L-functions Elliptic Curves and the Special Values of L-functions The Hasse-Weil L-function of an Elliptic Curve ICTS, 2021 The BSD Conjecture Introduction to Elliptic Curves: Lecture 3 Anupam Saikia Department of Mathematics, Indian Institute of Technology Guwahati Anupam Saikia IIT Guwahati August 3, 2021 1 / 32 Sections L-functions The Hasse-Weil L-function of an Elliptic Curve The BSD 1 L-functions Conjecture 2 The Hasse-Weil L-function of an Elliptic Curve 3 The BSD Conjecture Anupam Saikia IIT Guwahati August 3, 2021 1 / 32 Sections L-functions The Hasse-Weil L-function of an Elliptic Curve The BSD 1 L-functions Conjecture 2 The Hasse-Weil L-function of an Elliptic Curve 3 The BSD Conjecture Anupam Saikia IIT Guwahati August 3, 2021 2 / 32 The Riemann Zeta Function: a typical example We are familiar with the zeta series L-functions 1 X 1 The Hasse-Weil ζ(s) = ; s 2 : L-function of an ns C Elliptic Curve n=1 The BSD Conjecture The series converges for all complex numbers s with Re(s) > 1 giving us the the Riemann zeta function. The Riemann zeta function ζ(s) has analytic continuation to all of C, except for a simple pole of residue 1 at s = 1. The Riemann zeta function also satisfies a functional equation, relating its values at s and 1 − s. The Riemnann zeta function can be expressed as a Euler product Y 1 ζ(s) = ; Re(s) > 1: (1 − p−s) p Anupam Saikia IIT Guwahati August 3, 2021 3 / 32 The Riemann Zeta Function: a typical example We are familiar with the zeta series L-functions 1 X 1 The Hasse-Weil ζ(s) = ; s 2 : L-function of an ns C Elliptic Curve n=1 The BSD Conjecture The series converges for all complex numbers s with Re(s) > 1 giving us the the Riemann zeta function. The Riemann zeta function ζ(s) has analytic continuation to all of C, except for a simple pole of residue 1 at s = 1. The Riemann zeta function also satisfies a functional equation, relating its values at s and 1 − s. The Riemnann zeta function can be expressed as a Euler product Y 1 ζ(s) = ; Re(s) > 1: (1 − p−s) p Anupam Saikia IIT Guwahati August 3, 2021 3 / 32 The Riemann Zeta Function: a typical example We are familiar with the zeta series L-functions 1 X 1 The Hasse-Weil ζ(s) = ; s 2 : L-function of an ns C Elliptic Curve n=1 The BSD Conjecture The series converges for all complex numbers s with Re(s) > 1 giving us the the Riemann zeta function. The Riemann zeta function ζ(s) has analytic continuation to all of C, except for a simple pole of residue 1 at s = 1. The Riemann zeta function also satisfies a functional equation, relating its values at s and 1 − s. The Riemnann zeta function can be expressed as a Euler product Y 1 ζ(s) = ; Re(s) > 1: (1 − p−s) p Anupam Saikia IIT Guwahati August 3, 2021 3 / 32 The Riemann Zeta Function: a typical example We are familiar with the zeta series L-functions 1 X 1 The Hasse-Weil ζ(s) = ; s 2 : L-function of an ns C Elliptic Curve n=1 The BSD Conjecture The series converges for all complex numbers s with Re(s) > 1 giving us the the Riemann zeta function. The Riemann zeta function ζ(s) has analytic continuation to all of C, except for a simple pole of residue 1 at s = 1. The Riemann zeta function also satisfies a functional equation, relating its values at s and 1 − s. The Riemnann zeta function can be expressed as a Euler product Y 1 ζ(s) = ; Re(s) > 1: (1 − p−s) p Anupam Saikia IIT Guwahati August 3, 2021 3 / 32 The Riemann Zeta Function: a typical example We are familiar with the zeta series L-functions 1 X 1 The Hasse-Weil ζ(s) = ; s 2 : L-function of an ns C Elliptic Curve n=1 The BSD Conjecture The series converges for all complex numbers s with Re(s) > 1 giving us the the Riemann zeta function. The Riemann zeta function ζ(s) has analytic continuation to all of C, except for a simple pole of residue 1 at s = 1. The Riemann zeta function also satisfies a functional equation, relating its values at s and 1 − s. The Riemnann zeta function can be expressed as a Euler product Y 1 ζ(s) = ; Re(s) > 1: (1 − p−s) p Anupam Saikia IIT Guwahati August 3, 2021 3 / 32 Arithmetic Significance of Values of ζ(s) The values of the Riemann zeta function at at even positive