The j-function, the golden ratio, and rigid meromorphic cocycles

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1 The j-function, the golden ratio, and rigid meromorphic cocycles Henri Darmon, McGill University CNTA XV, July 2018

2 Reminiscences of CNTA 0 The 1987 CNTA in Quebec City was an exciting one for me personally, and also for number theory;

3 Reminiscences of CNTA 0 The 1987 CNTA in Quebec City was an exciting one for me personally, and also for number theory; Karl Rubin had just exhibited the first examples of elliptic curves (with CM) with finite Shafarevich-Tate groups, building on ideas of Thaine;

4 Reminiscences of CNTA 0 The 1987 CNTA in Quebec City was an exciting one for me personally, and also for number theory; Karl Rubin had just exhibited the first examples of elliptic curves (with CM) with finite Shafarevich-Tate groups, building on ideas of Thaine; I was just about to start graduate school, and these lectures made a strong impression on me;

5 Reminiscences of CNTA 0 The 1987 CNTA in Quebec City was an exciting one for me personally, and also for number theory; Karl Rubin had just exhibited the first examples of elliptic curves (with CM) with finite Shafarevich-Tate groups, building on ideas of Thaine; I was just about to start graduate school, and these lectures made a strong impression on me; It was at CNTA 0 that I really met for the first time many future colleagues: Claude, Jean-Marie, Damien, Hershy, Ram and Kumar Murty...

6 Andrew Granville Andrew Granville was also present at the meeting, having just finished writing his PhD under Paulo Ribenboim;

7 Andrew Granville Andrew Granville was also present at the meeting, having just finished writing his PhD under Paulo Ribenboim; Andrew at CNTA 0 Andrew Today

8 The j-function, the golden ratio, and rigid meromorphic cocycles Henri Darmon, McGill University CNTA, July 2018

9 This is joint work with Jan Vonk

10 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0},

11 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z),

12 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz.

13 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz. This function crops up in many parts of mathematics:

14 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz. This function crops up in many parts of mathematics: complex analysis (proof of the uniformisation theorem),

15 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz. This function crops up in many parts of mathematics: complex analysis (proof of the uniformisation theorem), elliptic curves and number theory,

16 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz. This function crops up in many parts of mathematics: complex analysis (proof of the uniformisation theorem), elliptic curves and number theory, finite group theory (monstrous moonshine),

17 The modular function j The modular function j(z) is the unique holomorphic function on H := {z C, Im(z) > 0}, satisfying j(z + 1) = j(z), j( 1/z) = j(z), j(q) = 1 q q + with q = e2πiz. This function crops up in many parts of mathematics: complex analysis (proof of the uniformisation theorem), elliptic curves and number theory, finite group theory (monstrous moonshine), etc.

18 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H.

19 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H. The theory of complex multiplication asserts that these values are algebraic integers, and enjoy many remarkable properties.

20 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H. The theory of complex multiplication asserts that these values are algebraic integers, and enjoy many remarkable properties. Examples: j(i) = 1728; j ( ) = 0; j ( ) = 3375.

21 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H. The theory of complex multiplication asserts that these values are algebraic integers, and enjoy many remarkable properties. Examples: j(i) = 1728; j ( ) = w, where j ( ) = 0; j ( ) = 3375.

22 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H. The theory of complex multiplication asserts that these values are algebraic integers, and enjoy many remarkable properties. Examples: j(i) = 1728; j ( ) = w, where j ( ) = 0; j ( ) = w w w = 0.

23 Special values, or singular moduli A singular modulus is a special value of j at a quadratic imaginary argument (CM point) in H. The theory of complex multiplication asserts that these values are algebraic integers, and enjoy many remarkable properties. Examples: j(i) = 1728; j ( ) = w, where j ( ) = 0; j ( ) = w w w = 0. It generates an interesting cubic extension of Q( 23).

24 Gauss composition and class groups For τ quadratic, there is a unique primitive integral binary quadratic form F (x, y) = Ax 2 + Bxy + Cy 2 satisfying A > 0 and F (τ, 1) = 0.

25 Gauss composition and class groups For τ quadratic, there is a unique primitive integral binary quadratic form F (x, y) = Ax 2 + Bxy + Cy 2 satisfying A > 0 and F (τ, 1) = 0. D := B 2 4AC is called the discriminant of τ.

26 Gauss composition and class groups For τ quadratic, there is a unique primitive integral binary quadratic form F (x, y) = Ax 2 + Bxy + Cy 2 satisfying A > 0 and F (τ, 1) = 0. D := B 2 4AC is called the discriminant of τ. The set of SL 2 (Z)-equivalence classes of primitive integral binary quadratic forms of discriminant D forms a group under Gauss s composition law, denoted G D : the class group of discriminant D.

27 Gauss composition and class groups For τ quadratic, there is a unique primitive integral binary quadratic form F (x, y) = Ax 2 + Bxy + Cy 2 satisfying A > 0 and F (τ, 1) = 0. D := B 2 4AC is called the discriminant of τ. The set of SL 2 (Z)-equivalence classes of primitive integral binary quadratic forms of discriminant D forms a group under Gauss s composition law, denoted G D : the class group of discriminant D. Class groups have been intensely studied since the time of Gauss, and are fundamental objects in number theory.

