The Mahler measure of elliptic curves

Transcription

1 The Mahler measure of elliptic curves Matilde Lalín (Université de Montréal) joint with Detchat Samart (University of Illinois at Urbana-Champaign) and Wadim Zudilin (University of Newcastle) mlalin The Geometry, Algebra, and Analysis of Algebraic Numbers BIRS October 5, 2015 M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

2 Mahler measure of multivariable polynomials P C[x ±1 1,..., x ±1 n ], the (logarithmic) Mahler measure is : m(p) = = (2πi) n 1 0 log P ( e 2πiθ 1,..., e 2πiθn ) dθ1... dθ n log P (x 1,..., x n ) dx 1 T n x 1... dx n x n By Jensen's formula, ( ) m a (x ki ) = log a + log max{1, k i }. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

3 Mahler measure is ubiquitous! Interesting questions about distribution of values Heights Volumes in hyperbolic space Entropy of certain arithmetic dynamical systems Special values of L-functions M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

4 The measures of a family of genus-one curves m(k) := m (x + 1x + y + 1y + k ) Numerical Guess (Boyd (1998)) s k Z (often) m (k)? = b N(k) s k k N 0, 4 b N(k) = N(k) 4π 2 L(E k, 2) = εl (E k, 0) X = 1 xy, (y x)(1 + xy) Y = 2x 2 y 2 E N(k) : Y 2 = X 3 + (k 2 /4 2)X 2 + X M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

5 Boyd's conjectures m (k)? = N(k) 4π 2 s k L(E k, 2) k s k N(k) k s k N(k) / / * * / / / M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

6 Some proven formulas for m(k) ( m 4 2 ) = b 64 Rodriguez-Villegas (1997) m(1) = b 15 Rogers & Zudilin (2010) m(2i) = b 40 Mellit (2011) m(2) = b 24 Rogers & Zudilin (2012) m(i) = 2b 17 Zudilin (2014) m( 2) = 1 4 b 56 Zudilin (2014) m(3) = 2b 21 Brunault (2015) m(12) = 2b 48 Brunault (2015) L. & Rogers (2007), L. (2010) m(5) = 6m(1), m(16) = 11m(1), m(3i) = 5m(1), m(8) = 4m(2), m(3 2) = 5 2 m(2). M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

7 The relationship with Beilinson's conjectures Deninger (1997): Understand these formulas in terms of Be linson's conjectures. Global information from local information through L-functions L X (0) Q reg(ξ) ξ K rank-one abelian group Example: Dirichlet class number formula, for F real quadratic eld, ζ (0) F Q log ξ ξ O F M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

8 The relationship with the regulator xy (x + 1x + y + 1y + k ) P(x, y) = a 2 (x)(y y 1 (x))(y y 2 (x)) E : P(x, y) = 0 elliptic curve m(p) m(a 2 (x)) = 1 log y y (2πi) 2 1 (x) + log y y 2 (x) dx dy T 2 x y M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

9 The relationship with the regulator m(p) m(a 2 (x)) = 1 log y y (2πi) 2 1 (x) + log y y 2 (x) dx dy T 2 x y Suppose that y 1 (x) 1 and y 2 (x) 1. By Jensen's formula, = 1 2πi T 1 log + y 1 (x) dx x = 1 2πi γ = {(x, y) E, x = 1, y 1} γ log y dx x M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

10 The regulator m(p) m(a 2 (x)) = 1 log y dx 2πi γ x = 1 η(x, y) 2π γ η(x, y) = log x d arg y log y d arg x multiplicative and antisymmetric Under favorable circumstances, γ η(x, y) is a function on K 2 (E) H 1 (E, Z). M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

11 The favorable cases Tame symbols are trivial ( γ η(x, y) is a function on K 2 (E) H 1 (E, Z)). E : P(x, y) = 0 admits a parametrization by modular units x(τ) and y(τ), products and quotients of Siegel functions: g a (τ) = q NB 2(a/N)/2 n 1 n a mod N (1 q n ) n 1 n a mod N q = e 2πiτ B 2 (x) = {x} 2 {x} + 1/6 (1 q n ) M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

