Deligne's Finite Fields Riemann Hypothesis uses Math-Induction?? #238; 2nd ed; Correcting Math
in [Logic]
|
Prev: Help with some exercises
Next: OPERATIONS AND SUPPLY MANAGEMENT 12-E JACOBS Test Bank and solution manual is available. Email me at allsolutionmanuals11[at]gmail.com if you need to buy this. All emails will be answered as soon as possible.
From: A on 26 Oct 2009 08:41 On Oct 26, 5:15 am, David Bernier <david...(a)videotron.ca> wrote: > Archimedes Plutonium wrote: > > > David Bernier wrote: > > >> There's Deligne's "La conjecture de Weil : I." , > >> downloadable from here: > >>http://www.numdam.org/numdam-bin/item?id=PMIHES_1974__43__273_0 > > >> Also, Barry Mazur's Zbl. review: > >>http://www.zentralblatt-math.org/zmath/en/search/?q=an:0287.14001 > >> --> "Display scanned Zentralblatt-MATH page with this review." > > >> David Bernier > > > I looked at both of those, one in French, and still not able to > > confirm > > whether Deligne's Finite Fields uses Mathematical-Induction in the > > "splitting fields" concept. > > > What is your opinion, David, as to whether Deligne's proof uses > > Math-Induction? > > I think the underpinnings of Deligne's proof, or algebraic geometry > a la Grothendieck, make use of mathematical induction. > > David Bernier > > > I think he does, although perhaps a tacit use. And that would make his > > overall proof structure as a Direct Method Existence proof. What you guys are talking about is the Riemann Hypothesis for zeta- functions of projective varieties over finite fields, which was the last part of the Weil conjectures to be proven, and was indeed proven by Deligne. The Wikipedia page on the Weil conjectures gives a reasonably precise formulation and a few examples, so it's worth looking at. Deligne's proof--as well as all other progress on the Weil conjectures--was based on the following trick: the zeta-function of an algebraic variety X over F_p is supposed to "count the points" of the variety, in the sense that if you evaluate the logarithmic derivative of the zeta-function at p^n, you get the number of points of the curve in F_{p^n} (up to some small, easily described constant). But you can actually recover the zeta-function of a variety X in a different way, as follows: since X is defined over F_p, there is a Frobenius (p-th power) map from X to X. Choose a nonnegative integer n, a positive integer m, and a prime l not equal to p. Then the constant sheaf Z/ (l^m)Z on the local etale site of X (there's probably a Wikipedia page on this as well, or otherwise, Milne's book "Etale Cohomology" is a wonderful reference) is defined, and we can take its nth cohomology group, H^n_{et}(X, Z/(l^m)Z). Since cohomology is functorial in X, the Frobenius map on X induces a Frobenius map on this cohomology group. Take the inverse limit of these groups over all m to get a group lim_m H^n_{et}(X, Z/(l^m)Z), which is a module over the l-adic integers; then tensor this module up with the rational numbers to get a vector space over Q_l, the l-adic rationals, which is sometimes written H^n_ {et}(X, Q_l) (although it ISN'T the same as the etale cohomology of X with coefficients in the constant sheaf Q_l), and which is called the l-adic cohomology of X. The Frobenius map has acted on every single object we've had along the way, meaning we still have a Frobenius map from H^n_{et}(X,Q_l) to H^n_{et}(X,Q_l); but these l-adic cohomology groups are Q_l-vector spaces, so the Frobenius map is just some linear operator on a vector space, so we can speak of its determinant. There is a formula, essentially due to Lefschetz, for the zeta-function of X in terms of these determinants! This is why Deligne was able to prove something very important about zeta-functions using purely algebraic (rather than analytic) methods--he was working with this l-adic cohomology and the Frobenius operations on it.