28 Abelian extensions and Kronecker s theorem The class group G D measures the failure of the principle of unique factorisation in the ring of integers of K = Q( D),

29 Abelian extensions and Kronecker s theorem The class group G D measures the failure of the principle of unique factorisation in the ring of integers of K = Q( D), and also arises in the study of abelian extensions of K:

30 Abelian extensions and Kronecker s theorem The class group G D measures the failure of the principle of unique factorisation in the ring of integers of K = Q( D), and also arises in the study of abelian extensions of K: If D is a fundamental discriminant, the group G D is canonically identified with Gal(H D /K), where H D is the Hilbert class field associated to D.

31 Abelian extensions and Kronecker s theorem The class group G D measures the failure of the principle of unique factorisation in the ring of integers of K = Q( D), and also arises in the study of abelian extensions of K: If D is a fundamental discriminant, the group G D is canonically identified with Gal(H D /K), where H D is the Hilbert class field associated to D. Theorem (Kronecker) For quadratic τ H of fundamental discriminant D < 0, the singular modulus j(τ) is an algebraic integer in H D, and generates this field.

32 Differences of singular moduli and their factorisations Gross, Zagier (1984). For all τ 1, τ 2 quadratic imaginary,

33 Differences of singular moduli and their factorisations Gross, Zagier (1984). For all τ 1, τ 2 quadratic imaginary, the quantity J(τ 1, τ 2 ) := j(τ 1 ) j(τ 2 ) is a smooth algebraic integer with an explicit factorisation.

34 Differences of singular moduli and their factorisations Gross, Zagier (1984). For all τ 1, τ 2 quadratic imaginary, the quantity J(τ 1, τ 2 ) := j(τ 1 ) j(τ 2 ) is a smooth algebraic integer with an explicit factorisation. The norm of J(τ 1, τ 2 ) to Z is divisible only by (large powers of) small primes.

35 Differences of singular moduli and their factorisations Gross, Zagier (1984). For all τ 1, τ 2 quadratic imaginary, the quantity J(τ 1, τ 2 ) := j(τ 1 ) j(τ 2 ) is a smooth algebraic integer with an explicit factorisation. The norm of J(τ 1, τ 2 ) to Z is divisible only by (large powers of) small primes. The equation J(τ 1, τ 2 ) + J(τ 2, τ 3 ) = J(τ 1, τ 3 )

36 Differences of singular moduli and their factorisations Gross, Zagier (1984). For all τ 1, τ 2 quadratic imaginary, the quantity J(τ 1, τ 2 ) := j(τ 1 ) j(τ 2 ) is a smooth algebraic integer with an explicit factorisation. The norm of J(τ 1, τ 2 ) to Z is divisible only by (large powers of) small primes. The equation J(τ 1, τ 2 ) + J(τ 2, τ 3 ) = J(τ 1, τ 3 ) expresses a powerful number as a sum of two powerful numbers.

37 An example D = 4 and 163 have class number one,

38 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ,

39 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ) 3, j = , (

40 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ) 3, j = , ( ) J, i 2 ( = ,

41 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ) 3, j = , ( ) J, i 2 ( = , = ,

42 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ) 3, j = , ( ) J, i 2 ( = , = , =

43 An example D = 4 and 163 have class number one, and, j(i) = 1728 = ) 3, j = , ( ) J, i 2 ( = , = , = Similar examples arise from any J(τ 1, τ 2 ) for which j(τ 1 ) and j(τ 2 ) are integers.

44 The abc conjecture The abc conjecture predicts that there can not be too many such instances.

45 The abc conjecture The abc conjecture predicts that there can not be too many such instances. Why doesn t complex multiplication contradict the abc conjecture?

46 The Gauss class number problem Theorem (Gauss; Heilbronn; Baker, Heegner, Stark) There are finitely many D < 0 with for which G D = {1}, namely 3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43 67, 163.

47 The Gauss class number problem Theorem (Gauss; Heilbronn; Baker, Heegner, Stark) There are finitely many D < 0 with for which G D = {1}, namely 3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43 67, 163. Hence there are finitely many quadratic imaginary τ for which j(τ) Z.

48 The Gauss class number problem Theorem (Gauss; Heilbronn; Baker, Heegner, Stark) There are finitely many D < 0 with for which G D = {1}, namely 3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43 67, 163. Hence there are finitely many quadratic imaginary τ for which j(τ) Z. The abc conjecture is saved!

49 The Gauss class number problem Theorem (Gauss; Heilbronn; Baker, Heegner, Stark) There are finitely many D < 0 with for which G D = {1}, namely 3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43 67, 163. Hence there are finitely many quadratic imaginary τ for which j(τ) Z. The abc conjecture is saved! Granville, Stark (2000): A strong form of the abc conjecture implies the non-existence of Siegel zeroes for L-functions attached to odd Dirichlet characters.

50 How to disprove the abc conjecture ;-) Real quadratic discriminants behave differently from negative ones.

51 How to disprove the abc conjecture ;-) Real quadratic discriminants behave differently from negative ones. Conjecture (Gauss) There are infinitely many D > 0 for which G D = {1}.