12 Integrating η Theorem (MellitBrunaultZudilin (2014)) For a, b, c Z, N ac, bc, i c/n η(g a, g b ) = 1 L(f (τ) f (i ), 2) 4π where f (τ) = f a,b;c(τ) is a weight 2 modular form f a,b;c := e a,bce b, ac e a, bce b,ac where e a,b is a weight 1 level N 2 Eisenstein series. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

13 The case m(3) Theorem (Brunault, L-Samart-Zudilin, 2015) m(3) = 2L (E, 0) x + 1 x + y + 1 y + 3 = 0 does not have a modular unit parametrization. Strategies Modify the curve via an isogeny to get another curve that can be parametrized by modular units, use this to prove m(3). Brunault extends MBZ for the integral of η(g a, g b ) to Siegel units and works with Siegel units. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

14 Extend the family P a,c(x) = a ( X = a xy, Y = a 2xy ( ( c 2 Y 2 = X X 2 + ) ( + y + 1 ) + c, y x + 1 x ( (y 1y a x 1 x ) ) 4 1 a2 X + a 2. )), M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

15 The 2-isogeny a ( x + 1 x ) ( + y + 1 ) 2(a2 1) + c = 0 c = y a2 + 1 There is a 2 isogeny ϕ between the latter and x + 1 x + y + 1 y + 4(a2 1) a = 0. Take a = 7, c = 3, E 7,3 is isogenous to E 1,3. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

16 A modular parametrization Furthermore: 7 ( x + 1 x ) ( + y + 1 ) + 3 = 0, y x(τ) = 1 η(τ)η(3τ) 7 η(7τ)η(21τ) ỹ(τ) = = 1 7 g 1 (τ)g 2 (τ)g 2 3 (τ)g 4 (τ)g 5 (τ)g 2 6 (τ)g 8 (τ)g 2 9 (τ)g 10 (τ), ( ) η(τ)η(21τ) 2 η(3τ)η(7τ) = (g 1 (τ)g 2 (τ)g 4 (τ)g 5 (τ)g 8 (τ)g 10 (τ)) 2, τ X 0 (21). M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

17 More diculties for the case m(3) P (x, y) = ( 7 x + 1 ) ( + y + 1 ) ,3 x y y ± (x) = ( 7(x + x 1 ) + 3)) ± ( 7(x + x 1 ) + 3)) Both y ± have sometimes absolute value > 1 m(p 7,3 ) = log 7 2. Solution: Instead of m(p 7,3 ) = m(y y +(x)) + m(y y (x)) look at m(y y + (x))? = 1 2 L (E 21, 0) log 7 m(y y (x))? = 1 2 L (E 21, 0) log 7 M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

18 Strategies for the proof Fix c = 2(a2 1) a Prove m(p 1,4(a 2 1)/(a 2 +1)) = 4m(y y (x)) 3 log a = 4m(y y + (x)) + log a. By dierentiating respect to a. Then set a = 7. 2 Prove 4m(y y (x)) = 4m(y y + (x)) + p log 7, p Z by working with the regulator. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

19 Isogeny and regulator q (1/8)Z, m(y y + (x)) = 1 2π m(y y (x)) = 1 2π γ + η(x, y) = 1 8π γ η(x, y) = 1 8π η(x, y ) + q log a, γ We work with the isogeny and the elliptic dilogarithm D E. γ η(x, y ) + (q 1) log a. The divisors of x, y, x, y are supported in torsion points of order at most 4. The boundaries of γ ±, γ are supported in torsion points of order at most 8. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

20 The nal step Prove m(y y (x)) = 1 2 L (E 21, 0) + 3 log 7. 8 The tame symbols are not trivial. γ η(x, y) is not a function on K 2 (E) H 1 (E, Z)). Solution: avoid integrating around the problematic points of X 0 (21). m(y y (x)) = 1 2π i 2/7 η( 7 x(τ), ỹ(τ)) M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

21 Other directions Other families (results by Mellit, Brunault, Rogers & Zudilin, Bertin) Higher genus (results by Brunault, Bertin & Zudilin) Three variables and L (E, 1) (relations between formulas, Boyd, L.) Predict s k. M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22

22 Thank you! M. Lalín*, D. Samart, W. Zudilin The Mahler measure of elliptic curves October 5, / 22