the of ramanujan - iiser) bhopalhome.iiserbhopal.ac.in/~kashyap/tau.pdfthe ˝ of ramanujan v. kumar...
TRANSCRIPT
The τ of Ramanujan
V. Kumar MurtyUniversity of Toronto
Indian Institute of Science Education and Research, BhopalBhopal, India
October 10, 2011
What is the τ function?
It is defined by the equality
q∞∏n=1
(1− qn)24 =∞∑n=1
τ(n)qn.
Thus, τ(1) = 1, τ(2) = −24, τ(3) = 252.
Why is it interesting?
I The main aim of this talk is to explain why Ramanujan’sstudy of the arithmetic properties of this function is not onlyinteresting, but one of the epoch-making insights of ourcentury.
I As with much of Ramanujan’s work, many different strandsare tied together here and we have to separate them tounderstand what is going on.
Why is it interesting?
I The main aim of this talk is to explain why Ramanujan’sstudy of the arithmetic properties of this function is not onlyinteresting, but one of the epoch-making insights of ourcentury.
I As with much of Ramanujan’s work, many different strandsare tied together here and we have to separate them tounderstand what is going on.
A table of the first few values
n τ(n) n τ(n) n τ(n)
1 1 11 534,612 21 -4,219,488
2 -24 12 -370,944 22 -12,830,688
3 252 13 -577,738 23 18,643,272
4 -1472 14 401,856 24 21,288,960
5 4830 15 1,217,160 25 -25,499,225
6 -6048 16 987,136 26 13,865,712
7 -16744 17 -6,905,934 27 -73,279,080
8 84480 18 2,727,432 28 24,647,168
9 -113,643 19 10,661,420 29 128,406,630
10 -115,920 20 -7,109,760 30 -29,211,840
Growth Estimate
I The table suggests the values are growing quite rapidly.
I Ramanujan proved that τ(n) < n6 so it is in fact at mostpolynomial growth.
I He conjectured a stronger estimate
|τ(n)| ≤ d(n)n11/2
where d(n) denotes the number of positive divisors of n.
Growth Estimate
I The table suggests the values are growing quite rapidly.
I Ramanujan proved that τ(n) < n6 so it is in fact at mostpolynomial growth.
I He conjectured a stronger estimate
|τ(n)| ≤ d(n)n11/2
where d(n) denotes the number of positive divisors of n.
Growth Estimate
I The table suggests the values are growing quite rapidly.
I Ramanujan proved that τ(n) < n6 so it is in fact at mostpolynomial growth.
I He conjectured a stronger estimate
|τ(n)| ≤ d(n)n11/2
where d(n) denotes the number of positive divisors of n.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
The Ramanujan Hypothesis
I The question attracted the attention and efforts of manyauthors who proved significant but incremental improvementsof Ramanujan’s estimate.
I Hardy gave it the name “The Ramanujan Hypothesis”.
I It was finally proved by Deligne by reducing it to the Weilconjectures.
I The Weil conjectures are a set of conjectures about zetafunctions associated to algebraic varieties over finite fields. Inparticular, they give an analogue of the Riemann Hypothesisfor such zeta functions.
I It is not at all evident that there is any connection to theRamanujan Hypothesis.
I We don’t have an alternate proof.
Convergence and HolomorphyI Thinking of q as a complex number with |q| < 1 and taking
the logarithm of∞∏n=1
(1− qn)24
we get
−24∞∑n=1
∞∑m=1
qnm
m= −24
∞∑m=1
1
m
qm
1− qm.
This converges for |q| < 1.
I For z ∈ C with positive imaginary part, q = e2πiz satisfies|q| < 1. Thus, the function
∆(z) = q∞∏n=1
(1− qn)24
is a holomorphic function on the upper half plane. Moreover,it is non-vanishing there.
Convergence and HolomorphyI Thinking of q as a complex number with |q| < 1 and taking
the logarithm of∞∏n=1
(1− qn)24
we get
−24∞∑n=1
∞∑m=1
qnm
m= −24
∞∑m=1
1
m
qm
1− qm.