52 How to disprove the abc conjecture ;-) Real quadratic discriminants behave differently from negative ones. Conjecture (Gauss) There are infinitely many D > 0 for which G D = {1}. Empirically, it seems like the majority of prime D > 0 somewhat over 3/4 have this class number one property.

53 How to disprove the abc conjecture ;-) Real quadratic discriminants behave differently from negative ones. Conjecture (Gauss) There are infinitely many D > 0 for which G D = {1}. Empirically, it seems like the majority of prime D > 0 somewhat over 3/4 have this class number one property. One might disprove the abc conjecture by extending the definition of J(τ 1, τ 2 ) so that it can be evaluated at real quadratic, and not just imaginary quadratic, arguments!

54 The motivating question ( ) Let ϕ := be the golden ratio.

55 The motivating question ( ) Let ϕ := be the golden ratio. Question. What is j(ϕ)...?

56 The motivating question ( ) Let ϕ := be the golden ratio. Question. What is j(ϕ)...? Answer.

57 The motivating question ( ) Let ϕ := be the golden ratio. Question. What is j(ϕ)...? Answer. It depends on p.

58 The motivating question ( ) Let ϕ := be the golden ratio. Question. What is j(ϕ)...? Answer. It depends on p. Here, p is a prime congruent to 2 or 3 modulo 5.

59 The motivating question ( ) Let ϕ := be the golden ratio. Question. What is j(ϕ)...? Answer. It depends on p. Here, p is a prime congruent to 2 or 3 modulo 5. The rest of the talk will try to explain this cryptic answer.

60 p-adic methods Imaginary and real quadratic fields differ in that their usual completions are isomorphic to C and R respectively.

61 p-adic methods Imaginary and real quadratic fields differ in that their usual completions are isomorphic to C and R respectively. After fixing a prime p, one can consider the p-adic distance, whereby two rational numbers are decreed to be close if their difference is divisible by a high power of p.

62 p-adic methods Imaginary and real quadratic fields differ in that their usual completions are isomorphic to C and R respectively. After fixing a prime p, one can consider the p-adic distance, whereby two rational numbers are decreed to be close if their difference is divisible by a high power of p. The completion of Q with respect to this new metric is called the field Q p of p-adic numbers.

63 p-adic methods Imaginary and real quadratic fields differ in that their usual completions are isomorphic to C and R respectively. After fixing a prime p, one can consider the p-adic distance, whereby two rational numbers are decreed to be close if their difference is divisible by a high power of p. The completion of Q with respect to this new metric is called the field Q p of p-adic numbers. The field Q p plays the role of R, while the counterpart of C is C p := Q p.

64 Drinfeld s p-adic upper-half plane Likewise, the complex upper half plane H with its action of SL 2 (R) by Möbius transformations, admits a good p-adic analogue.

65 Drinfeld s p-adic upper-half plane Likewise, the complex upper half plane H with its action of SL 2 (R) by Möbius transformations, admits a good p-adic analogue. Drinfeld. The p-adic analytic space H p := P 1 (C p ) P 1 (Q p ) is called the p-adic upper half-plane.

66 Drinfeld s p-adic upper-half plane Likewise, the complex upper half plane H with its action of SL 2 (R) by Möbius transformations, admits a good p-adic analogue. Drinfeld. The p-adic analytic space H p := P 1 (C p ) P 1 (Q p ) is called the p-adic upper half-plane. It is analogous to the union H + H = P 1 (C) P 1 (R) of two copies of the Poincaré upper half plane.

67 Drinfeld s p-adic upper-half plane Likewise, the complex upper half plane H with its action of SL 2 (R) by Möbius transformations, admits a good p-adic analogue. Drinfeld. The p-adic analytic space H p := P 1 (C p ) P 1 (Q p ) is called the p-adic upper half-plane. It is analogous to the union H + H = P 1 (C) P 1 (R) of two copies of the Poincaré upper half plane. Tate, Mumford: a rigid analytic function on H p is a C p -valued function whose restriction to every good ( affinoid ) subset S is a uniform limit of rational functions having poles outside of S.

68 Drinfeld s p-adic upper-half plane Likewise, the complex upper half plane H with its action of SL 2 (R) by Möbius transformations, admits a good p-adic analogue. Drinfeld. The p-adic analytic space H p := P 1 (C p ) P 1 (Q p ) is called the p-adic upper half-plane. It is analogous to the union H + H = P 1 (C) P 1 (R) of two copies of the Poincaré upper half plane. Tate, Mumford: a rigid analytic function on H p is a C p -valued function whose restriction to every good ( affinoid ) subset S is a uniform limit of rational functions having poles outside of S. This is the proper generalisation of holomorphic functions on H.

69 RM points Unlike its complex counterpart, H p contains plenty of real quadratic arguments.

70 RM points Unlike its complex counterpart, H p contains plenty of real quadratic arguments. E.g., the golden ratio belongs to H 2, H 3, H 5, H 7, H 13...,

71 RM points Unlike its complex counterpart, H p contains plenty of real quadratic arguments. E.g., the golden ratio belongs to H 2, H 3, H 5, H 7, H 13..., (but not to H 11, H 19,...).