integers are L-functions related to Bernoulli numbers, and have interesting arithmetic The Hasse-Weil interpretations in terms of ideal class groups of cyclotomic fields. We L-function of an Elliptic Curve have (2π)2nB The BSD ζ(2n) = (−1)n+1 2n ; n 2 ; Conjecture 2(2n)! N where Bk denotes the k-th Bernoulli number defined by 1 t X tk = B : et − 1 k k! k=0 Let A denote the p-Sylow subgroup of the class group of the cyclotomic (i) i field Q(ζp) for an odd prime p, and A denote χ component of A where χ is the cyclotomic character. (i) Herbrand’s Theorem: If p j #A , then p j Bp−i for 3 ≤ i ≤ p − 2, i odd. Ribet’s Theorem: The converse also holds. Lichtenbaum’s conjecture gives arithmetic interpretation for values of ζ(s) at odd negative integers. Anupam Saikia IIT Guwahati August 3, 2021 4 / 32 Arithmetic Significance of Values of ζ(s) The values of the Riemann zeta function at at even positive integers are L-functions related to Bernoulli numbers, and have interesting arithmetic The Hasse-Weil interpretations in terms of ideal class groups of cyclotomic fields. We L-function of an Elliptic Curve have (2π)2nB The BSD ζ(2n) = (−1)n+1 2n ; n 2 ; Conjecture 2(2n)! N where Bk denotes the k-th Bernoulli number defined by 1 t X tk = B : et − 1 k k! k=0 Let A denote the p-Sylow subgroup of the class group of the cyclotomic (i) i field Q(ζp) for an odd prime p, and A denote χ component of A where χ is the cyclotomic character. (i) Herbrand’s Theorem: If p j #A , then p j Bp−i for 3 ≤ i ≤ p − 2, i odd. Ribet’s Theorem: The converse also holds. Lichtenbaum’s conjecture gives arithmetic interpretation for values of ζ(s) at odd negative integers. Anupam Saikia IIT Guwahati August 3, 2021 4 / 32 Arithmetic Significance of Values of ζ(s) The values of the Riemann zeta function at at even positive integers are L-functions related to Bernoulli numbers, and have interesting arithmetic The Hasse-Weil interpretations in terms of ideal class groups of cyclotomic fields. We L-function of an Elliptic Curve have (2π)2nB The BSD ζ(2n) = (−1)n+1 2n ; n 2 ; Conjecture 2(2n)! N where Bk denotes the k-th Bernoulli number defined by 1 t X tk = B : et − 1 k k! k=0 Let A denote the p-Sylow subgroup of the class group of the cyclotomic (i) i field Q(ζp) for an odd prime p, and A denote χ component of A where χ is the cyclotomic character. (i) Herbrand’s Theorem: If p j #A , then p j Bp−i for 3 ≤ i ≤ p − 2, i odd. Ribet’s Theorem: The converse also holds. Lichtenbaum’s conjecture gives arithmetic interpretation for values of ζ(s) at odd negative integers. Anupam Saikia IIT Guwahati August 3, 2021 4 / 32 Arithmetic Significance of Values of ζ(s) The values of the Riemann zeta function at at even positive integers are L-functions related to Bernoulli numbers, and have interesting arithmetic The Hasse-Weil interpretations in terms of ideal class groups of cyclotomic fields. We L-function of an Elliptic Curve have (2π)2nB The BSD ζ(2n) = (−1)n+1 2n ; n 2 ; Conjecture 2(2n)! N where Bk denotes the k-th Bernoulli number defined by 1 t X tk = B : et − 1 k k! k=0 Let A denote the p-Sylow subgroup of the class group of the cyclotomic (i) i field Q(ζp) for an odd prime p, and A denote χ component of A where χ is the cyclotomic character. (i) Herbrand’s Theorem: If p j #A , then p j Bp−i for 3 ≤ i ≤ p − 2, i odd. Ribet’s Theorem: The converse also holds. Lichtenbaum’s conjecture gives arithmetic interpretation for values of ζ(s) at odd negative integers. Anupam Saikia IIT Guwahati August 3, 2021 4 / 32 Arithmetic Significance of Values of ζ(s) The values of the Riemann zeta function at at even positive integers are L-functions related to Bernoulli numbers, and have interesting arithmetic The Hasse-Weil interpretations in terms of ideal class groups of cyclotomic fields. We L-function of an Elliptic Curve have (2π)2nB The BSD ζ(2n) = (−1)n+1 2n ; n 2 ; Conjecture 2(2n)! N where Bk denotes the k-th Bernoulli number defined by 1 t X tk = B : et − 1 k k! k=0 Let A denote the p-Sylow subgroup of the class group of the cyclotomic (i) i field Q(ζp) for an odd prime p, and A denote χ component of A where χ is the cyclotomic character.