This converges for |q| < 1.I For z ∈ C with positive imaginary part, q = e2πiz satisfies|q| < 1. Thus, the function
∆(z) = q∞∏n=1
(1− qn)24
is a holomorphic function on the upper half plane. Moreover,it is non-vanishing there.
The Discriminant Function
I Replacing z by z + 1 doesn’t change the value of q = e2πiz so
∆(z + 1) = ∆(z).
I Less evident is that replacing z by −1/z also gives rise to asymmetry
∆(−1/z) = z12∆(z).
I These properties, together with the holomorphy and the“vanishing at ∞” makes ∆ a modular (cusp) form of weight12 for the group SL2(Z).
The Discriminant Function
I Replacing z by z + 1 doesn’t change the value of q = e2πiz so
∆(z + 1) = ∆(z).
I Less evident is that replacing z by −1/z also gives rise to asymmetry
∆(−1/z) = z12∆(z).
I These properties, together with the holomorphy and the“vanishing at ∞” makes ∆ a modular (cusp) form of weight12 for the group SL2(Z).
The Discriminant Function
I Replacing z by z + 1 doesn’t change the value of q = e2πiz so
∆(z + 1) = ∆(z).
I Less evident is that replacing z by −1/z also gives rise to asymmetry
∆(−1/z) = z12∆(z).
I These properties, together with the holomorphy and the“vanishing at ∞” makes ∆ a modular (cusp) form of weight12 for the group SL2(Z).
The Discriminant Function
I ∆(z) had been studied by other authors before Ramanujan,notably Dedekind.
I The transformation properties were known and used by theseauthors.
I ∆(z) occurs in many places such as the Kronecker LimitFormula, construction of Ramachandra units, transformationproperties of Dedekind sums, ...
The Discriminant Function
I ∆(z) had been studied by other authors before Ramanujan,notably Dedekind.
I The transformation properties were known and used by theseauthors.
I ∆(z) occurs in many places such as the Kronecker LimitFormula, construction of Ramachandra units, transformationproperties of Dedekind sums, ...
The Discriminant Function
I ∆(z) had been studied by other authors before Ramanujan,notably Dedekind.
I The transformation properties were known and used by theseauthors.
I ∆(z) occurs in many places such as the Kronecker LimitFormula, construction of Ramachandra units, transformationproperties of Dedekind sums, ...
∆(z) and sums of squares
I Ramanujan might have noticed it in the context of formulasfor the number of ways of writing an integer as a sum ofsquares.
I Clasically, Jacobi and others had explicit formulae for writinga number as a sum of 2 squares and 4 squares in terms offunctions like the function σ(n), the sum of the positivedivisors of n.
I The ∆ function arises when you ask about writing a numberas a sum of 24 squares. Here one gets an explicit formula interms of elementary functions like σk(n), the sum of the k-thpowers of the positive divisors of n (for various values of k)and one “mysterious” function, namely the τ function.
I In other words, the τ function appears as an “error” term.
∆(z) and sums of squares
I Ramanujan might have noticed it in the context of formulasfor the number of ways of writing an integer as a sum ofsquares.
I Clasically, Jacobi and others had explicit formulae for writinga number as a sum of 2 squares and 4 squares in terms offunctions like the function σ(n), the sum of the positivedivisors of n.
I The ∆ function arises when you ask about writing a numberas a sum of 24 squares. Here one gets an explicit formula interms of elementary functions like σk(n), the sum of the k-thpowers of the positive divisors of n (for various values of k)and one “mysterious” function, namely the τ function.
I In other words, the τ function appears as an “error” term.
∆(z) and sums of squares
I Ramanujan might have noticed it in the context of formulasfor the number of ways of writing an integer as a sum ofsquares.
I Clasically, Jacobi and others had explicit formulae for writinga number as a sum of 2 squares and 4 squares in terms offunctions like the function σ(n), the sum of the positivedivisors of n.
I The ∆ function arises when you ask about writing a numberas a sum of 24 squares. Here one gets an explicit formula interms of elementary functions like σk(n), the sum of the k-thpowers of the positive divisors of n (for various values of k)and one “mysterious” function, namely the τ function.
I In other words, the τ function appears as an “error” term.
∆(z) and sums of squares
I Ramanujan might have noticed it in the context of formulasfor the number of ways of writing an integer as a sum ofsquares.
