Shifted Convolution L-Series Values of Elliptic Curves

Similar documents
Abstract. Gauss s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form

Mock modular forms and their shadows

Elliptic curves and modularity

A Motivated Introduction to Modular Forms

Introduction to Modular Forms

Representations of integers as sums of an even number of squares. Özlem Imamoḡlu and Winfried Kohnen

Class Number Type Relations for Fourier Coefficients of Mock Modular Forms

Cusp forms and the Eichler-Shimura relation

RANKIN-COHEN BRACKETS AND VAN DER POL-TYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION

arxiv: v3 [math.nt] 28 Jul 2012

25 Modular forms and L-functions

PAlmetto Number Theory Series

LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES

Distribution of Fourier coefficients of primitive forms

EICHLER-SHIMURA THEORY FOR MOCK MODULAR FORMS KATHRIN BRINGMANN, PAVEL GUERZHOY, ZACHARY KENT, AND KEN ONO

Applications of modular forms to partitions and multipartitions

Mock Modular Forms and Class Number Relations

SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS

Basic Background on Mock Modular Forms and Weak Harmonic Maass Forms

A MIXED MOCK MODULAR SOLUTION OF KANEKO ZAGIER EQUATION. k(k + 1) E 12

ETA-QUOTIENTS AND ELLIPTIC CURVES

Eisenstein Series and Modular Differential Equations

RIMS. Ibukiyama Zhuravlev. B.Heim

The Arithmetic of Noncongruence Modular Forms. Winnie Li. Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan

denote the Dirichlet character associated to the extension Q( D)/Q, that is χ D

Analytic Number Theory

Modular forms, combinatorially and otherwise

Ramanujan s Deathbed Letter. Larry Rolen. Emory University

REGULARIZED INNER PRODUCTS AND WEAKLY HOLOMORPHIC HECKE EIGENFORMS

On Rankin-Cohen Brackets of Eigenforms

Before we prove this result, we first recall the construction ( of) Suppose that λ is an integer, and that k := λ+ 1 αβ

The Galois Representation Associated to Modular Forms (Part I)

SPECIAL VALUES OF SHIFTED CONVOLUTION DIRICHLET SERIES

4 LECTURES ON JACOBI FORMS. 1. Plan

Computer methods for Hilbert modular forms

MOCK MODULAR FORMS AS p-adic MODULAR FORMS

Periods and generating functions in Algebraic Geometry

On congruences for the coefficients of modular forms and some applications. Kevin Lee James. B.S. The University of Georgia, 1991

Rank-one Twists of a Certain Elliptic Curve

Ramanujan s last prophecy: quantum modular forms

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3

Quantum Mock Modular Forms Arising From eta-theta Functions

p-adic iterated integrals, modular forms of weight one, and Stark-Heegner points.

OVERPARTITION M 2-RANK DIFFERENCES, CLASS NUMBER RELATIONS, AND VECTOR-VALUED MOCK EISENSTEIN SERIES

LECTURE 2 FRANZ LEMMERMEYER

Tables of elliptic curves over number fields

Congruence Subgroups

SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS

Arithmetic properties of harmonic weak Maass forms for some small half integral weights

AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION

The Arithmetic of Elliptic Curves

Rational points on elliptic curves. cycles on modular varieties

CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION

Ramanujan s first letter to Hardy: 5 + = 1 + e 2π 1 + e 4π 1 +

TWISTS OF ELLIPTIC CURVES. Ken Ono

CHUNG PANG MOK. The correct form of Proposition 2.2 of [M1] should be stated as follows:

Distribution Agreement. In presenting this thesis or dissertation as a partial fulllment of the requirements for

RANKIN-COHEN BRACKETS AND SERRE DERIVATIVES AS POINCARÉ SERIES. φ k M = 1 2

EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

SOME CONGRUENCES FOR TRACES OF SINGULAR MODULI

Laval University, Québec September 2010

PARITY OF THE PARTITION FUNCTION. (Communicated by Don Zagier)

TATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES. Ken Ono. X(E N ) is a simple