72 RM points Unlike its complex counterpart, H p contains plenty of real quadratic arguments. E.g., the golden ratio belongs to H 2, H 3, H 5, H 7, H 13..., (but not to H 11, H 19,...). Definition A point on τ H p is called a real multiplication (RM) point if it belongs to H p K for some real quadratic field K.

73 RM points Unlike its complex counterpart, H p contains plenty of real quadratic arguments. E.g., the golden ratio belongs to H 2, H 3, H 5, H 7, H 13..., (but not to H 11, H 19,...). Definition A point on τ H p is called a real multiplication (RM) point if it belongs to H p K for some real quadratic field K. Goal: Define a convincing p-adic avatar of j which can be meaningfully evaluated at RM points, leading to singular moduli for real quadratic τ H p.

74 Rigid meromorphic functions on H p Question: What is this p-adic analogue of the j-function?

75 Rigid meromorphic functions on H p Question: What is this p-adic analogue of the j-function? Classical setting: j(z) is an SL 2 (Z)-invariant meromorphic function on H.

76 Rigid meromorphic functions on H p Question: What is this p-adic analogue of the j-function? Classical setting: j(z) is an SL 2 (Z)-invariant meromorphic function on H. The p-adic setting: A rigid meromorphic function is a ratio of rigid analytic functions.

77 Rigid meromorphic functions on H p Question: What is this p-adic analogue of the j-function? Classical setting: j(z) is an SL 2 (Z)-invariant meromorphic function on H. The p-adic setting: A rigid meromorphic function is a ratio of rigid analytic functions. Let M := the ring of rigid meromorphic functions on H p.

78 SL 2 (Z)-invariant functions First futile attempt: Study rigid meromorphic functions that are invariant under SL 2 (Z) SL 2 (Q p ).

79 SL 2 (Z)-invariant functions First futile attempt: Study rigid meromorphic functions that are invariant under SL 2 (Z) SL 2 (Q p ). This is a futile, because: The actions of SL 2 (Z) on H p is not discrete in the p-adic topology.

80 SL 2 (Z)-invariant functions First futile attempt: Study rigid meromorphic functions that are invariant under SL 2 (Z) SL 2 (Q p ). This is a futile, because: The actions of SL 2 (Z) on H p is not discrete in the p-adic topology. The subgroup of translations z z + n, with n Z, already has non-discrete orbits!

81 SL 2 (Z)-invariant functions First futile attempt: Study rigid meromorphic functions that are invariant under SL 2 (Z) SL 2 (Q p ). This is a futile, because: The actions of SL 2 (Z) on H p is not discrete in the p-adic topology. The subgroup of translations z z + n, with n Z, already has non-discrete orbits! There are no non-constant SL 2 (Z)-invariant elements in M:

82 SL 2 (Z)-invariant functions First futile attempt: Study rigid meromorphic functions that are invariant under SL 2 (Z) SL 2 (Q p ). This is a futile, because: The actions of SL 2 (Z) on H p is not discrete in the p-adic topology. The subgroup of translations z z + n, with n Z, already has non-discrete orbits! There are no non-constant SL 2 (Z)-invariant elements in M: H 0 (SL 2 (Z), M) = C p.

83 The p-modular group In fact, it turns out to be profitable to consider functions that have good transformation properties under an even larger subgroup of SL 2 (Q p ):

84 The p-modular group In fact, it turns out to be profitable to consider functions that have good transformation properties under an even larger subgroup of SL 2 (Q p ): the p-modular group Γ := SL 2 (Z[1/p]).

85 The p-modular group In fact, it turns out to be profitable to consider functions that have good transformation properties under an even larger subgroup of SL 2 (Q p ): the p-modular group Γ := SL 2 (Z[1/p]). This does nothing to alleviate the problems of the previous slide: H 0 (Γ, M) = C p.

86 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M.

87 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p higher cohomology! is uninteresting, consider its

88 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p is uninteresting, consider its higher cohomology! Definition A rigid meromorphic cocycle is a class in H 1 (Γ, M )

89 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p is uninteresting, consider its higher cohomology! Definition A rigid meromorphic cocycle is a class in H 1 (Γ, M ) whose restriction to the parabolic subgroups of Γ is C p -valued.

90 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p is uninteresting, consider its higher cohomology! Definition A rigid meromorphic cocycle is a class in H 1 (Γ, M ) whose restriction to the parabolic subgroups of Γ is C p -valued. H 1 f (Γ, M ) := the group of rigid meromorphic cocycles.

91 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p is uninteresting, consider its higher cohomology! Definition A rigid meromorphic cocycle is a class in H 1 (Γ, M ) whose restriction to the parabolic subgroups of Γ is C p -valued. H 1 f (Γ, M ) := the group of rigid meromorphic cocycles. J(γ 1 γ 2 )(z) = J(γ 1 )(z) J(γ 2 )(γ 1 1 z).