I Clasically, Jacobi and others had explicit formulae for writinga number as a sum of 2 squares and 4 squares in terms offunctions like the function σ(n), the sum of the positivedivisors of n.
I The ∆ function arises when you ask about writing a numberas a sum of 24 squares. Here one gets an explicit formula interms of elementary functions like σk(n), the sum of the k-thpowers of the positive divisors of n (for various values of k)and one “mysterious” function, namely the τ function.
I In other words, the τ function appears as an “error” term.
Ramanujan’s great insight
I Ramanujan’s original contribution and great insight was torealize that the coefficients τ(n) might be interesting in andof themselves and not just as an “error” term.
I It is the beginning of the modern arithmetic theory ofautomorphic forms.
I Work of Hardy before coming into contact with Ramanujanmight be termed as all related to GL1 theory.
I Ramanujan’s insight opened the door to the study of GL2theory by number theorists, including Hardy.
Ramanujan’s great insight
I Ramanujan’s original contribution and great insight was torealize that the coefficients τ(n) might be interesting in andof themselves and not just as an “error” term.
I It is the beginning of the modern arithmetic theory ofautomorphic forms.
I Work of Hardy before coming into contact with Ramanujanmight be termed as all related to GL1 theory.
I Ramanujan’s insight opened the door to the study of GL2theory by number theorists, including Hardy.
Ramanujan’s great insight
I Ramanujan’s original contribution and great insight was torealize that the coefficients τ(n) might be interesting in andof themselves and not just as an “error” term.
I It is the beginning of the modern arithmetic theory ofautomorphic forms.
I Work of Hardy before coming into contact with Ramanujanmight be termed as all related to GL1 theory.
I Ramanujan’s insight opened the door to the study of GL2theory by number theorists, including Hardy.
Ramanujan’s great insight
I Ramanujan’s original contribution and great insight was torealize that the coefficients τ(n) might be interesting in andof themselves and not just as an “error” term.
I It is the beginning of the modern arithmetic theory ofautomorphic forms.
I Work of Hardy before coming into contact with Ramanujanmight be termed as all related to GL1 theory.
I Ramanujan’s insight opened the door to the study of GL2theory by number theorists, including Hardy.
Related Products
I ∆(z) has a product structure. We can look at other productsthat have interesting coefficients when expanded.
I Euler studied the product
∞∏n=1
(1− qn)
and proved that it is equal to∑n∈Z
(−1)nq(3n2+3n)/2.
I Note that this product does not have a nice transformationproperty like ∆ (in particular, it is not a modular form).
Related Products
I ∆(z) has a product structure. We can look at other productsthat have interesting coefficients when expanded.
I Euler studied the product
∞∏n=1
(1− qn)
and proved that it is equal to∑n∈Z
(−1)nq(3n2+3n)/2.
I Note that this product does not have a nice transformationproperty like ∆ (in particular, it is not a modular form).
Related Products
I ∆(z) has a product structure. We can look at other productsthat have interesting coefficients when expanded.
I Euler studied the product
∞∏n=1
(1− qn)
and proved that it is equal to∑n∈Z
(−1)nq(3n2+3n)/2.
I Note that this product does not have a nice transformationproperty like ∆ (in particular, it is not a modular form).
Euler and Jacobi
I Jacobi studied the cube
∞∏n=1
(1− qn)3
which is also not a modular form.
I He proved that it is equal to
∞∑n=0
(2n + 1)qn(n+1)/2.
I By raising Euler’s formula to the 24-th power, or Jacobi’sformula to the 8-th power, we get an expression for the τfunction.
Euler and Jacobi
I Jacobi studied the cube
∞∏n=1
(1− qn)3
which is also not a modular form.
I He proved that it is equal to
∞∑n=0
(2n + 1)qn(n+1)/2.
I By raising Euler’s formula to the 24-th power, or Jacobi’sformula to the 8-th power, we get an expression for the τfunction.
Euler and Jacobi
I Jacobi studied the cube
∞∏n=1
(1− qn)3
which is also not a modular form.
I He proved that it is equal to
∞∑n=0
(2n + 1)qn(n+1)/2.