RESEARCH ANNOUNCEMENTS PROJECTIONS OF C AUTOMORPHIC FORMS BY JACOB STURM 1

Congruences for the Coefficients of Modular Forms and Applications to Number Theory

DETERMINATION OF GL(3) CUSP FORMS BY CENTRAL VALUES OF GL(3) GL(2) L-FUNCTIONS, LEVEL ASPECT

PARITY OF THE COEFFICIENTS OF KLEIN S j-function

When 2 and 3 are invertible in A, L A is the scheme

Moonshine: Lecture 3. Moonshine: Lecture 3. Ken Ono (Emory University)

Divisibility of the 5- and 13-regular partition functions

Equations for Hilbert modular surfaces

On the low-lying zeros of elliptic curve L-functions

Elliptic Curves and the abc Conjecture

MODULAR SYMBOLS PAUL E. GUNNELLS

Euler s concordant forms

REGULARIZED INNER PRODUCTS AND WEAKLY HOLOMORPHIC HECKE EIGENFORMS

CONGRUENCES FOR BROKEN k-diamond PARTITIONS

arxiv: v1 [math.nt] 28 Jan 2010

Galois Representations

RIMS L. Title: Abstract:,,

On the arithmetic of modular forms

Modular forms and the Hilbert class field

ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS. Ken Ono. 1. Introduction

RECIPES FOR TERNARY DIOPHANTINE EQUATIONS OF SIGNATURE (p, p, k)

Workshop Automorphic Galois Representations, L-functions and Arithmetic Columbia June 17th-22nd, 2006 Saturday, June 17th 9:30 Welcome breakfast in

Independence of Heegner Points Joseph H. Silverman (Joint work with Michael Rosen)

I. An overview of the theory of Zeta functions and L-series. K. Consani Johns Hopkins University

AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION

The Galois Representation Attached to a Hilbert Modular Form

TITLES & ABSTRACTS OF TALKS

EULER S CONCORDANT FORMS. Ken Ono

On the zeros of certain modular forms

Kleine AG: Travaux de Shimura

Elliptic curves with large analytic order of the Tate-Shafarevich group

DIVISIBILITY AND DISTRIBUTION OF PARTITIONS INTO DISTINCT PARTS

BSD and the Gross-Zagier Formula

Congruences for Fishburn numbers modulo prime powers

REDUCTION OF CM ELLIPTIC CURVES AND MODULAR FUNCTION CONGRUENCES

Shimura Degrees, New Modular Degrees, and Congruence Primes

MATH 797MF PROBLEM LIST

Transcription:

Shifted Convolution L-Series Values of Elliptic Curves Nitya Mani (joint with Asra Ali) December 18, 2017

Preliminaries

Modular Forms for Γ 0 (N)

Modular Forms for Γ 0 (N) Definition The congruence subgroup Γ 0 (N) SL 2 (Z) is {( ) ( ) a b Γ 0 (N) := c d 0 } (mod N). Definition A function f : H C is a modular form of weight k for Γ 0 (N) if f is holomorphic, f (z) = (cz + d) k f ( ) az+b cz+d for all γ Γ 0 (N), f is holomorphic at the cusps: the orbits of P 1 (Q) under Γ 0 (N).

Elliptic Curves are Modular

Elliptic Curves are Modular Consider z = x + iy H where H = {z C y > 0} Any elliptic curve E/Q has associated weight 2 newform f E (z) = a E (n)q n n=1 a E (1) = 1, q = e 2πiz

Elliptic Curves are Modular Consider z = x + iy H where H = {z C y > 0} Any elliptic curve E/Q has associated weight 2 newform f E (z) = a E (n)q n n=1 a E (1) = 1, q = e 2πiz f E a modular form for Γ 0 (N E ), N E conductor of E

Elliptic Curves are Modular Consider z = x + iy H where H = {z C y > 0} Any elliptic curve E/Q has associated weight 2 newform f E (z) = a E (n)q n n=1 a E (1) = 1, q = e 2πiz f E a modular form for Γ 0 (N E ), N E conductor of E Modularity Theorem + Eichler-Shimura Theory (Taylor, Wiles, Conrad, et al.)

