z ) := q n =1 n =1

Background Results Sketch of Proof The Ramanujan Tau Function Ramanujan tau function: ∞ ∞ (1 − q n ) 24 = τ ( n ) q n = q − 24 q 2 +252 q 3 + · · · . � � ∆( z ) := q n =1 n =1 The multiplicativity of the Ramanujan tau function follows from the fact that ∆( z ) is a newform.

Background Results Sketch of Proof The Ramanujan Tau Function Ramanujan tau function: ∞ ∞ (1 − q n ) 24 = τ ( n ) q n = q − 24 q 2 +252 q 3 + · · · . � � ∆( z ) := q n =1 n =1 The multiplicativity of the Ramanujan tau function follows from the fact that ∆( z ) is a newform. Conjecture (Lehmer) For all n ≥ 1 , τ ( n ) � = 0.

Background Results Sketch of Proof The Sato-Tate Law Theorem (Deligne, 1974) If f is a newform as above, then for each prime p we have k − 1 2 . | a f ( p ) | ≤ 2 p

Background Results Sketch of Proof The Sato-Tate Law Theorem (Deligne, 1974) If f is a newform as above, then for each prime p we have k − 1 2 . | a f ( p ) | ≤ 2 p By the Deligne bound, a f ( p ) = 2 p ( k − 1) / 2 cos( θ p ) for some angle θ p ∈ [0 , π ].

Background Results Sketch of Proof The Sato-Tate Law Theorem (Deligne, 1974) If f is a newform as above, then for each prime p we have k − 1 2 . | a f ( p ) | ≤ 2 p By the Deligne bound, a f ( p ) = 2 p ( k − 1) / 2 cos( θ p ) for some angle θ p ∈ [0 , π ]. Natural question: What is the distribution of the sequence { θ p } ?

Background Results Sketch of Proof The Sato-Tate Law (Continued) Theorem(Barnet-Lamb, Geraghty, Harris, Taylor) Let f ( z ) ∈ S new (Γ 0 ( N )) be a non-CM newform. If F : [0 , π ] → C k is a continuous function, then � π 1 � lim F ( θ p ) = F ( θ ) d µ ST π ( x ) x →∞ 0 p ≤ x where d µ ST = 2 π sin 2 ( θ ) d θ is the Sato-Tate measure. Further π f , I ( x ) := # { p ≤ x : θ p ∈ I } ∼ µ ST ( I ) Li ( x ) .

Background Results Sketch of Proof Symmetric Power L -functions We begin by writing ∞ ∞ a f ( m ) q m = k − 1 � � 2 λ f ( m ) q m . f ( z ) = m m =1 m =1

Background Results Sketch of Proof Symmetric Power L -functions We begin by writing ∞ ∞ a f ( m ) q m = k − 1 � � 2 λ f ( m ) q m . f ( z ) = m m =1 m =1 From this normalization, we have 1 − e i θ p p − s � − 1 � 1 − e − i θ p p − s � − 1 � � L ( s , f ) = , p and the n -th symmetric power L -function     n 1 − e ij θ p e ( j − n ) i θ p p − s � − 1 � � � � L ( s , Sym n f ) = L p ( s ) − 1   j =0 p ∤ N p | N

Background Results Sketch of Proof Symmetric Power L -functions We begin by writing ∞ ∞ a f ( m ) q m = k − 1 � � 2 λ f ( m ) q m . f ( z ) = m m =1 m =1 From this normalization, we have 1 − e i θ p p − s � − 1 � 1 − e − i θ p p − s � − 1 � � L ( s , f ) = , p and the n -th symmetric power L -function     n 1 − e ij θ p e ( j − n ) i θ p p − s � − 1 � � � � L ( s , Sym n f ) = L p ( s ) − 1   j =0 p ∤ N p | N To pass to arithmetic progressions, we consider L ( s , Sym n f ⊗ χ ).

Background Results Sketch of Proof Previous Work Define π f , I ( x ) = # { p ≤ x : θ p ∈ I } and let µ ST ( I ) denote the Sato-Tate measure of a subinterval I ⊂ [0 , π ].

Background Results Sketch of Proof Previous Work Define π f , I ( x ) = # { p ≤ x : θ p ∈ I } and let µ ST ( I ) denote the Sato-Tate measure of a subinterval I ⊂ [0 , π ]. Rouse and Thorner (2017): under certain analytic hypotheses on the symmetric power L -functions, | π f , I ( x ) − µ ST ( I ) Li ( x ) | ≤ 3 . 33 x 3 / 4 − 3 x 3 / 4 log log x +202 x 3 / 4 log q ( f ) log x log x for all x ≥ 2, where q ( f ) = N ( k − 1)

Background Results Sketch of Proof Previous Work Define π f , I ( x ) = # { p ≤ x : θ p ∈ I } and let µ ST ( I ) denote the Sato-Tate measure of a subinterval I ⊂ [0 , π ]. Rouse and Thorner (2017): under certain analytic hypotheses on the symmetric power L -functions, | π f , I ( x ) − µ ST ( I ) Li ( x ) | ≤ 3 . 33 x 3 / 4 − 3 x 3 / 4 log log x +202 x 3 / 4 log q ( f ) log x log x for all x ≥ 2, where q ( f ) = N ( k − 1) Rouse-Thorner also leads to an explicit upper bound for the Lang-Trotter conjecture, which predicts the asymptotic of the number of primes for which a f ( p ) = c for a fixed constant c .

Background Results Sketch of Proof Assumptions on Symmetric Power L -functions We make some reasonable assumptions about the twisted Symmetric Power L -functions associated to a newform f , including:

Background Results Sketch of Proof Assumptions on Symmetric Power L -functions We make some reasonable assumptions about the twisted Symmetric Power L -functions associated to a newform f , including: The Generalized Riemann Hypothesis for the twisted symmetric power L -functions L ( s , Sym n f ⊗ χ ).

Background Results Sketch of Proof Assumptions on Symmetric Power L -functions We make some reasonable assumptions about the twisted Symmetric Power L -functions associated to a newform f , including: The Generalized Riemann Hypothesis for the twisted symmetric power L -functions L ( s , Sym n f ⊗ χ ). The existence of an analytic continuation of L ( s , Sym n f ⊗ χ ) to an entire function on C (and a corresponding functional equation).

Background Results Sketch of Proof Assumptions on Symmetric Power L -functions We make some reasonable assumptions about the twisted Symmetric Power L -functions associated to a newform f , including: The Generalized Riemann Hypothesis for the twisted symmetric power L -functions L ( s , Sym n f ⊗ χ ). The existence of an analytic continuation of L ( s , Sym n f ⊗ χ ) to an entire function on C (and a corresponding functional equation). Assumptions about the form of the above completed L -function, including its gamma factor and conductor.