I By raising Euler’s formula to the 24-th power, or Jacobi’sformula to the 8-th power, we get an expression for the τfunction.
A formula for the τ function
Using the identities of Euler and Jacobi, we get
(n−1)τ(n) =∑
1≤|m|≤an
(n − 1− 25m
2(3m + 1)
)(n − m
2(3m + 1)
)where
an =1
6
(1 + (1 + 24n)
12
).
This is due to Lehmer. There is another formula due toRamanujan that expresses τ(n) as an alternating weighted sum ofτ(m) with m�
√n.
A computational question
I The formula above shows that the value of τ(n) can becomputed in O(
√n) steps.
I Is there a faster way? In particular, can τ(n) be computed inpolynomial time?
A computational question
I The formula above shows that the value of τ(n) can becomputed in O(
√n) steps.
I Is there a faster way? In particular, can τ(n) be computed inpolynomial time?
Multiplicative Property
I Something that is not at all evident is that τ is multiplicative:
τ(mn) = τ(m)τ(n) if (m, n) = 1.
I Moreover, for powers of a fixed prime p, there is the degreetwo recursion
τ(pa+1) = τ(p)τ(pa)− p11τ(pa−1).
I Ramanujan conjectured both of these properties in his 1916paper. They were later proved by Mordell and then generalizedby Hecke in his theory of Hecke operators, a fundamentalcomponent of the modern theory of automorphic forms.
Multiplicative Property
I Something that is not at all evident is that τ is multiplicative:
τ(mn) = τ(m)τ(n) if (m, n) = 1.
I Moreover, for powers of a fixed prime p, there is the degreetwo recursion
τ(pa+1) = τ(p)τ(pa)− p11τ(pa−1).
I Ramanujan conjectured both of these properties in his 1916paper. They were later proved by Mordell and then generalizedby Hecke in his theory of Hecke operators, a fundamentalcomponent of the modern theory of automorphic forms.
Multiplicative Property
I Something that is not at all evident is that τ is multiplicative:
τ(mn) = τ(m)τ(n) if (m, n) = 1.
I Moreover, for powers of a fixed prime p, there is the degreetwo recursion
τ(pa+1) = τ(p)τ(pa)− p11τ(pa−1).
I Ramanujan conjectured both of these properties in his 1916paper. They were later proved by Mordell and then generalizedby Hecke in his theory of Hecke operators, a fundamentalcomponent of the modern theory of automorphic forms.
Back to Computation
I A result of Edixhoven (generalizing the famous algorithm ofSchoof for elliptic curves) shows that τ(p) can be computedin polynomial time.
I This means that τ(n) can be computed with the sameefficiency as factoring. With current knowledge, this meanssub-exponential time
O(exp((log n)12+ε).
Back to Computation
I A result of Edixhoven (generalizing the famous algorithm ofSchoof for elliptic curves) shows that τ(p) can be computedin polynomial time.
I This means that τ(n) can be computed with the sameefficiency as factoring. With current knowledge, this meanssub-exponential time
O(exp((log n)12+ε).
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Is the converse true? That is, given a method to computeτ(n) can n be factored?
I For an RSA modulus, n = pq with p and q large primes.
I Thenτ(n2) = (τ(p)2 − p11)(τ(q)2 − q11)
I Simplifying,
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I This means that we know τ(p)2q11 + τ(q)2p11.
I We also know the product τ(p)2q11τ(q)2p11 = τ(n)2n11.
I Hence, we know the individual terms τ(p)2q11 and τ(q)2p11.
τ(n) and factoring
I Dividing by n11, we get τ(p)2/p11 and τ(q)2/q11.
I From this, we can determine p and q.
I Summarizing, if n is an RSA modulus and we know τ(n) andτ(n2), we can factor n.
I Is there a general algorithm?
τ(n) and factoring
I Dividing by n11, we get τ(p)2/p11 and τ(q)2/q11.
I From this, we can determine p and q.
I Summarizing, if n is an RSA modulus and we know τ(n) andτ(n2), we can factor n.
I Is there a general algorithm?
τ(n) and factoring
I Dividing by n11, we get τ(p)2/p11 and τ(q)2/q11.
I From this, we can determine p and q.
