Number Theory
Zeta Functions of Varieties
The Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Prize Problems, carrying a $1,000,000 prize. It connects an analytic property of an $L$-function (the order of vanishing at $s=1$) to an algebraic one (the rank of the group of rational points). Zeta functions of varieties are the language in which this connection is stated.
- Deligne's theorem (Fields Medal 1978): the Weil conjectures about zeta functions of algebraic varieties over finite fields are the geometric analogue of the Riemann hypothesis. The proof introduced etale cohomology - an algebraic counterpart to topological invariants.
- Elliptic curve cryptography (ECC): the number of points $|E(\mathbb{F}_p)|$ is controlled by Hasse's bound $|a_p| \leq 2\sqrt{p}$ - a consequence of the Weil conjectures, and exactly what makes ECC key sizes predictable.
- Wiles' proof of Fermat's Last Theorem (1995) proceeds by showing every semistable elliptic curve over $\mathbb{Q}$ is modular - a statement about matching $L$-functions of curves with $L$-functions of modular forms.
Цели урока
- Construct the Weil zeta function of a variety over a finite field and compute it for simple examples
- State the Weil conjectures and explain their connection to the topology of varieties
- Describe $L$-functions of elliptic curves and state the BSD conjecture
Предварительные знания
- Dirichlet L-functions and the Riemann zeta function
- Elliptic curves over finite fields
- Cohomology in topology (Betti numbers)
The Weil zeta function
For a variety $X$ over a finite field $\mathbb{F}_q$, the Weil zeta function is $Z(X, T) = \exp\left(\sum_{n=1}^\infty \frac{|X(\mathbb{F}_{q^n})|}{n} T^n\right)$. The Weil conjectures (1949): (1) rationality - $Z$ is a rational function of $T$; (2) functional equation; (3) relation of poles/zeros to Betti numbers; (4) analogue of the Riemann hypothesis - zeros lie on circles $|T| = q^{-k/2}$. Deligne proved (4) in 1974.
Betti numbers $b_i = \dim H^i(X, \mathbb{Q}_\ell)$ (etale cohomology) determine the structure of the zeta function: $Z(X,T) = \prod_{i=0}^{2d} P_i(T)^{(-1)^{i+1}}$ where $P_i$ has degree $b_i$. For an elliptic curve ($d=1$): $b_0=1$, $b_1=2$, $b_2=1$ - the numerator has degree 2, the denominator factors of degrees 1 and 1.
The Birch-Swinnerton-Dyer conjecture
The Hasse-Weil $L$-function of a curve $E$: $L(E, s) = \prod_p L_p(E, p^{-s})^{-1}$, where $L_p$ encodes $a_p$ at primes of good reduction. BSD conjecture: $\mathrm{ord}_{s=1} L(E, s) = \mathrm{rank}(E(\mathbb{Q}))$. In other words: $E$ has infinitely many rational points if and only if $L(E,1) = 0$. Partial results: Kolyvagin-Wiles-Taylor (2001), Skinner-Urban (2014) for analytic rank $\leq 1$.
BSD remains an open problem. What is proved: if analytic rank is 0 (i.e. $L(E,1) \neq 0$) then algebraic rank is also 0 (finitely many rational points). And if analytic rank is 1, then algebraic rank is $\geq 1$. The full conjecture for rank $\geq 2$ is open.
Bryan Birch and Peter Swinnerton-Dyer formulated the conjecture in 1965 based on
Bryan Birch and Peter Swinnerton-Dyer formulated the conjecture in 1965 based on computations on EDSAC-2, one of Cambridge's early computers. Deligne proved the Weil conjectures in 1974, earning a Fields Medal and later the Wolf Prize. Modern proofs of Fermat's Last Theorem use $L$-functions at a crucial step, connecting elliptic curves to modular forms.
The Weil Zeta Function
Andre Weil formulated 4 conjectures about point counts of algebraic varieties over F_q in 1948: rationality, functional equation, Riemann Hypothesis analogue, and connection with Betti numbers of a lift to characteristic zero. Bernard Dwork proved rationality (1960) by p-adic methods, Alexander Grothendieck and Michael Artin developed etale cohomology in the 1960s, and Pierre Deligne finally proved the Riemann Hypothesis analogue (1974), receiving the Fields Medal in 1978 for this work.
What does Deligne's proof of the Weil conjectures establish?
L-functions and the Birch--Swinnerton-Dyer Conjecture
In 1967 Robert Langlands wrote his famous letter to Andre Weil sketching a program that unifies representation theory and arithmetic. Hasse-Weil L-functions L(E,s) of elliptic curves, Dirichlet L-functions, and the Riemann zeta itself fit into a single framework of automorphic L-functions. The most celebrated open question about elliptic L-functions is the Birch--Swinnerton-Dyer conjecture, one of the Clay Millennium Problems.
What does the Birch--Swinnerton-Dyer conjecture predict?
Zeta function of an elliptic curve
For $E: y^2 = x^3 - x$ over $\mathbb{F}_p$: $Z(E/\mathbb{F}_p, T) = \frac{(1-\alpha T)(1-\bar\alpha T)}{(1-T)(1-pT)}$ where $\alpha\bar\alpha = p$ and $\alpha + \bar\alpha = a_p = p+1 - |E(\mathbb{F}_p)|$. By Hasse's theorem $|a_p| \leq 2\sqrt{p}$. For $p=5$: $|E(\mathbb{F}_5)| = 4$, $a_5 = 2$. The numerator $(1-\alpha T)(1-\bar\alpha T) = 1 - a_p T + pT^2$ encodes all information about the curve's points.
Итоги
- The Weil zeta function encodes the number of points of a variety over extensions of finite fields.
- The Weil conjectures (proved by Deligne) link the zeros of the zeta function to the topology of the variety.
- The BSD conjecture connects the analytic rank of an $L$-function to the algebraic rank of the group of rational points.
Connections to other topics
The Riemann zeta function (nt-10) is the simplest $L$-function: $\zeta(s) = \prod_p (1-p^{-s})^{-1}$ encodes primes. Elliptic curves (nt-25, nt-29) are geometric objects with a rich $L$-function. Wiles' theorem connects $L$-functions of elliptic curves to $L$-functions of modular forms via the Shimura-Taniyama correspondence.
- Nt 10 — related
- Nt 25 — related
- Nt 29 — related
Вопросы для размышления
- Hasse's formula gives $|E(\mathbb{F}_p)| = p + 1 - a_p$ with $|a_p| \leq 2\sqrt{p}$. This means a curve over $\mathbb{F}_p$ has approximately $p$ points for any $p$. How does this property matter for ECC security: why does having $\sim p$ points make the discrete logarithm computationally hard?