The fourth moment of Dirichlet L -functions along a coset and the Weyl bound Ian Petrow ETH Z¨ urich Joint work with Matthew P. Young Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 1 / 16
The subconvexity problem Given π an automorphic form, let C ( π ) be its analytic conductor. Example: χ a Dirichlet character modulo q and | · | it : n �→ n it C ( χ. | · | it ) = (1 + | t | ) q . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 2 / 16
The subconvexity problem Given π an automorphic form, let C ( π ) be its analytic conductor. Example: χ a Dirichlet character modulo q and | · | it : n �→ n it C ( χ. | · | it ) = (1 + | t | ) q . Trivial “convexity” bound: L (1 / 2 , π ) ≪ C ( π ) 1 / 4+ ε Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 2 / 16
The subconvexity problem Given π an automorphic form, let C ( π ) be its analytic conductor. Example: χ a Dirichlet character modulo q and | · | it : n �→ n it C ( χ. | · | it ) = (1 + | t | ) q . Trivial “convexity” bound: L (1 / 2 , π ) ≪ C ( π ) 1 / 4+ ε GRH ⇒ Generalized Lindel¨ of hypothesis: L (1 / 2 , π ) ≪ C ( π ) ε Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 2 / 16
The subconvexity problem Given π an automorphic form, let C ( π ) be its analytic conductor. Example: χ a Dirichlet character modulo q and | · | it : n �→ n it C ( χ. | · | it ) = (1 + | t | ) q . Trivial “convexity” bound: L (1 / 2 , π ) ≪ C ( π ) 1 / 4+ ε GRH ⇒ Generalized Lindel¨ of hypothesis: L (1 / 2 , π ) ≪ C ( π ) ε Subconvexity problem: show that there exists δ > 0 so that L (1 / 2 , π ) ≪ C ( π ) 1 / 4 − δ Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 2 / 16
The subconvexity problem Given π an automorphic form, let C ( π ) be its analytic conductor. Example: χ a Dirichlet character modulo q and | · | it : n �→ n it C ( χ. | · | it ) = (1 + | t | ) q . Trivial “convexity” bound: L (1 / 2 , π ) ≪ C ( π ) 1 / 4+ ε GRH ⇒ Generalized Lindel¨ of hypothesis: L (1 / 2 , π ) ≪ C ( π ) ε Subconvexity problem: show that there exists δ > 0 so that L (1 / 2 , π ) ≪ C ( π ) 1 / 4 − δ Michel-Venkatesh (2010): π on GL 1 or GL 2 with unspecified δ > 0. Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 2 / 16
Subconvexity results Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 3 / 16
Subconvexity results First subconvexity result: Weyl (1922): ζ (1 1 6 + ε . 2 + it ) ≪ (1 + | t | ) Based on the method of Weyl differencing; invariance of continuous functions under translation. Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 3 / 16
Subconvexity results First subconvexity result: Weyl (1922): ζ (1 1 6 + ε . 2 + it ) ≪ (1 + | t | ) Based on the method of Weyl differencing; invariance of continuous functions under translation. Burgess (1962) χ primitive modulo q : 3 16 + ε L (1 / 2 , χ ) ≪ q Throws away correlation between character sums on many very short intervals, but uses H¨ older and RH for curves over finite fields. Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 3 / 16
Subconvexity results First subconvexity result: Weyl (1922): ζ (1 1 6 + ε . 2 + it ) ≪ (1 + | t | ) Based on the method of Weyl differencing; invariance of continuous functions under translation. Burgess (1962) χ primitive modulo q : 3 16 + ε L (1 / 2 , χ ) ≪ q Throws away correlation between character sums on many very short intervals, but uses H¨ older and RH for curves over finite fields. The exponent 3 / 16 re-occurs often in modern incarnations of these problems (Blomer-Harcos-Michel, Blomer-Harcos, Han Wu). Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 3 / 16
The Weyl exponent and Conrey-Iwaniec Until recently, the exponent 1 / 6 only known in special cases related to quadratic characters (Conrey-Iwaniec, Ivi´ c, Young, P.-Young) Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 4 / 16
The Weyl exponent and Conrey-Iwaniec Until recently, the exponent 1 / 6 only known in special cases related to quadratic characters (Conrey-Iwaniec, Ivi´ c, Young, P.-Young) Conrey-Iwaniec (2000): if χ 2 = 1 with odd (sq.-free) conductor q 1 6 + ε , L (1 / 2 , χ ) ≪ q using input from automorphic forms and Deligne’s RH for varieties . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 4 / 16