I Summarizing, if n is an RSA modulus and we know τ(n) andτ(n2), we can factor n.
I Is there a general algorithm?
τ(n) and factoring
I Dividing by n11, we get τ(p)2/p11 and τ(q)2/q11.
I From this, we can determine p and q.
I Summarizing, if n is an RSA modulus and we know τ(n) andτ(n2), we can factor n.
I Is there a general algorithm?
τ(n) and primality
I If p is prime, we have
τ(p2) = τ(p)2 − p11.
I If for some n, we have
τ(n2) = τ(n)2 − n11
is n prime?
τ(n) and primality
I If p is prime, we have
τ(p2) = τ(p)2 − p11.
I If for some n, we have
τ(n2) = τ(n)2 − n11
is n prime?
τ(n) and primality
I We can certainly see that n cannot be of the form pq.
I Indeed, if n = pq,
τ(n2) = (τ(p)2 − p11)(τ(q)2 − q11).
I Expanding and rearranging, this is
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I The conditionτ(n2) = τ(n)2 − n11
would force p6|τ(p) and q6|τ(q) and this in turn forcesτ(p) = τ(q) = 0.
I But then τ(n) = 0 and so τ(n2) = τ(p2)τ(q2) = n11.
I In particular, τ(n2) 6= τ(n)2 − n11.
τ(n) and primality
I We can certainly see that n cannot be of the form pq.
I Indeed, if n = pq,
τ(n2) = (τ(p)2 − p11)(τ(q)2 − q11).
I Expanding and rearranging, this is
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I The conditionτ(n2) = τ(n)2 − n11
would force p6|τ(p) and q6|τ(q) and this in turn forcesτ(p) = τ(q) = 0.
I But then τ(n) = 0 and so τ(n2) = τ(p2)τ(q2) = n11.
I In particular, τ(n2) 6= τ(n)2 − n11.
τ(n) and primality
I We can certainly see that n cannot be of the form pq.
I Indeed, if n = pq,
τ(n2) = (τ(p)2 − p11)(τ(q)2 − q11).
I Expanding and rearranging, this is
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I The conditionτ(n2) = τ(n)2 − n11
would force p6|τ(p) and q6|τ(q) and this in turn forcesτ(p) = τ(q) = 0.
I But then τ(n) = 0 and so τ(n2) = τ(p2)τ(q2) = n11.
I In particular, τ(n2) 6= τ(n)2 − n11.
τ(n) and primality
I We can certainly see that n cannot be of the form pq.
I Indeed, if n = pq,
τ(n2) = (τ(p)2 − p11)(τ(q)2 − q11).
I Expanding and rearranging, this is
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I The conditionτ(n2) = τ(n)2 − n11
would force p6|τ(p) and q6|τ(q) and this in turn forcesτ(p) = τ(q) = 0.
I But then τ(n) = 0 and so τ(n2) = τ(p2)τ(q2) = n11.
I In particular, τ(n2) 6= τ(n)2 − n11.
τ(n) and primality
I We can certainly see that n cannot be of the form pq.
I Indeed, if n = pq,
τ(n2) = (τ(p)2 − p11)(τ(q)2 − q11).
I Expanding and rearranging, this is
τ(n2) = τ(n)2 + n11 − (τ(p)2q11 + τ(q)2p11).
I The conditionτ(n2) = τ(n)2 − n11
would force p6|τ(p) and q6|τ(q) and this in turn forcesτ(p) = τ(q) = 0.
I But then τ(n) = 0 and so τ(n2) = τ(p2)τ(q2) = n11.
I In particular, τ(n2) 6= τ(n)2 − n11.
Parity of the τ -function
The congruence
(1− qn)24 ≡ (1 + q8n)3 (mod 2)
and Jacobi’s identity
∞∏n=1
(1 + q8n)3 =∞∑
m=0
q4m(m+1)
gives the congruence
∞∑n=1
τ(n)qn ≡ q∞∑
m=0
q4m(m+1) ≡∞∑
m=0
q(2m+1)2 (mod 2).
In particular, τ(n) is odd if and only if n is an odd square. Inparticular, τ(p) is even for every prime p.