Elliptic Curves are Modular Consider z = x + iy H where H = {z C y > 0} Any elliptic curve E/Q has associated weight 2 newform f E (z) = a E (n)q n n=1 a E (1) = 1, q = e 2πiz f E a modular form for Γ 0 (N E ), N E conductor of E Modularity Theorem + Eichler-Shimura Theory (Taylor, Wiles, Conrad, et al.) p prime: a E (p) = p + 1 #E(F p )

Shifted Convolution L-series Modular form: f E (z) = n=1 a E (n)q n Dirichlet L-series: L(E, s) = L(f E, s) = n=1 a E (n) n s

Shifted Convolution L-series Modular form: f E (z) = n=1 a E (n)q n Dirichlet L-series: L(E, s) = L(f E, s) = n=1 a E (n) n s Shifted convolution L-series: ( 1 D fe (h; s) = a E (n + h)a E (n) (n + h) s 1 ) n s n=1

Shifted Convolution L-series Modular form: f E (z) = n=1 a E (n)q n Dirichlet L-series: L(E, s) = L(f E, s) = n=1 a E (n) n s Shifted convolution L-series: ( 1 D fe (h; s) = a E (n + h)a E (n) (n + h) s 1 ) n s n=1 Generating function (s = 1): L fe (z) = D fe (h; 1)q h h=1

Previous Work Shifted convolution L-series defined by Hoffstein, Hulse, Reznikov (generalized Rankin-Selberg convolutions)

Previous Work Shifted convolution L-series defined by Hoffstein, Hulse, Reznikov (generalized Rankin-Selberg convolutions) Values are essentially coefficients of mixed mock modular forms; recent work on p-adic properties and asymptotic behavior (Beckwith, Bringmann, Mertens, Ono)

Goals

Goals Given E/Q, we have weight 2 newform: f E = a E (n)q n, a E (1) = 1 n=1 Understand variants of L(f E, s) at s = 1

Goals Given E/Q, we have weight 2 newform: f E = a E (n)q n, a E (1) = 1 n=1 Understand variants of L(f E, s) at s = 1 Closed form for D fe (h; 1) values for some E/Q

Weierstrass ζ-function

Weierstrass ζ-function E C/Λ E, where Λ E a 2-dimensional complex lattice

Weierstrass ζ-function E C/Λ E, where Λ E a 2-dimensional complex lattice Weierstrass ζ-function: ζ(λ E ; z) = 1 z + w Λ E \{0} ( 1 z w + 1 w + z ) w 2

Weierstrass ζ-function E C/Λ E, where Λ E a 2-dimensional complex lattice Weierstrass ζ-function: ζ(λ E ; z) = 1 z + w Λ E \{0} ( 1 z w + 1 w + z ) w 2 d dz ζ(λ E ; z) = (Λ E ; z) (Eisenstein) Lattice-invariant function Z E (z) Z E (z) = ζ(λ E ; z) S(Λ E )z S(Λ E ) = lim s 0 w Λ E \{0} π vol(λ E ) z 1 w 2 w 2s

Weierstrass Mock Modular Form

Weierstrass Mock Modular Form Eichler integral of associated newform f E : E fe (z) = n=1 a E (n) q n n

Weierstrass Mock Modular Form Eichler integral of associated newform f E : E fe (z) = n=1 a E (n) q n n Harmonic Maaß form ẐE (z): Ẑ E (z) = Z E (E fe (z))

Weierstrass Mock Modular Form Eichler integral of associated newform f E : E fe (z) = n=1 a E (n) q n n Harmonic Maaß form ẐE (z): Ẑ E (z) = Z E (E fe (z)) Can split ẐE (z): ẐE (z) = Ẑ+ E (z) + Ẑ E (z)

Weierstrass Mock Modular Form Eichler integral of associated newform f E : E fe (z) = n=1 a E (n) q n n Harmonic Maaß form ẐE (z): Ẑ E (z) = Z E (E fe (z)) Can split ẐE (z): ẐE (z) = Ẑ+ E (z) + Ẑ E (z) If genus(x 0 (N)) = 1, Ẑ+ E (z) is holomorphic; called the Weierstrass mock modular form Z + E (E f E (z)) = ζ(λ E ; E fe (z)) S(Λ E )E fe (z)

Example (N = 27) N = 27 and E 27 : y 2 + y = x 3 7

Example (N = 27) N = 27 and E 27 : y 2 + y = x 3 7 Weight 2 modular form associated with E 27 : f E27 = q 2q 4 q 7 + 5q 13 + 4q 16 7q 19 + O(q 20 ).