92 Rigid meromorphic modular cocycles Let M be the multiplicative group of non-zero elements of M. Key idea: Since H 0 (Γ, M ) = C p is uninteresting, consider its higher cohomology! Definition A rigid meromorphic cocycle is a class in H 1 (Γ, M ) whose restriction to the parabolic subgroups of Γ is C p -valued. H 1 f (Γ, M ) := the group of rigid meromorphic cocycles. J(γ 1 γ 2 )(z) = J(γ 1 )(z) J(γ 2 )(γ 1 1 z). Thesis: rigid meromorphic cocycles play the role of meromorphic modular functions in extending CM theory to real quadratic fields.

93 Evaluating a modular cocycle at an RM point

94 Evaluating a modular cocycle at an RM point An element τ H p is an RM point if and only if Stab Γ (τ) = ±γ τ is an infinite group of rank one.

95 Evaluating a modular cocycle at an RM point An element τ H p is an RM point if and only if is an infinite group of rank one. Definition Stab Γ (τ) = ±γ τ If J H 1 (Γ, M ) is a rigid meromorphic cocycle, and τ H p is an RM point, then the value of J at τ is J[τ] := J(γ τ )(τ) C p { }.

96 Evaluating a modular cocycle at an RM point An element τ H p is an RM point if and only if is an infinite group of rank one. Definition Stab Γ (τ) = ±γ τ If J H 1 (Γ, M ) is a rigid meromorphic cocycle, and τ H p is an RM point, then the value of J at τ is J[τ] := J(γ τ )(τ) C p { }. The quantity J[τ] is a well-defined numerical invariant,

97 Evaluating a modular cocycle at an RM point An element τ H p is an RM point if and only if is an infinite group of rank one. Definition Stab Γ (τ) = ±γ τ If J H 1 (Γ, M ) is a rigid meromorphic cocycle, and τ H p is an RM point, then the value of J at τ is J[τ] := J(γ τ )(τ) C p { }. The quantity J[τ] is a well-defined numerical invariant, independent of the cocycle representing the class of J,

98 Evaluating a modular cocycle at an RM point An element τ H p is an RM point if and only if is an infinite group of rank one. Definition Stab Γ (τ) = ±γ τ If J H 1 (Γ, M ) is a rigid meromorphic cocycle, and τ H p is an RM point, then the value of J at τ is J[τ] := J(γ τ )(τ) C p { }. The quantity J[τ] is a well-defined numerical invariant, independent of the cocycle representing the class of J, and J[γτ] = J[τ], for all γ Γ.

99 Explicit rigid meromorphic cocycles For any τ H p, the orbit Γτ is dense in H p.

100 Explicit rigid meromorphic cocycles For any τ H p, the orbit Γτ is dense in H p. Suppose that τ is an RM point in H p.

101 Explicit rigid meromorphic cocycles For any τ H p, the orbit Γτ is dense in H p. Suppose that τ is an RM point in H p. The set Σ τ := {w Γτ such that ww < 0} is discrete.

102 Explicit rigid meromorphic cocycles For any τ H p, the orbit Γτ is dense in H p. Suppose that τ is an RM point in H p. The set Σ τ := {w Γτ such that ww < 0} is discrete. Theorem (Jan Vonk, D) For each real quadratic τ H p, there is a unique J τ H 1 (Γ, M /C p ) for which

103 Explicit rigid meromorphic cocycles For any τ H p, the orbit Γτ is dense in H p. Suppose that τ is an RM point in H p. The set Σ τ := {w Γτ such that ww < 0} is discrete. Theorem (Jan Vonk, D) For each real quadratic τ H p, there is a unique J τ H 1 (Γ, M /C p ) for which J τ (S)(z) (z w) sgn(w), S := 0 1. w Σ τ 1 0

104 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i.

105 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem:

106 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem: Theorem (Vonk, D) If J H 1 (Γ, M ), then there exists A divisor Div(Γ\H RM p );

107 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem: Theorem (Vonk, D) If J H 1 (Γ, M ), then there exists A divisor Div(Γ\Hp RM ); An analytic cocycle J 0 H 1 (Γ, A /C p );

108 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem: Theorem (Vonk, D) If J H 1 (Γ, M ), then there exists A divisor Div(Γ\Hp RM ); An analytic cocycle J 0 H 1 (Γ, A /C p ); such that J = J J 0 (mod C p ).

109 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem: Theorem (Vonk, D) If J H 1 (Γ, M ), then there exists A divisor Div(Γ\Hp RM ); An analytic cocycle J 0 H 1 (Γ, A /C p ); such that J = J J 0 (mod C p ). determines J completely.

110 The classification theorem If = i n i[τ i ] Div(Γ\H p ), write J := i J n i τ i. A hint that rigid meromorphic cocycles are intimately tied to the arithmetic of real quadratic fields is given by the following classification theorem: Theorem (Vonk, D) If J H 1 (Γ, M ), then there exists A divisor Div(Γ\Hp RM ); An analytic cocycle J 0 H 1 (Γ, A /C p ); such that J = J J 0 (mod C p ). determines J completely. (It is called the divisor of J).

111 Fields of definition The previous discussion allows us to associate to an arbitrary J H 1 f (Γ, M ) a field of definition.