Relations discovered by Ramanujan
Ramanujan proved other congruences
I τ(p) ≡ 1 + p (mod 3)
I τ(p) ≡ p + p10 (mod 52)
I τ(p) ≡ p + p4 (mod 7)
I τ(p) ≡ 1 + p11 (mod 691)
Relations discovered by Ramanujan
Ramanujan proved other congruences
I τ(p) ≡ 1 + p (mod 3)
I τ(p) ≡ p + p10 (mod 52)
I τ(p) ≡ p + p4 (mod 7)
I τ(p) ≡ 1 + p11 (mod 691)
Relations discovered by Ramanujan
Ramanujan proved other congruences
I τ(p) ≡ 1 + p (mod 3)
I τ(p) ≡ p + p10 (mod 52)
I τ(p) ≡ p + p4 (mod 7)
I τ(p) ≡ 1 + p11 (mod 691)
Relations discovered by Ramanujan
Ramanujan proved other congruences
I τ(p) ≡ 1 + p (mod 3)
I τ(p) ≡ p + p10 (mod 52)
I τ(p) ≡ p + p4 (mod 7)
I τ(p) ≡ 1 + p11 (mod 691)
Galois Representations
I In the theory of Serre and Swinnerton-Dyer, these arise fromreducibility properties of certain families of Galoisrepresentations
ρ` : Gal(Q/Q) −→ GL2(Z`).
Here ` runs over rational primes.
I Deligne’s proof of the Ramanujan Hypothesis also constructsa family of such representations with the property that ρ` isunramified at all primes other than ` and for p 6= `, aFrobenius automorphism at p has the characteristicpolynomial
T 2 − τ(p)T + p11.
Galois Representations
I In the theory of Serre and Swinnerton-Dyer, these arise fromreducibility properties of certain families of Galoisrepresentations
ρ` : Gal(Q/Q) −→ GL2(Z`).
Here ` runs over rational primes.
I Deligne’s proof of the Ramanujan Hypothesis also constructsa family of such representations with the property that ρ` isunramified at all primes other than ` and for p 6= `, aFrobenius automorphism at p has the characteristicpolynomial
T 2 − τ(p)T + p11.
Reciprocity Laws
I To understand the significance of this, we have to recognizethat a fundamental theme in number theory is a reciprocitylaw: a law that determines the factorization of primes.
I If we consider a field K ⊇ Q of finite degree over Q, there isan analogue for K of the ordinary integers Z denoted OK .
I A prime ideal pZ in Z can be lifted to an ideal pOK which ingeneral will fail to be prime. For example if K = Q(
√−1),
then OK = Z[√−1] and
2OK = ((1 +√−1)OK )2
is the square of a prime ideal.
Reciprocity Laws
I To understand the significance of this, we have to recognizethat a fundamental theme in number theory is a reciprocitylaw: a law that determines the factorization of primes.
I If we consider a field K ⊇ Q of finite degree over Q, there isan analogue for K of the ordinary integers Z denoted OK .
I A prime ideal pZ in Z can be lifted to an ideal pOK which ingeneral will fail to be prime. For example if K = Q(
√−1),
then OK = Z[√−1] and
2OK = ((1 +√−1)OK )2
is the square of a prime ideal.
Reciprocity Laws
I To understand the significance of this, we have to recognizethat a fundamental theme in number theory is a reciprocitylaw: a law that determines the factorization of primes.
I If we consider a field K ⊇ Q of finite degree over Q, there isan analogue for K of the ordinary integers Z denoted OK .
I A prime ideal pZ in Z can be lifted to an ideal pOK which ingeneral will fail to be prime. For example if K = Q(
√−1),
then OK = Z[√−1] and
2OK = ((1 +√−1)OK )2
is the square of a prime ideal.
Reciprocity Laws
I A law that specifies how “primes factorize” is called areciprocity law.
I The most famous of these is the quadratic reciprocity law. Inthe special case of Q(
√−1) it says that pOK remains a prime
ideal if and only if p ≡ 3 (mod 4).
I In general, if K/Q is a Galois extension with Abelian group,the reciprocity law can be stated in terms of congruences: psplits in a certain way depending on its congruence classmodulo the “conductor” of K .
Reciprocity Laws
I A law that specifies how “primes factorize” is called areciprocity law.
I The most famous of these is the quadratic reciprocity law. Inthe special case of Q(
√−1) it says that pOK remains a prime
ideal if and only if p ≡ 3 (mod 4).
I In general, if K/Q is a Galois extension with Abelian group,the reciprocity law can be stated in terms of congruences: psplits in a certain way depending on its congruence classmodulo the “conductor” of K .
Reciprocity Laws
I A law that specifies how “primes factorize” is called areciprocity law.
I The most famous of these is the quadratic reciprocity law. Inthe special case of Q(
√−1) it says that pOK remains a prime
ideal if and only if p ≡ 3 (mod 4).
I In general, if K/Q is a Galois extension with Abelian group,the reciprocity law can be stated in terms of congruences: psplits in a certain way depending on its congruence classmodulo the “conductor” of K .
Reciprocity Laws
I In the general (non-Abelian) case, a reciprocity law is specifiednot by a congruence satisfied by p, but rather the Frobeniusconjugacy class in the Galois group associated to the prime.
I In particular, if the Galois group of K/Q is GL2(Z/`Z), givingthe characteristic polynomial (almost) specifies the conjugacyclass.
Reciprocity Laws
I In the general (non-Abelian) case, a reciprocity law is specifiednot by a congruence satisfied by p, but rather the Frobeniusconjugacy class in the Galois group associated to the prime.
I In particular, if the Galois group of K/Q is GL2(Z/`Z), givingthe characteristic polynomial (almost) specifies the conjugacyclass.
Reciprocity Laws
I In particular, reducing the representations of Deligne modulo `gives
ρ` : Gal(Q/Q) −→ GL2(Z/`Z)
and so the image is finite.
I The fixed field of the kernel of this representation is a finiteextension K`/Q.
I Moreover, Frobp belongs to a conjugacy class withcharacteristic polynomial
T 2 − (τ(p) (mod `))T + (p11 (mod `)).
Reciprocity Laws
I In particular, reducing the representations of Deligne modulo `gives
ρ` : Gal(Q/Q) −→ GL2(Z/`Z)
and so the image is finite.
I The fixed field of the kernel of this representation is a finiteextension K`/Q.
I Moreover, Frobp belongs to a conjugacy class withcharacteristic polynomial
T 2 − (τ(p) (mod `))T + (p11 (mod `)).
Reciprocity Laws
I In particular, reducing the representations of Deligne modulo `gives
ρ` : Gal(Q/Q) −→ GL2(Z/`Z)
and so the image is finite.
I The fixed field of the kernel of this representation is a finiteextension K`/Q.
I Moreover, Frobp belongs to a conjugacy class withcharacteristic polynomial
T 2 − (τ(p) (mod `))T + (p11 (mod `)).
Reciprocity Laws
This is not a special case. In fact by Serre’s Conjecture (nowproved by Khare and Wintenberger), an odd two dimensionalrepresentation
ρ` : Gal(Q/Q) −→ GL2(Z/`Z)
has to come by reducing modulo ` the Galois representationassociated to a modular form.
Ramanujan’s Congruences
I In this context, if for a given `, it happens that
ρ` = 1 ⊕ χ11
is a reducible representation given by the two charactersabove, then
τ(p) ≡ 1 + p11 (mod `).
I In particular, this happens for ` = 3, 691. Other congruencesare explained similarly.
I Thus, the question of congruences has been transformed intoa question of studying the reducibility of the Galoisrepresentations associated to ∆.
Ramanujan’s Congruences
I In this context, if for a given `, it happens that
ρ` = 1 ⊕ χ11
is a reducible representation given by the two charactersabove, then
τ(p) ≡ 1 + p11 (mod `).
I In particular, this happens for ` = 3, 691. Other congruencesare explained similarly.
I Thus, the question of congruences has been transformed intoa question of studying the reducibility of the Galoisrepresentations associated to ∆.
Ramanujan’s Congruences
I In this context, if for a given `, it happens that
ρ` = 1 ⊕ χ11
is a reducible representation given by the two charactersabove, then
τ(p) ≡ 1 + p11 (mod `).
I In particular, this happens for ` = 3, 691. Other congruencesare explained similarly.
I Thus, the question of congruences has been transformed intoa question of studying the reducibility of the Galoisrepresentations associated to ∆.
Nonlinear relations discovered by Ramanujan
The 1916 paper in which Ramanujan formulated his famousconjectures about the τ function, actually begins with thefollowing quadratic relation:
σ1(1)σ3(n)+σ1(3)σ3(n−1)+· · ·+σ1(2n+1)σ3(0) =1
240σ5(n+1).
Here,σk(n) =
∑d |n,d>0
dk
and
σk(0) =1
2ζ(−k).
Other non-linear relations
I The expression above can be seen as a quadratic relationamongst Eisenstein series.
I There are also relations involving cusp forms, in particular ∆.Lahiri was the first to give a relation of this kind.
I Many quadratic identities have been produced by S. Gun, B.Ramakrishnan and B. Sahu, as well as other authors.
I One such identity is
τ(n) = n2σ7(n)− 540n−1∑m=1
m(n −m)σ3(m)σ3(n −m).
I What do they mean?
Other non-linear relations
I The expression above can be seen as a quadratic relationamongst Eisenstein series.
I There are also relations involving cusp forms, in particular ∆.Lahiri was the first to give a relation of this kind.
I Many quadratic identities have been produced by S. Gun, B.Ramakrishnan and B. Sahu, as well as other authors.
I One such identity is
τ(n) = n2σ7(n)− 540n−1∑m=1
m(n −m)σ3(m)σ3(n −m).
I What do they mean?
Other non-linear relations
I The expression above can be seen as a quadratic relationamongst Eisenstein series.
I There are also relations involving cusp forms, in particular ∆.Lahiri was the first to give a relation of this kind.
I Many quadratic identities have been produced by S. Gun, B.Ramakrishnan and B. Sahu, as well as other authors.
I One such identity is
τ(n) = n2σ7(n)− 540n−1∑m=1
m(n −m)σ3(m)σ3(n −m).
I What do they mean?
Other non-linear relations
I The expression above can be seen as a quadratic relationamongst Eisenstein series.
I There are also relations involving cusp forms, in particular ∆.Lahiri was the first to give a relation of this kind.
I Many quadratic identities have been produced by S. Gun, B.Ramakrishnan and B. Sahu, as well as other authors.
I One such identity is
τ(n) = n2σ7(n)− 540n−1∑m=1
m(n −m)σ3(m)σ3(n −m).
I What do they mean?
Other non-linear relations
I The expression above can be seen as a quadratic relationamongst Eisenstein series.
I There are also relations involving cusp forms, in particular ∆.Lahiri was the first to give a relation of this kind.
I Many quadratic identities have been produced by S. Gun, B.Ramakrishnan and B. Sahu, as well as other authors.
I One such identity is
τ(n) = n2σ7(n)− 540n−1∑m=1
m(n −m)σ3(m)σ3(n −m).
I What do they mean?
Quadratic Relations in the Representation Ring
I If we consider the ring of mod ` representations of modularforms (under ⊗ product), the above identities can be seen asquadratic relations in this ring.
I Can we classify all such relations?
I In understanding quadratic relations, we seem to be wherecongruences were before the Serre/Swinnerton-Dyer theory.
I Much work to be done!
Quadratic Relations in the Representation Ring
I If we consider the ring of mod ` representations of modularforms (under ⊗ product), the above identities can be seen asquadratic relations in this ring.
I Can we classify all such relations?
I In understanding quadratic relations, we seem to be wherecongruences were before the Serre/Swinnerton-Dyer theory.
I Much work to be done!
Quadratic Relations in the Representation Ring
I If we consider the ring of mod ` representations of modularforms (under ⊗ product), the above identities can be seen asquadratic relations in this ring.
I Can we classify all such relations?
I In understanding quadratic relations, we seem to be wherecongruences were before the Serre/Swinnerton-Dyer theory.
I Much work to be done!
Quadratic Relations in the Representation Ring
I If we consider the ring of mod ` representations of modularforms (under ⊗ product), the above identities can be seen asquadratic relations in this ring.
I Can we classify all such relations?
I In understanding quadratic relations, we seem to be wherecongruences were before the Serre/Swinnerton-Dyer theory.
I Much work to be done!