The Weyl exponent and Conrey-Iwaniec Until recently, the exponent 1 / 6 only known in special cases related to quadratic characters (Conrey-Iwaniec, Ivi´ c, Young, P.-Young) Conrey-Iwaniec (2000): if χ 2 = 1 with odd (sq.-free) conductor q 1 6 + ε , L (1 / 2 , χ ) ≪ q using input from automorphic forms and Deligne’s RH for varieties . H it ( m , ψ ) = { Maass newforms of level m char. ψ and spec. par. it } . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 4 / 16
The Weyl exponent and Conrey-Iwaniec Until recently, the exponent 1 / 6 only known in special cases related to quadratic characters (Conrey-Iwaniec, Ivi´ c, Young, P.-Young) Conrey-Iwaniec (2000): if χ 2 = 1 with odd (sq.-free) conductor q 1 6 + ε , L (1 / 2 , χ ) ≪ q using input from automorphic forms and Deligne’s RH for varieties . H it ( m , ψ ) = { Maass newforms of level m char. ψ and spec. par. it } . � T L (1 / 2 , π ⊗ χ ) 3 + � � � | L (1 / 2 + it , χ ) | 6 ℓ ( t ) dt − T | t j |≤ T m | q π ∈ H itj ( m , 1) ≪ T B q 1+ ε . B < ∞ unspecified, ℓ ( t ) = t 2 (4 + t 2 ) − 1 . L (1 / 2 , π ⊗ χ ) ≥ 0 by Guo. Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 4 / 16
Fact (Atkin-Li 1978, or use local Langands for GL 2 ): If m | q , χ conductor q , π ∈ H it ( m , χ 2 ), then π ⊗ χ ∈ H it ( q 2 , 1). Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 5 / 16
Fact (Atkin-Li 1978, or use local Langands for GL 2 ): If m | q , χ conductor q , π ∈ H it ( m , χ 2 ), then π ⊗ χ ∈ H it ( q 2 , 1). Theorem (P.-Young (2018)) Let χ be primitive of conductor q cube-free and not quadratic. � T L (1 / 2 , π ⊗ χ ) 3 + | L (1 / 2 + it , χ ) | 6 dt � � � − T π ∈ H itj ( m ,χ 2 ) | t j |≤ T m | q ≪ T B q 1+ ε . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 5 / 16
Fact (Atkin-Li 1978, or use local Langands for GL 2 ): If m | q , χ conductor q , π ∈ H it ( m , χ 2 ), then π ⊗ χ ∈ H it ( q 2 , 1). Theorem (P.-Young (2018)) Let χ be primitive of conductor q cube-free and not quadratic. � T L (1 / 2 , π ⊗ χ ) 3 + | L (1 / 2 + it , χ ) | 6 dt � � � − T π ∈ H itj ( m ,χ 2 ) | t j |≤ T m | q ≪ T B q 1+ ε . Theorem (P.-Young (2018)) Let χ be primitive of conductor q cube-free and T ≫ q ε . � T +1 L (1 / 2 , π ⊗ χ ) 3 + | L (1 / 2 + it , χ ) | 6 dt � � � T π ∈ H itj ( m ,χ 2 ) T < | t j |≤ T +1 m | q ≪ ( Tq ) 1+ ε . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 5 / 16
Fact (Atkin-Li 1978, or use local Langands for GL 2 ): If m | q , χ conductor q , π ∈ H it ( m , χ 2 ), then π ⊗ χ ∈ H it ( q 2 , 1). Theorem (P.-Young (2019)) Let χ be primitive of conductor q cube-free and not quadratic. � T L (1 / 2 , π ⊗ χ ) 3 + | L (1 / 2 + it , χ ) | 6 dt � � � − T π ∈ H itj ( m ,χ 2 ) | t j |≤ T m | q ≪ T B q 1+ ε . Theorem (P.-Young (2019)) Let χ be primitive of conductor q cube-free and T ≫ q ε . � T +1 L (1 / 2 , π ⊗ χ ) 3 + | L (1 / 2 + it , χ ) | 6 dt � � � T π ∈ H itj ( m ,χ 2 ) T < | t j |≤ T +1 m | q ≪ ( Tq ) 1+ ε . Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 6 / 16
The Weyl Bound Corollary (P.-Young 2019) For all primitive χ modulo q we have 1 6 + ε . L (1 / 2 + it , χ ) ≪ ((1 + | t | ) q ) Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 7 / 16
The Weyl Bound Corollary (P.-Young 2019) For all primitive χ modulo q we have 1 6 + ε . L (1 / 2 + it , χ ) ≪ ((1 + | t | ) q ) In other language: For any Hecke character χ on GL 1 over Q we have 1 6 + ε . L (1 / 2 , χ ) ≪ C ( χ ) Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 7 / 16
The Weyl Bound Corollary (P.-Young 2019) For all primitive χ modulo q we have 1 6 + ε . L (1 / 2 + it , χ ) ≪ ((1 + | t | ) q ) In other language: For any Hecke character χ on GL 1 over Q we have 1 6 + ε . L (1 / 2 , χ ) ≪ C ( χ ) Why did the cube-free hypothesis come up, and how to remove it? Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 7 / 16
Summary of proof Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 8 / 16
Summary of proof Apply: Approximate functional equation to expand L (1 / 2 , π ⊗ χ ) 1 Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 8 / 16
Summary of proof Apply: Approximate functional equation to expand L (1 / 2 , π ⊗ χ ) 1 Bruggeman-Kuznetsov formula 2 (for newforms, using explicit orthonormal basis of S ( q , χ 2 )) Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 8 / 16
Summary of proof Apply: Approximate functional equation to expand L (1 / 2 , π ⊗ χ ) 1 Bruggeman-Kuznetsov formula 2 (for newforms, using explicit orthonormal basis of S ( q , χ 2 )) Poisson summation (Voronoi formula for Eis. series on GL 3 ) 3 Ian Petrow (ETH Z¨ urich) The 4th moment and the Weyl bound 8 / 16
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