Shifted Convolution L-Series Values of Elliptic Curves

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

2 Preliminaries

3 Modular Forms for Γ 0 (N)

4 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).

5 Elliptic Curves are Modular

6 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

7 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

8 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.)

9 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 )

10 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

11 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

12 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

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

14 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)

15 Goals

16 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

17 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

18 Weierstrass ζ-function

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

20 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

21 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

22 Weierstrass Mock Modular Form

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

24 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))

25 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)

26 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)

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

28 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 q 16 7q 19 + O(q 20 ).

29 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 q 16 7q 19 + O(q 20 ). Weight 0 mock modular form Ẑ+ E 27 (z) associated to f E27 : Ẑ + E 27 (z) = q q q q q q14 + O(q 17 )

30 Main Results

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

32 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}

33 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}

34 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

35 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]

36 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

37 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)).

38 Proof Idea: Holomorphic Projection

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

40 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))

41 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

42 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

43 Holomorphic Projection

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

45 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

46 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))

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

48 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

49 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)

50 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)

51 Examples

52 Example [Holomorphic Projection] (CM Curves) N π hol ( ) ( ) ( ( ) ) σ(3n)q 3n ( d q 3n ) σ(n)q 3n 36 n=1 36 n=1 d 3n,3 d n=1 ( ) ( ( ) ) σ(4n)q 4n d q 4n 48 n=1 48 n=1 ( ( ) ) d 4n,2 d d q 4n 15 ( ) σ(n)q 4n 2 n=1 d 4n,4 d 2 7 n=1 ( ) ( ( ) ) σ(6n)q 6n d q 6n 72 n=1 36 n=1 ( ( ) ) d 6n,2 d d q 6n + 85 ( ) σ(n)q 6n 6 n=1 d 6n,3 d 72 5 n=1 ( ) ( ( ) ) σ(7n)q 7n d q 7n 56 n=1 56 n=1 d 7n,7 d

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

54 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 ) = gives Ẑ + E (z) = q q q q q q 5 + O(q 6 ).

55 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 ) = gives Ẑ + E (z) = q q q q q q 5 + O(q 6 ) coefficients of f E to compute L fe (z) = n=1 D f E (h; 1)q h L fe (z) = q q q q q 5 + O(q 6 )

56 Example (N = 11) dim(m 2 (Γ 0 (11)) CE 2 ) = 2 with basis F 0 11,2, F 11,2 : F 11,2 = q q q q q q6 + O(q 7 )

57 Example (N = 11) dim(m 2 (Γ 0 (11)) CE 2 ) = 2 with basis F 0 11,2, F 11,2 : F 11,2 = q q q q q q6 + O(q 7 ) Apply Theorem 1 with α = , β 1 = 1, β 2 = 0: vol(λ E ) π ( ) (f E Ẑ+ E ) αf E F11,2 = q q q q q 5 + O(q 6 ) = L fe (z)

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

59 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 q 16 7q 19 + O(q 20 ).

60 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 q 16 7q 19 + O(q 20 ). Mock modular form: Ẑ + E 27 (z) = q q q q q q14 + O(q 17 ).

61 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 q 16 7q 19 + O(q 20 ). Mock modular form: Ẑ + E 27 (z) = q q q q q q14 + O(q 17 ). Holomorphic projection: π hol (f E Ẑ+ E 27 )(z) = 1 + 3q 9 + 9q 18 12q Normalized F27,2 : F27,2 = 1 + 3q 9 + 9q 18 12q 27...

62 Conclusion

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

64 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.

65 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.

66 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

67 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.

68 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.

69 Questions?

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

Abstract. Gauss s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form GAUSS S 2 F HYPERGEOMETRIC FUNCTION AND THE CONGRUENT NUMBER ELLIPTIC CURVE AHMAD EL-GUINDY AND KEN ONO Abstract Gauss s hypergeometric function gives a modular parameterization of period integrals of