112 Fields of definition The previous discussion allows us to associate to an arbitrary J H 1 f (Γ, M ) a field of definition. Definition The field of definition of J, denoted H J, is the compositum of H D as D ranges over the discriminants of all the RM points in the support of the divisor of J.

113 The main conjecture of real multiplication The main assertion of complex multiplication:

114 The main conjecture of real multiplication The main assertion of complex multiplication: Theorem (Kronecker,...) Let J be a meromorphic modular function on SL 2 (Z)\H with fourier coefficients in a field H J.

115 The main conjecture of real multiplication The main assertion of complex multiplication: Theorem (Kronecker,...) Let J be a meromorphic modular function on SL 2 (Z)\H with fourier coefficients in a field H J. For all imaginary quadratic τ H of discriminant D, the value J(τ) belongs to the compositum of H J and H D.

116 The main conjecture of real multiplication The main assertion of complex multiplication: Theorem (Kronecker,...) Let J be a meromorphic modular function on SL 2 (Z)\H with fourier coefficients in a field H J. For all imaginary quadratic τ H of discriminant D, the value J(τ) belongs to the compositum of H J and H D. Conjecture (Jan Vonk, D) Let J be a rigid meromorphic cocycle, and let H J denote its field of definition.

117 The main conjecture of real multiplication The main assertion of complex multiplication: Theorem (Kronecker,...) Let J be a meromorphic modular function on SL 2 (Z)\H with fourier coefficients in a field H J. For all imaginary quadratic τ H of discriminant D, the value J(τ) belongs to the compositum of H J and H D. Conjecture (Jan Vonk, D) Let J be a rigid meromorphic cocycle, and let H J denote its field of definition. For all real quadratic τ H p of discriminant D, the value J[τ] belongs to the compositum of H J and H D.

118 Monstrous primes and principal divisors Definition A prime p is said to be monstrous if it satisfies the following equivalent conditions:

119 Monstrous primes and principal divisors Definition A prime p is said to be monstrous if it satisfies the following equivalent conditions: p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71.

120 Monstrous primes and principal divisors Definition A prime p is said to be monstrous if it satisfies the following equivalent conditions: p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. p divides the cardinality of the monster sporadic simple group;

121 Monstrous primes and principal divisors Definition A prime p is said to be monstrous if it satisfies the following equivalent conditions: p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. p divides the cardinality of the monster sporadic simple group; the quotient X 0 (p)/w p has genus zero.

122 Monstrous primes and principal divisors Definition A prime p is said to be monstrous if it satisfies the following equivalent conditions: p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. p divides the cardinality of the monster sporadic simple group; the quotient X 0 (p)/w p has genus zero. Proposition If p is a monstrous prime and τ is any RM point on H p, then (τ) (pτ) is a principal divisor.

123 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio.

124 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal.

125 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal. Let J ϕ + be the associated rigid meromorphic cocycle.

126 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal. Let J ϕ + be the associated rigid meromorphic cocycle. The RM point τ = 223 of discriminant 223 has class number 6, and J + ϕ [ 223] appears to satisfy:

127 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal. Let J ϕ + be the associated rigid meromorphic cocycle. The RM point τ = 223 of discriminant 223 has class number 6, and J + ϕ [ 223] appears to satisfy: p = x x x x x x ,

128 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal. Let J ϕ + be the associated rigid meromorphic cocycle. The RM point τ = 223 of discriminant 223 has class number 6, and J + ϕ [ 223] appears to satisfy: p = x x x x x x , p = x x x x x x ,

129 A cocycle attached to the golden ratio Recall that ϕ = is the golden ratio. If p = 2, 3, 5, 7, 13, 17, 23, or 47 is a monstrous prime which is ±1 (mod 5), the divisor (ϕ) (pϕ) is principal. Let J ϕ + be the associated rigid meromorphic cocycle. The RM point τ = 223 of discriminant 223 has class number 6, and J + ϕ [ 223] appears to satisfy: p = x x x x x x , p = x x x x x x , p = 47. 4x 6 + 4x 5 + x 4 2x 3 + x 2 + 4x + 4.

130 Differences of RM singular moduli If p = 2, 3, 5, 7, or 13, then any τ H RM p is principal.

131 Differences of RM singular moduli If p = 2, 3, 5, 7, or 13, then any τ H RM p is principal. Definition: The quantity J p (τ 1, τ 2 ) := J τ1 [τ 2 ] is called the p-adic intersection number of τ 1 and τ 2.

132 Differences of RM singular moduli If p = 2, 3, 5, 7, or 13, then any τ H RM p is principal. Definition: The quantity J p (τ 1, τ 2 ) := J τ1 [τ 2 ] is called the p-adic intersection number of τ 1 and τ 2. Conjecture (Jan Vonk, D) The quantity J p (τ 1, τ 2 ) belongs to the compositum H 1 H 2 of the ring class fields of K 1 = Q(τ 1 ) and K 2 = Q(τ 2 ) attached to τ 1 and τ 2 respectively,

133 Differences of RM singular moduli If p = 2, 3, 5, 7, or 13, then any τ H RM p is principal. Definition: The quantity J p (τ 1, τ 2 ) := J τ1 [τ 2 ] is called the p-adic intersection number of τ 1 and τ 2. Conjecture (Jan Vonk, D) The quantity J p (τ 1, τ 2 ) belongs to the compositum H 1 H 2 of the ring class fields of K 1 = Q(τ 1 ) and K 2 = Q(τ 2 ) attached to τ 1 and τ 2 respectively, and behaves in all important respects just like the classical J (τ 1, τ 2 ) := J(τ 1, τ 2 ) = j(τ 1 ) j(τ 2 ) of Gross-Zagier.

134 Gross-Zagier factorisations Let τ 1 and τ 2 H be CM points with (negative) discriminants D 1 and D 2, and let O D1 and O D2 be the associated quadratic orders.

135 Gross-Zagier factorisations Let τ 1 and τ 2 H be CM points with (negative) discriminants D 1 and D 2, and let O D1 and O D2 be the associated quadratic orders. For each prime q that is non-split in Q(τ 1 ) and Q(τ 2 ), Gross and Zagier define an explicit purely algebraic quantity e q (τ 1, τ 2 ) in terms of the embeddings of O D1 and O D2 into the definite quaternion algebra B q ramified at q and.

136 Gross-Zagier factorisations Let τ 1 and τ 2 H be CM points with (negative) discriminants D 1 and D 2, and let O D1 and O D2 be the associated quadratic orders. For each prime q that is non-split in Q(τ 1 ) and Q(τ 2 ), Gross and Zagier define an explicit purely algebraic quantity e q (τ 1, τ 2 ) in terms of the embeddings of O D1 and O D2 into the definite quaternion algebra B q ramified at q and. Conjecture (Gross-Zagier) If q is split in either Q(τ 1 ) or Q(τ 2 ), then ord q J (τ 1, τ 2 ) = 0.

137 Gross-Zagier factorisations Let τ 1 and τ 2 H be CM points with (negative) discriminants D 1 and D 2, and let O D1 and O D2 be the associated quadratic orders. For each prime q that is non-split in Q(τ 1 ) and Q(τ 2 ), Gross and Zagier define an explicit purely algebraic quantity e q (τ 1, τ 2 ) in terms of the embeddings of O D1 and O D2 into the definite quaternion algebra B q ramified at q and. Conjecture (Gross-Zagier) If q is split in either Q(τ 1 ) or Q(τ 2 ), then ord q J (τ 1, τ 2 ) = 0. Otherwise, ord q J (τ 1, τ 2 ) = e q (τ 1, τ 2 ).

138 Factorisations of real quadratic singular moduli For τ 1, τ 2 H RM p, one can likewise define an arithmetic quantity e qp (τ 1, τ 2 ) involving the indefinite quaternion algebra B qp ramified at q and p.

139 Factorisations of real quadratic singular moduli For τ 1, τ 2 Hp RM, one can likewise define an arithmetic quantity e qp (τ 1, τ 2 ) involving the indefinite quaternion algebra B qp ramified at q and p. The group Γ pq = (O Bqp ) 1 SL 2(R) acts discretely and co-compactly on H;

140 Factorisations of real quadratic singular moduli For τ 1, τ 2 Hp RM, one can likewise define an arithmetic quantity e qp (τ 1, τ 2 ) involving the indefinite quaternion algebra B qp ramified at q and p. The group Γ pq = (O Bqp ) 1 SL 2(R) acts discretely and co-compactly on H; The Riemann surface Γ pq \H is called the Shimura curve X qp attached to the pair (p, q).

141 Factorisations of real quadratic singular moduli For τ 1, τ 2 Hp RM, one can likewise define an arithmetic quantity e qp (τ 1, τ 2 ) involving the indefinite quaternion algebra B qp ramified at q and p. The group Γ pq = (O Bqp ) 1 SL 2(R) acts discretely and co-compactly on H; The Riemann surface Γ pq \H is called the Shimura curve X qp attached to the pair (p, q). To τ 1 and τ 2 are associated hyperbolic geodesics γ 1 and γ 2 on X qp.

142 Factorisations of real quadratic singular moduli For τ 1, τ 2 Hp RM, one can likewise define an arithmetic quantity e qp (τ 1, τ 2 ) involving the indefinite quaternion algebra B qp ramified at q and p. The group Γ pq = (O Bqp ) 1 SL 2(R) acts discretely and co-compactly on H; The Riemann surface Γ pq \H is called the Shimura curve X qp attached to the pair (p, q). To τ 1 and τ 2 are associated hyperbolic geodesics γ 1 and γ 2 on X qp. e qp (τ 1, τ 2 ) := weighted intersection of γ 1 and γ 2 on X qp.

143 A Gross-Zagier-style factorisation Let τ 1 H D 1 p and τ 2 H D 2 p.

144 A Gross-Zagier-style factorisation Let τ 1 H D 1 p and τ 2 H D 2 p. Conjecture (Jan Vonk, D) Let q be a prime.

145 A Gross-Zagier-style factorisation Let τ 1 H D 1 p and τ 2 H D 2 p. Conjecture (Jan Vonk, D) Let q be a prime. If q is split in either Q(τ 1 ) or Q(τ 2 ), then ord q J p (τ 1, τ 2 ) = 0.