Example (N = 27) N = 27 and E 27 : y 2 + y = x 3 7 Weight 2 modular form associated with E 27 : f E27 = q 2q 4 q 7 + 5q 13 + 4q 16 7q 19 + O(q 20 ). Weight 0 mock modular form Ẑ+ E 27 (z) associated to f E27 : Ẑ + E 27 (z) = q 1 + 1 2 q2 + 1 5 q5 + 3 4 q8 6 11 q11 1 2 q14 + O(q 17 )

Main Results

Elliptic Curves E/Q is a strong Weil elliptic curve If E has conductor N E, then genus(x 0 (N E )) = 1

Elliptic Curves E/Q is a strong Weil elliptic curve If E has conductor N E, then genus(x 0 (N E )) = 1 Two situations: 1. N E is squarefree N {11, 14, 15, 17, 19, 21}

Elliptic Curves E/Q is a strong Weil elliptic curve If E has conductor N E, then genus(x 0 (N E )) = 1 Two situations: 1. N E is squarefree N {11, 14, 15, 17, 19, 21} 2. E has complex multiplication N {27, 32, 36, 49}

Theorem 1 E strong Weil curve, genus(x 0 (N)) = 1, N squarefree Recall L fe (z) = h=1 D f E (h; 1)q h

Theorem 1 E strong Weil curve, genus(x 0 (N)) = 1, N squarefree Recall L fe (z) = h=1 D f E (h; 1)q h Theorem (A-M 17) L fe (z; 1) = vol(λ E ) π ( (f E (z) Ẑ+ E (z)) αf E (z) i F N,2 (z) ) F N,2 (z) is the Eisenstein series for Γ 0(N) nonvanishing only at the cusp α = (f E Ẑ+ E )[1] π vol Λ E D fe (1; 1) F N,2 [1]

Theorem 2 E is a strong Weil elliptic curve with complex multiplication where genus(x 0 (N)) = 1 Recall L fe (z) = h=1 D f E (h; 1)q h F N,2 is the Eisenstein series for Γ 0(N) nonvanishing only at the cusp

Theorem 2 E is a strong Weil elliptic curve with complex multiplication where genus(x 0 (N)) = 1 Recall L fe (z) = h=1 D f E (h; 1)q h F N,2 is the Eisenstein series for Γ 0(N) nonvanishing only at the cusp Theorem (A-M 17) With L fe (z) defined as above, L fe (z; 1) = vol(λ E )( (fe (z) π Ẑ+ E )(z) F N,2 (z)).

Proof Idea: Holomorphic Projection

Motivation Consider f : H C continuous, transforms like a modular form for Γ 0 (N) of weight k 2, has moderate growth at cusps

Motivation Consider f : H C continuous, transforms like a modular form for Γ 0 (N) of weight k 2, has moderate growth at cusps Induces a linear functional based on the Petersson Inner Product g g, f for g S k (Γ 0 (N))

Motivation Consider f : H C continuous, transforms like a modular form for Γ 0 (N) of weight k 2, has moderate growth at cusps Induces a linear functional based on the Petersson Inner Product g g, f for g S k (Γ 0 (N)) = There is some f S k (Γ 0 (N)) such that, f =, f

Motivation Consider f : H C continuous, transforms like a modular form for Γ 0 (N) of weight k 2, has moderate growth at cusps Induces a linear functional based on the Petersson Inner Product g g, f for g S k (Γ 0 (N)) = There is some f S k (Γ 0 (N)) such that, f =, f f is essentially the holomorphic projection of f

