[PPT] - Hypergeometric evaluations of L -values of an elliptic curve Wadim PowerPoint Presentation

SLIDE 1

Hypergeometric evaluations

f L-values of an elliptic curve

Wadim Zudilin 17–22 December 2012 Ramanujan-125 Conference “The Legacy of Srinivasa Ramanujan” (University of Delhi, New Delhi, India)

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 1 / 25

SLIDE 2

Ramanujan’s closed forms

One of (so many!) Ramanujan’s fames is an enormous production of highly nontrivial closed form evaluations of the values of certain “useful” series and functions. By a closed form here we normally mean identifying the quantities in question with certain algebraic numbers or with values of hypergeometric functions

mFm−1

a1, a2, . . . , am b2, . . . , bm

z
=

∞

n=0

(a1)n(a2)n · · · (am)n (b2)n · · · (bm)n zn n! where (a)n = Γ(a + n) Γ(a) =

n−1

j=0

(a + j) denotes the Pochhammer symbol (the shifted factorial).

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 2 / 25

SLIDE 3

Efficient formulae

An elegant “side” effect of such evaluations is computationally efficient formulae for mathematical constants, like 1 π = 32 √ 2

∞

n=0

(4n)! n!4 (1103 + 26390n) 1 3964n+2 , G = L(χ−4, 2) =

∞

n=0

(−1)n (2n + 1)2 = π

∞

n=0

2n n 2 (1/4)2n+1 2n + 1 . Catalan’s constant G is one of the simplest arithmetic quantities whose irrationality is still unproven.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 3 / 25

SLIDE 4

Zeta values

Similar expressions for zeta values, ζ(s) =

∞

n=1

1 ns where s = 2, 3, . . . , were

btained more recently by others.
R. Ap´

ery (1978) made use of acceleration formulae ζ(2) = 3

∞

n=1

1 n22n

n

and

ζ(3) = 5 2

∞

n=1

(−1)n−1 n32n

n

in his proof of the irrationality of ζ(2) and ζ(3).

The computationally efficient acceleration formula ζ(3) = 1 2

∞

n=1

(−1)n−1 5n2 + 8(5n − 2)2 n52n

n

5 is due to T. Amdeberhan and D. Zeilberger (1997).

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 4 / 25

SLIDE 5

Gamma values

An example of a slightly different type, π 51/4Γ( 3

4)4 = ∞

n=0

Bn

− 1

20 n where Bn =

n

j=0

n j 4 , is due to J. Guillera and Z. (2012). Note that it is, roughly speaking, a “half” of Ramanujan-type formula 5 2π =

∞

n=0

Bn (1 + 3n)

− 1

20 n which is established recently by S. Cooper.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 5 / 25

SLIDE 6

Periods

In order to “unify” such representations, M. Kontsevich and D. Zagier (2001) introduced the numerical class of periods. A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in Rn given by polynomial inequalities with rational coefficients. Without much harm, the three appearances of the adjective “rational” can be replaced by “algebraic”. The set of periods P is countable and admits a ring structure. It contains a lot of “important” numbers, mathematical constants like π, Catalan’s constant and zeta values.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 6 / 25

SLIDE 7

Extended periods

The extended period ring P := P[1/π] = P[(2πi)−1] (rather than the period ring P itself) contains even more natural examples, like values of generalised hypergeometric functions mFm−1 at algebraic points and special L-values. For example, a general theorem due to Beilinson and Deninger–Scholl states that the (non-critical) value of the L-series attached to a cusp form f (τ) of weight k at a positive integer m ≥ k belongs to P. In spite of the effective nature of the proof of the theorem, computing these L-values as periods remains a difficult problem even for particular examples. Many such computations are motivated by (conjectural) evaluations of the logarithmic Mahler measures of multi-variate polynomials.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 7 / 25

SLIDE 8

Elliptic curves

In the talk we will limit those “special L-values” to the L-values of elliptic curves. An elliptic curve can be defined in many different ways. Usually, it is a plane curve defined by y2 = x3 + ax + b, a Weierstrass equation. Although a and b can be treated as real or complex numbers, we will assume for all practical purposes that they are in Z.

Example. y2 = x3 − x is an elliptic curve (of conductor 32).

The integrality of a and b makes counting possible, not only over Z but

ver any finite field Fpn.

The count can be further related to a Dirichlet-type generating function L(E, s) =

∞

n=1

an ns .

