Theoretical basics Modeling L-functions statistics Conclusion

Modeling Elliptic Curve L-Functions with Random Matrix theory Florian B OUGUET Ecole Normale Supérieure de Cachan - Antenne de Bretagne

July - August 2011

Research work supervised by Nina C.S NAITH University of Bristol

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Key date 1973: encounter of Hugh L. Montgomery and Freeman J. Dyson over a cup of tea at the Princeton Institute for Advanced Studies. Discovery of the very first link between random matrix theory and analytic number theory.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

1

Theoretical basics Elliptic curves L-functions

2

Modeling L-functions statistics Choice of the model Validity of the model

3

Conclusion

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Elliptic curve) An elliptic curve over Q is a non-singular curve E ⊆ R2 defined by an equation of the form: E : y 2 = x 3 + ax + b, where a, b ∈ Q.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Elliptic curve) An elliptic curve over Q is a non-singular curve E ⊆ R2 defined by an equation of the form: E : y 2 = x 3 + ax + b, where a, b ∈ Q.

E : y2 = x3 − x + 1 F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Discriminant of an elliptic curve) ∆E = −16(4a3 + 27b2 ). E is non-singular if, and only if, ∆E 6= 0.

E : y2 = x3 − x + 1

E : y2 = x3

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Goal We want to describe the set of the rational solutions of E, that is to say the pairs (x, y ) ∈ Q2 lying on E.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Goal We want to describe the set of the rational solutions of E, that is to say the pairs (x, y ) ∈ Q2 lying on E. To this end, it is possible to supply a group law to E. Without going into details, we create a "point at infinity" called O, lying on any vertical line.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Goal We want to describe the set of the rational solutions of E, that is to say the pairs (x, y ) ∈ Q2 lying on E. To this end, it is possible to supply a group law to E. Without going into details, we create a "point at infinity" called O, lying on any vertical line. Proposition (E, +) is an abelian group, with O as the identity element.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Rational solutions of E: E(Q)

(E ∩ Q2 ) ∪ {O} = {(x, y ) ∈ Q2 y 2 = x 3 + ax + b} ∪ {O}. =

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Rational solutions of E: E(Q)

(E ∩ Q2 ) ∪ {O} = {(x, y ) ∈ Q2 y 2 = x 3 + ax + b} ∪ {O}. =

Proposition (E(Q), +) is an abelian subgroup of (E, +).

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Theorem (Mordell’s theorem) Let E be an elliptic curve. Then E(Q) is a finitely-generated abelian group.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Theorem (Mordell’s theorem) Let E be an elliptic curve. Then E(Q) is a finitely-generated abelian group. E(Q) = Zn ⊕ G.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Theorem (Mordell’s theorem) Let E be an elliptic curve. Then E(Q) is a finitely-generated abelian group. E(Q) = Zn ⊕ G.

Definition (Rank of an elliptic curve) rank(E)=rank(E(Q))=n.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Conjecture (Birch and Swinnerton-Dyer conjecture) The rank of an elliptic curve E over Q is the order of the zero of its Hasse-Weil L-function LE (s) at s = 1/2. One of the Millenium Prize Problems. Special case of Hilbert’s tenth problem.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Before introducing the Hasse-Weil L-function of E, we need to define a major component of E: its conductor QE .

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Before introducing the Hasse-Weil L-function of E, we need to define a major component of E: its conductor QE . Definition (Conductor of E) QE measures the arithmetic complexity of the curve. QE ∈ N.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Riemann zeta function) ζ(s) : C → C analytically continues the series ∞ X 1 . ns n=1

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Riemann zeta function) ζ(s) : C → C analytically continues the series ∞ X 1 . ns n=1

Definition (L-function) L(s) : C → C is an L-function if there exists (an ) ∈ CN such as L(s) is the analytic continuation of the series ∞ X an n=1

F. B OUGUET

ns

.

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Hasse-Weil L-function of an elliptic curve) The Hasse-Weil L-function of an elliptic curve E is the following function: Y
1 − ap p−s + ε(p)p1−2s , LE (s) = p∈P

  ε(p) = 0 or 1, depending on ∆E and p. ap = p + 1 − ](E(FP )). where  E(Fp ) = {(x, y ) ∈ F2p y 2 ≡ x 3 + ax + b (mod p)}.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Definition (Hasse-Weil L-function of an elliptic curve) The Hasse-Weil L-function of an elliptic curve E is the following function: Y
1 − ap p−s + ε(p)p1−2s , LE (s) = p∈P

  ε(p) = 0 or 1, depending on ∆E and p. ap = p + 1 − ](E(FP )). where  E(Fp ) = {(x, y ) ∈ F2p y 2 ≡ x 3 + ax + b (mod p)}. Proposition LE (s) is an L-function. LE (s) converges by definition over { 3/2}. LE (s) can be extended by analytic continuation to the whole complex plane.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

We define D ⊂ Z the set of the fundamental discriminants. Without going into details, those integers have good arithmetical properties. Definition (Quadratic twist) Given an elliptic curve over Q E : y 2 = x 3 + ax + b and d ∈ D non-divisor of QE , we define the quadratic twist of E by d as Ed : dy 2 = x 3 + ax + b.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

We define D ⊂ Z the set of the fundamental discriminants. Without going into details, those integers have good arithmetical properties. Definition (Quadratic twist) Given an elliptic curve over Q E : y 2 = x 3 + ax + b and d ∈ D non-divisor of QE , we define the quadratic twist of E by d as Ed : dy 2 = x 3 + ax + b. Proposition Let LE (s) be the Hasse-Weil L-function of E: LE (s) = the following formula holds: LEd (s) = LE (s, χd ) =

∞ X an χd (n) n=1

where χd (n) =

d n




ns

P∞

an n=1 ns .

Then

,

is the Kronecker symbol. F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Thanks to functional equations satisfied by the Hasse-Weil L-functions, one of the following case holds for d ∈ D: LE (s, χd ) is called even, and the order of its zero at s = 1/2 is an even number LE (s, χd ) is called odd, and the order of its zero at s = 1/2 is an odd number.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Conjecture (Riemann hypothesis) The non-trivial zeros of ζ(s) have real part 1/2. One of the Millenium Prize Problems. Hilbert’s eighth problem.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Elliptic curves L-functions

Conjecture (Riemann hypothesis) The non-trivial zeros of ζ(s) have real part 1/2. One of the Millenium Prize Problems. Hilbert’s eighth problem. Conjecture (Generalized Riemann hypothesis) The non-trivial zeros of an L-function have real part 1/2.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Choice of the model Validity of the model

Let us introduce the elliptic curve we are going to work with : E19 E19 : y 2 = x 3 + x 2 − 9x − 15. This curve is called E19 because its conductor is QE19 = 19.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Choice of the model Validity of the model

Let us introduce the elliptic curve we are going to work with : E19 E19 : y 2 = x 3 + x 2 − 9x − 15. This curve is called E19 because its conductor is QE19 = 19. We introduce as well: L and LX L = {LE19 (s, χd ) d ∈ D, LE19 (s, χd ) is even}, LX = {LE19 (s, χd ) ∈ L |d| ≤ X }. So far, we have computed L-functions zeros whose fundamental discriminants vary between 0 and -300,000.

F. B OUGUET

Modeling Elliptic Curve L-Functions with Random Matrix theory

Theoretical basics Modeling L-functions statistics Conclusion

Choice of the model Validity of the model

Expectations After a suitable rescaling, in the limit of large height on the line {