From: David Bernier on 26 Oct 2009 17:01
A wrote: > On Oct 26, 5:15 am, David Bernier <david...(a)videotron.ca> wrote: >> Archimedes Plutonium wrote: >> >>> David Bernier wrote: >>>> There's Deligne's "La conjecture de Weil : I." , >>>> downloadable from here: >>>> http://www.numdam.org/numdam-bin/item?id=PMIHES_1974__43__273_0 >>>> Also, Barry Mazur's Zbl. review: >>>> http://www.zentralblatt-math.org/zmath/en/search/?q=an:0287.14001 >>>> --> "Display scanned Zentralblatt-MATH page with this review." >>>> David Bernier >>> I looked at both of those, one in French, and still not able to >>> confirm >>> whether Deligne's Finite Fields uses Mathematical-Induction in the >>> "splitting fields" concept. >>> What is your opinion, David, as to whether Deligne's proof uses >>> Math-Induction? >> I think the underpinnings of Deligne's proof, or algebraic geometry >> a la Grothendieck, make use of mathematical induction. >> >> David Bernier >> >>> I think he does, although perhaps a tacit use. And that would make his >>> overall proof structure as a Direct Method Existence proof. > > > > What you guys are talking about is the Riemann Hypothesis for zeta- > functions of projective varieties over finite fields, which was the > last part of the Weil conjectures to be proven, and was indeed proven > by Deligne. The Wikipedia page on the Weil conjectures gives a > reasonably precise formulation and a few examples, so it's worth > looking at. > > Deligne's proof--as well as all other progress on the Weil > conjectures--was based on the following trick: the zeta-function of an > algebraic variety X over F_p is supposed to "count the points" of the > variety, in the sense that if you evaluate the logarithmic derivative > of the zeta-function at p^n, you get the number of points of the curve > in F_{p^n} (up to some small, easily described constant). But you can > actually recover the zeta-function of a variety X in a different way, > as follows: since X is defined over F_p, there is a Frobenius (p-th > power) map from X to X. Choose a nonnegative integer n, a positive > integer m, and a prime l not equal to p. Then the constant sheaf Z/ > (l^m)Z on the local etale site of X (there's probably a Wikipedia page > on this as well, or otherwise, Milne's book "Etale Cohomology" is a > wonderful reference) is defined, and we can take its nth cohomology > group, H^n_{et}(X, Z/(l^m)Z). Since cohomology is functorial in X, the > Frobenius map on X induces a Frobenius map on this cohomology group. > Take the inverse limit of these groups over all m to get a group lim_m > H^n_{et}(X, Z/(l^m)Z), which is a module over the l-adic integers; > then tensor this module up with the rational numbers to get a vector > space over Q_l, the l-adic rationals, which is sometimes written H^n_ > {et}(X, Q_l) (although it ISN'T the same as the etale cohomology of X > with coefficients in the constant sheaf Q_l), and which is called the > l-adic cohomology of X. The Frobenius map has acted on every single > object we've had along the way, meaning we still have a Frobenius map > from H^n_{et}(X,Q_l) to H^n_{et}(X,Q_l); but these l-adic cohomology > groups are Q_l-vector spaces, so the Frobenius map is just some linear > operator on a vector space, so we can speak of its determinant. There > is a formula, essentially due to Lefschetz, for the zeta-function of X > in terms of these determinants! This is why Deligne was able to prove > something very important about zeta-functions using purely algebraic > (rather than analytic) methods--he was working with this l-adic > cohomology and the Frobenius operations on it. Thanks for your explanations. You write: "purely algebraic (rather than analytic) methods", and I guess that's one of the main ideas I'll try to remember about Deligne's proof. I had a look at the Wikipedia article on the Hilbert-Polya Conjecture about a yet unknown spectral interpretation for the imaginary parts of the non-trivial zeros of the Riemann zeta function: < http://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture > If we call A the conjectured (unbounded) hermitian operator, one thing I don't understand is why, if gamma is in the spectrum of A, then -gamma is also in the spectrum of A. I don't see how this comes out of the Hilbert-Polya Conjecture as sketched in Wikipedia. So I looked for an operator-theoretic idea. If B is an operator on a Hilbert space H, C = BB* is a positive hermitian operator. Suppose H is finite-dimensional. Then we can take D = sqrt( - C), and if lambda>0 is an eigenvalue of C, one of i*sqrt(lambda) , -i*sqrt(lambda) will be an eigenvalue of D [ but not both for the same D]. So it doesn't work how I would have wanted. Or maybe the sought-for operator A in the Hilbert-Polya Conjecture is positive hermitian with spectrum the positive imaginary parts of the non-trivial zeta zeros ... David Bernier |