Holomorphic Projection

Holomorphic Projection If f (z) is holomorphic, π hol f = f (z)

Holomorphic Projection If f (z) is holomorphic, π hol f = f (z) { M k (Γ 0 (N)) k > 2 π hol (f (z)) M 2 (Γ 0 (N)) CE 2 k = 2

Holomorphic Projection If f (z) is holomorphic, π hol f = f (z) { M k (Γ 0 (N)) k > 2 π hol (f (z)) M 2 (Γ 0 (N)) CE 2 k = 2 g, f = g, π hol f for g S k (Γ 0 (N))

Holomorphic Projection Consider a weight 2 newform f E S 2 (Γ 0 (N)) associated to an elliptic curve E

Holomorphic Projection Consider a weight 2 newform f E S 2 (Γ 0 (N)) associated to an elliptic curve E and the Weierstrass mock modular form Ẑ+ E

Holomorphic Projection Consider a weight 2 newform f E S 2 (Γ 0 (N)) associated to an elliptic curve E and the Weierstrass mock modular form Ẑ+ E Proposition The holomorphic projection is given by π hol (Ẑ+ E f E ) = Ẑ+ E f E h=1 q h ( 1 a E (n + h)a E (n) (n + h) 1 ) n n=1 }{{} D fe (h;1)

Holomorphic Projection Consider a weight 2 newform f E S 2 (Γ 0 (N)) associated to an elliptic curve E and the Weierstrass mock modular form Ẑ+ E Proposition The holomorphic projection is given by π hol (Ẑ+ E f E ) = Ẑ+ E f E h=1 q h and π hol (Ẑ+ E f E ) M 2 (Γ 0 (N)) CE 2. ( 1 a E (n + h)a E (n) (n + h) 1 ) n n=1 }{{} D fe (h;1)

Examples

Example [Holomorphic Projection] (CM Curves) N π hol ( ) ( 1 27 1 24 ) ( ( ) ) σ(3n)q 3n 1 1 + 12 ( d q 3n + 1 + 3 ) σ(n)q 3n 36 n=1 36 n=1 d 3n,3 d n=1 ( 1 32 1 24 ) ( ( ) ) σ(4n)q 4n + 47 1 + 24 d q 4n 48 n=1 48 n=1 ( ( ) ) d 4n,2 d 7 1 + 8 d q 4n 15 ( 1 + 24 ) σ(n)q 4n 2 n=1 d 4n,4 d 2 7 n=1 ( 1 36 1 24 ) ( ( ) ) σ(6n)q 6n 1 1 + 24 d q 6n 72 n=1 36 n=1 ( ( ) ) d 6n,2 d 1 1 + 12 d q 6n + 85 ( 1 + 24 ) σ(n)q 6n 6 n=1 d 6n,3 d 72 5 n=1 ( 1 49 1 24 ) ( ( ) ) σ(7n)q 7n + 55 1 + 4 d q 7n 56 n=1 56 n=1 d 7n,7 d

Example (N = 11) Modular curve X 0 (11), dim(x 0 (11)) = 1: E : y 2 + y = x 3 x 2 10x 20.

Example (N = 11) Modular curve X 0 (11), dim(x 0 (11)) = 1: E : y 2 + y = x 3 x 2 10x 20. S(Λ E ) = 0.38124... gives Ẑ + E (z) = q 1 + 1 + 0.9520... q + 1.547... q 2 + 0.3493... q 3 + 1.976... q 4 2.609... q 5 + O(q 6 ).

Example (N = 11) Modular curve X 0 (11), dim(x 0 (11)) = 1: E : y 2 + y = x 3 x 2 10x 20. S(Λ E ) = 0.38124... gives Ẑ + E (z) = q 1 + 1 + 0.9520... q + 1.547... q 2 + 0.3493... q 3 + 1.976... q 4 2.609... q 5 + O(q 6 ). 10 5 coefficients of f E to compute L fe (z) = n=1 D f E (h; 1)q h L fe (z) = 0.7063... q + 1.562... q 2 + 0.0944... q 3 + 1.237... q 4 2.026... q 5 + O(q 6 )