146 A Gross-Zagier-style factorisation Let τ 1 H D 1 p and τ 2 H D 2 p. Conjecture (Jan Vonk, D) Let q be a prime. If q is split in either Q(τ 1 ) or Q(τ 2 ), then ord q J p (τ 1, τ 2 ) = 0. Otherwise, ord q J p (τ 1, τ 2 ) = e qp (τ 1, τ 2 ).

147 An example James Rickards has developed and implemented efficient algorithms for computing the q-weighted topological intersection numbers e qp (τ 1, τ 2 ) of real quadratic geodesics on the Shimura curve X qp.

148 An example: D 1 = 13, D 2 = 285 = , p = 2

149 An example: D 1 = 13, D 2 = 285 = , p = 2 J 2 (τ 1, τ 2 ) satisfies (to 800 digits of 2-adic precision)

150 An example: D 1 = 13, D 2 = 285 = , p = 2 J 2 (τ 1, τ 2 ) satisfies (to 800 digits of 2-adic precision) x x x x ,

151 An example: D 1 = 13, D 2 = 285 = , p = 2 J 2 (τ 1, τ 2 ) satisfies (to 800 digits of 2-adic precision) x x x x , James Rickards: The topological intersection e q2 (τ 1, τ 2 ) is zero except for the primes q in the following table: q e q2 (τ 1, τ 2 )

152 An example: D 1 = 13, D 2 = 285 = , p = 2 J 2 (τ 1, τ 2 ) satisfies (to 800 digits of 2-adic precision) x x x x , James Rickards: The topological intersection e q2 (τ 1, τ 2 ) is zero except for the primes q in the following table: q e q2 (τ 1, τ 2 ) Jan Vonk, D: =

153 An example: D 1 = 13, D 2 = 285 = , p = 2 J 2 (τ 1, τ 2 ) satisfies (to 800 digits of 2-adic precision) x x x x , James Rickards: The topological intersection e q2 (τ 1, τ 2 ) is zero except for the primes q in the following table: q e q2 (τ 1, τ 2 ) Jan Vonk, D: = ,

154 Conclusion We have described a convincing conjectural extension of the theory of complex multiplication, and of differences of singular moduli, to real quadratic fields.

155 Conclusion We have described a convincing conjectural extension of the theory of complex multiplication, and of differences of singular moduli, to real quadratic fields. Whither the abc conjecture?

156 Some key difference with the classical setting Meromorphic functions form a ring: they can be added and multiplied.

157 Some key difference with the classical setting Meromorphic functions form a ring: they can be added and multiplied. Rigid meromorphic cocycles can only be multiplied!

158 Some key difference with the classical setting Meromorphic functions form a ring: they can be added and multiplied. Rigid meromorphic cocycles can only be multiplied! The (trivial) identity J (τ 1, τ 2 ) + J (τ 2, τ 3 ) = J (τ 1, τ 2 ) has no counterpart for J p (τ 1, τ 2 ).

159 Some key difference with the classical setting Meromorphic functions form a ring: they can be added and multiplied. Rigid meromorphic cocycles can only be multiplied! The (trivial) identity J (τ 1, τ 2 ) + J (τ 2, τ 3 ) = J (τ 1, τ 2 ) has no counterpart for J p (τ 1, τ 2 ). The abc conjecture is saved!

160 Some key difference with the classical setting Meromorphic functions form a ring: they can be added and multiplied. Rigid meromorphic cocycles can only be multiplied! The (trivial) identity J (τ 1, τ 2 ) + J (τ 2, τ 3 ) = J (τ 1, τ 2 ) has no counterpart for J p (τ 1, τ 2 ). The abc conjecture is saved! Likewise there seem to be no applications to Siegel zeroes for even Dirichlet characters, à la Granville-Stark.

161 Theoretical evidence The evidence for this theory so far is largely experimental, but there has been some recent progress on the theoretical front.

162 Theoretical evidence The evidence for this theory so far is largely experimental, but there has been some recent progress on the theoretical front. Definition The cuspidal value of a rigid meromorphic cocycle J is the quantity J[ ] := J(γ ), where γ ( ) =.

163 Theoretical evidence The evidence for this theory so far is largely experimental, but there has been some recent progress on the theoretical front. Definition The cuspidal value of a rigid meromorphic cocycle J is the quantity Theorem (Vonk, D) J[ ] := J(γ ), where γ ( ) =. J[ ] is algebraic. More precisely, a power of it belongs to H J.

164 Theoretical evidence The evidence for this theory so far is largely experimental, but there has been some recent progress on the theoretical front. Definition The cuspidal value of a rigid meromorphic cocycle J is the quantity Theorem (Vonk, D) J[ ] := J(γ ), where γ ( ) =. J[ ] is algebraic. More precisely, a power of it belongs to H J. The proof rests on a remarkable recent work of Samit Dasgupta and Mahesh Kakde, building on the proof of the rank one p-adic Gross-Stark conjecture by Dasgupta-Darmon-Pollack (2006).

165 Thank you for your attention.

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