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 8 / 25

SLIDE 9

L-series of elliptic curves

The critical line for the function is Re s = 1, and L(E, s) =

∞

n=1

an ns can be analytically continued to C where it satisfies a functional equation which relates L(E, s) to L(E, 2 − s). Computing the coefficients an is not a simple task in general... However, thanks to the modularity theorem due A. Wiles, R. Taylor and others, the L-series can be identified with L(f , s) for a cusp form of weight 2 and level N, the conductor of the elliptic curve.

Example. The L-series of y2 = x3 − x (and of any elliptic curve of

conductor 32) can be generated by

∞

n=1

anqn = q

∞

m=1

(1 − q4m)2(1 − q8m)2.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 9 / 25

SLIDE 10

Computing L-values

Computing L(E, 1) is “easy”: it is either 0 or the period of elliptic curve E. Computing L(E, k) for k ≥ 2 is highly non-trivial. The already mentioned results of Beilinson generalised later by Denninger–Scholl show that any such L-value can be expressed as a period. Several examples are explicitly given for k = 2, mainly motivated by showing particular cases of Beilinson’s conjectures in K-theory and Boyd’s (conjectural) evaluations of Mahler measures. In spite of the algorithmic nature of Beilinson’s method and in view of its complexity, no examples were produced so far for a single L(E, 3).

M. Rogers and Z. in 2010–11 created an elementary alternative to

Beilinson–Denninger–Scholl to prove some conjectural Mahler evaluations.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 10 / 25

SLIDE 11

Examples from joint work with Rogers

Because the resulting Mahler measures can be expressed entirely via hypergeometric functions, our joint results with Rogers can be stated as follows: 10 π2 L(E20, 2) = 5 4 log 2 − 3 64 4F3 4

3, 5 3, 1, 1

2, 2, 2

27

32

,

12 π2 L(E24, 2) = 3F2 1

2, 1 2, 1 2

1, 3

2

1

4

=

∞

n=0

2n n 2 (1/8)2n 2n + 1 , 15 π2 L(E15, 2) = 3F2 1

2, 1 2, 1 2

1, 3

2

1

16

=

∞

n=0

2n n 2 (1/16)2n 2n + 1 . The last two formulae resemble Ramanujan’s evaluation 4 π G =

∞

n=0

2n n 2 (1/4)2n 2n + 1 from one of the first slides.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 11 / 25

SLIDE 12

Hypergeometric evaluations of L(E32, k)

Our original method with Rogers was used for L(E, 2) only, but it is general enough to serve for L(E, k) with k ≥ 3. Theorem For an elliptic curve E of conductor 32, L(E, 2) = π 16 1 1 + √ 1 − x2 (1 − x2)1/4 dx 1 dy 1 − x2(1 − y 2) = π1/2Γ( 1

4)2

96 √ 2

3F2

1, 1, 1

2 7 4, 3 2

1
+ π1/2Γ( 3

4)2

8 √ 2

3F2

1, 1, 1

2 5 4, 3 2

1
,

L(E, 3) = π2 128 1 (1 + √ 1 − x2)2 (1 − x2)3/4 dx 1 1 dy dw 1 − x2(1 − y 2)(1 − w 2) = π3/2Γ( 1

4)2

768 √ 2

4F3

1, 1, 1, 1

2 7 4, 3 2, 3 2

1
+ π3/2Γ( 3

4)2

32 √ 2

4F3

1, 1, 1, 1

2 5 4, 3 2, 3 2

1
+ π3/2Γ( 1

4)2

256 √ 2

4F3

1, 1, 1, 1

2 3 4, 3 2, 3 2

1
.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 12 / 25

SLIDE 13

Dedekind’s eta-function

Below we sketch the hardest (and newest) case of L(E, 3). As mentioned earlier, the L-series of an elliptic curve of conductor 32 coincides with the L-series attached to the cusp form f (τ) =

∞

n=1

anqn = q

∞

m=1

(1 − q4m)2(1 − q8m)2 = η2

4η2 8,

where q = e2πiτ for τ from the upper half-plane Im τ > 0, η(τ) := q1/24

∞

m=1

(1 − qm) =

∞

n=−∞

(−1)nq(6n+1)2/24 is Dedekind’s eta-function with its modular involution η(−1/τ) = √ −iτη(τ), and ηk = η(kτ) for short.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 13 / 25

SLIDE 14

Integral for L(E, 3)

Taking the differential operator δ = 1 2πi d dτ = q d dq and it inverse δ−1 : f → q f dq q (normalised by 0 at τ = i∞ or q = 0), we write L(E, 3) = L(f , 3) =

∞

n=1

an n3 = (δ−3f )|q=1 = 1 2 1 f log2 q dq q = 4π3 ∞ f (it)t2 dt.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 14 / 25

SLIDE 15

Eisenstein-series decomposition

Note the (Lambert series) expansion η4

8

η2

4

=

m≥1

−4 m

qm

1 − q2m =

m,n≥1

n odd

−4 m

qmn =
m,n≥1

a(m)b(n)qmn, where a(m) := −4 m

,

b(n) := n mod 2, and −4

m

denotes the quadratic residue character modulo 4.

Then f (it) = η2

4η2 8

τ=it = η4

8

η2

4

η4

4

η2

8

τ=it

= η4

8

η2

4

τ=it

· 1 2t η4

8

η2

4

τ=i/(32t)

= 1 2t

m1,n1≥1

b(m1)a(n1)e−2πm1n1t

m2,n2≥1

b(m2)a(n2)e−2πm2n2/(32t). where t > 0 and the modular involution of Dedekind’s eta-function was used.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 15 / 25

SLIDE 16

Principal trick

Furthermore, L(E, 3) = 2π3 ∞

m1,n1,m2,n2≥1

b(m1)a(n1)b(m2)a(n2) × exp

−2π
m1n1t + m2n2

32t

t dt

= 2π3

m1,n1,m2,n2≥1

b(m1)a(n1)b(m2)a(n2) × ∞ exp

−2π
m1n1t + m2n2

32t

t dt.

Here comes the crucial transformation of purely analytical origin: we make the change of variable t = n2u/n1. This does not change the form of the exponential factor but affects the differential, and we obtain. . .

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 16 / 25

SLIDE 17

Principal trick (continued)

. . . and we obtain L(E, 3) = 2π3

m1,n1,m2,n2≥1

b(m1)a(n1)b(m2)a(n2) × ∞ exp

−2π
m1n1t + m2n2

32t

t dt

= 2π3

m1,n1,m2,n2≥1

b(m1)a(n1)b(m2)a(n2)n2

2

n2

1

× ∞ exp

−2π
m1n2u + m2n1

32u

u du

= 2π3 ∞

m1,n2≥1

b(m1)a(n2)n2

2e−2πm1n2u

×

m2,n1≥1

b(m2)a(n1) n2

1

e−2πm2n1/(32u) u du.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 17 / 25

SLIDE 18

More Eisenstein series

Furthermore,

m,n≥1

b(m)a(n)n2qmn =

m,n≥1

m odd

−4 n

n2qmn = η8

2η4 8

η6

4

,

m,n≥1

b(m)a(n)m2qmn =

m,n≥1

m odd

−4 n

m2qmn = η18

4

η8

2η4 8

, so that r(τ) =

m,n≥1

b(m)a(n) n2 qmn = δ−2 η18

4

η8

2η4 8

.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 18 / 25

SLIDE 19

Back to L(E, 3)

Continuing the previous computation, L(E, 3) = 2π3 ∞ η8

2η4 8

η6

4

τ=iu

· r(i/(32u)) u du (we apply the involution to the eta quotient) = π3 8 ∞ η4

4η8 16

η6

8

r(τ)

τ=i/(32u)

du u2 (we change the variable u = 1/(32v)) = 4π3 ∞ η4

4η8 16

η6

8

r(τ)

τ=iv

dv. The real challenge of the latter expression is the Eisenstein series r(τ) of weight −1.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 19 / 25

SLIDE 20

Ramanujan’s formula

There is a standard recipe of expressing Eisenstein series of negative weight via solutions of non-homogeneous linear differential equations. It is an efficient way to write r(τ) as a “period”, however a complicated way. Accidentally, the Eisenstein series r(τ) of weight −1 possesses a different treatment because of a special formula due to Ramanujan: r(τ) =

m,n≥1

m odd

−4 n qmn n2 = ˜ x G(−˜ x2) 4F(−˜ x2) , where ˜ x(τ) = 4η4

8/η4 2,

F(−˜ x2) = 2F1 1

2, 1 2

1

−˜

x2

= 2

π 1 dy

(1 − y 2)(1 + ˜

x2y 2) = η4

2

η2

4

and G(z) = 3F2 1, 1, 1

3 2, 3 2

z
=

1 1 dy dw 1 − z(1 − y 2)(1 − w 2).

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 20 / 25

SLIDE 21

L(E, 3) as a period

Choosing the modular function x(τ) = 4η4

2η8 8/η12 4

to parameterise everything and noting that ˜ x = x/ √ 1 − x2 we may now write L(E, 3) as L(E, 3) = π3 64 ∞ s(x(τ)) x(τ) 1 − x(τ)2 G

−

x(τ)2 1 − x(τ)2

δx
τ=iv

dv, where s(x) = 16η10

4 η8 16

η8

2η10 8

= (1 − √ 1 − x2)2 x(1 − x2)3/4 . After performing the modular substitution x = x(τ) we finally arrive at L(E, 3) = π2 128 1 (1 − √ 1 − x2)2 (1 − x2)3/4 dx 1 1 dy dw 1 − x2(1 − (1 − y2)(1 − w2)). There is still some work to do in order to identify the resulted integral with the linear combination of hypergeometric functions in the theorem.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 21 / 25

SLIDE 22

Hypergeometric evaluations of L(E32, k)

Theorem For an elliptic curve E of conductor 32, L(E, 2) = π 16 1 1 + √ 1 − x2 (1 − x2)1/4 dx 1 dy 1 − x2(1 − y2) = π1/2Γ( 1

4)2

96 √ 2

3F2

1, 1, 1

2 7 4, 3 2

1
+ π1/2Γ( 3

4)2

8 √ 2

3F2

1, 1, 1

2 5 4, 3 2

1
,

L(E, 3) = π2 128 1 (1 + √ 1 − x2)2 (1 − x2)3/4 dx 1 1 dy dw 1 − x2(1 − y2)(1 − w2) = π3/2Γ( 1

4)2

768 √ 2

4F3

1, 1, 1, 1

2 7 4, 3 2, 3 2

1
+ π3/2Γ( 3

4)2

32 √ 2

4F3

1, 1, 1, 1

2 5 4, 3 2, 3 2

1
+ π3/2Γ( 1

4)2

256 √ 2

4F3

1, 1, 1, 1

2 3 4, 3 2, 3 2

1
.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 22 / 25

SLIDE 23

A general formula?

The theorem, in fact, produces amazingly similar hypergeometric forms of L(E, 2) and L(E, 3). In the notation Fk(a) := πk−1/2Γ(a) 23k−1Γ(a + 1

2) k+1Fk

k times
1, . . . , 1, 1

2

a + 1

2, 3 2, . . . , 3 2

k − 1 times
1
,

relations for L(E, 2) and L(E, 3) can be alternatively written as L(E, 2) = F2( 5

4) + F2( 3 4)

and L(E, 3) = F3( 5

4) + 2F3( 3 4) + F3( 1 4).

In view of the known formula L(E, 1) = π−1/2Γ( 1

4)2

8 √ 2 = π−1/2Γ( 1

4)2

24 √ 2

3F2

1, 1

2 7 4

1
= 2F1( 5

4),

we can conclude that, for k = 1, 2 or 3, the L-value L(E, k) can be written as a (simple) Q-linear combination of Fk( 7

4 − m 2 ) for m = 1, . . . , k.

However this pattern does not seem to work for k > 3.

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 23 / 25

SLIDE 24

Generalisations

The potentials of our method with Rogers are still “in press.” One of the latest news is period evaluations of Ramanujan’s zeta function L(∆, s) by Rogers, where ∆(τ) = η(τ)24 = q

∞

m=1

(1 − qm)24 =

∞

n=1

τ(n)qn, for s = k ≥ 12. For example, he shows that L(∆, 12) = − 128π11 8241 · 11! 1 F(z)5F(1 − z)5 × 2 + 251z + 876z2 + 251z3 + 2z4 1 − z log z dz, where as before F(z) = 2F1 1

2, 1 2

1

z
= 2

π 1 dy

(1 − y 2)(1 − zy 2)

. And there are still many more conjectures on Boyd’s list. . .

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 24 / 25

SLIDE 25

Merci

Thank you!

Wadim Zudilin (CARMA, UoN) Evaluations of L-values of an elliptic curve 17–22 December 2012 25 / 25