Example (N = 11) dim(m 2 (Γ 0 (11)) CE 2 ) = 2 with basis F 0 11,2, F 11,2 : F 11,2 = 1 + 1 5 q + 3 5 q2 + 4 5 q3 + 7 5 q4 + 6 5 q5 + 12 5 q6 + O(q 7 )

Example (N = 11) dim(m 2 (Γ 0 (11)) CE 2 ) = 2 with basis F 0 11,2, F 11,2 : F 11,2 = 1 + 1 5 q + 3 5 q2 + 4 5 q3 + 7 5 q4 + 6 5 q5 + 12 5 q6 + O(q 7 ) Apply Theorem 1 with α =.0016 0, β 1 = 1, β 2 = 0: vol(λ E ) π ( ) (f E Ẑ+ E ) αf E F11,2 = 0.706... q + 1.562... q 2 + 0.0930... q 3 + 1.234... q 4 2.024... q 5 + O(q 6 ) = L fe (z)

Example (N = 27) Strong Weil curve with N E = 27: E 27 = y 2 + y = x 3 7.

Example (N = 27) Strong Weil curve with N E = 27: E 27 = y 2 + y = x 3 7. Weight 2 newform: f E27 = q 2q 4 q 7 + 5q 13 + 4q 16 7q 19 + O(q 20 ).

Example (N = 27) Strong Weil curve with N E = 27: E 27 = y 2 + y = x 3 7. Weight 2 newform: f E27 = q 2q 4 q 7 + 5q 13 + 4q 16 7q 19 + O(q 20 ). Mock modular form: Ẑ + E 27 (z) = q 1 + 1 2 q2 + 1 5 q5 + 3 4 q8 6 11 q11 1 2 q14 + O(q 17 ).

Example (N = 27) Strong Weil curve with N E = 27: E 27 = y 2 + y = x 3 7. Weight 2 newform: f E27 = q 2q 4 q 7 + 5q 13 + 4q 16 7q 19 + O(q 20 ). Mock modular form: Ẑ + E 27 (z) = q 1 + 1 2 q2 + 1 5 q5 + 3 4 q8 6 11 q11 1 2 q14 + O(q 17 ). Holomorphic projection: π hol (f E Ẑ+ E 27 )(z) = 1 + 3q 9 + 9q 18 12q 27.... Normalized F27,2 : F27,2 = 1 + 3q 9 + 9q 18 12q 27...

Conclusion

Summary For nice elliptic curves E, give an explicit characterization of shifted convolution L-series values D fe (h; 1) using elliptic curve invariants.

Summary For nice elliptic curves E, give an explicit characterization of shifted convolution L-series values D fe (h; 1) using elliptic curve invariants. Enable fast computation of L fe (z) coefficients.

Summary For nice elliptic curves E, give an explicit characterization of shifted convolution L-series values D fe (h; 1) using elliptic curve invariants. Enable fast computation of L fe (z) coefficients. Understand properties of Weierstrass mock modular form Ẑ + E (z) in terms of f E, invariants of E/Q.

Summary For nice elliptic curves E, give an explicit characterization of shifted convolution L-series values D fe (h; 1) using elliptic curve invariants. Enable fast computation of L fe (z) coefficients. Understand properties of Weierstrass mock modular form Ẑ + E (z) in terms of f E, invariants of E/Q. Understand support, arithmetic properties of Fourier coefficients of f E, Ẑ+ E for CM curves E

Summary For nice elliptic curves E, give an explicit characterization of shifted convolution L-series values D fe (h; 1) using elliptic curve invariants. Enable fast computation of L fe (z) coefficients. Understand properties of Weierstrass mock modular form Ẑ + E (z) in terms of f E, invariants of E/Q. Understand support, arithmetic properties of Fourier coefficients of f E, Ẑ+ E for CM curves E Future work: generalize to other elliptic curves E, understand other L-series values + interesting variants.

Acknowledgements Thanks to the NSF and the Emory REU, in particular Ken Ono and Michael Mertens for all of their support with our projects. Thanks to the anonymous Archiv der Mathematik referee for valuable feedback on this work.

Questions?

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback