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Twisted L-Functions and Monodromy. (AM-150)

Twisted L-Functions and Monodromy. (AM-150)

Nicholas M. Katz
Copyright Date: 2002
Pages: 237
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  • Book Info
    Twisted L-Functions and Monodromy. (AM-150)
    Book Description:

    For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves?

    Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves.

    The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy.Twisted L-Functions and Monodromyis essential reading for anyone interested in number theory and algebraic geometry.

    eISBN: 978-1-4008-2488-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-2)
  3. Introduction
    (pp. 3-22)

    The present work grew out of an entirely unsuccessful attempt to answer some basic questions about elliptic curves overQ. Start with an elliptic curve E overQ, say given by a Weierstrass equation

    ${\text{E:}}\quad \;{{\text{y}}^2} = 4{{\text{x}}^3} - {\text{ax}} - {\text{b}}$,

    with a, b integers and${{\text{a}}^3} - 27{{\text{b}}^2} \ne \;0$. By Mordell’s theorem [Mor], the group${\text{E(Q)}}$ofQ–rational points is a finitely generated abelian group. The dimension of theQ–vector space${\text{E(Q)}} \otimes {{\text{Z}}^{\text{Q}}}$is called the Mordell–Weil rank, or simply the rank, of E. Thus we get a function

    $\{ ({\rm{a}},{\rm{b}}){\rm{in}}{{\rm{Z}}^2}{\rm{with}}{{\rm{a}}^3} - 27{{\rm{b}}^2} \ne 0\} {\rm{ }} \to {\rm{ }}\{ {\rm{nonnegativeintegers}}\} $

    defined by

    $({\text{a,}}\;{\text{b)}} \mapsto {\text{the rank of the curve}}\;{{\text{y}}^2} = 4{{\text{x}}^3} - {\text{ax}} - {\text{b}}$.

    It is remarkable how little we know about this function. For example, we...

  4. Part I: Background Material

    • Chapter 1: “Abstract” Theorems of Big Monodromy
      (pp. 23-42)

      (1.0.1) It will be convenient to introduce two generalizations of the notion of pseudoreflection. Suppose we are given a finite–dimensional vector space V over a field K. We write GL(V) for${\text{Au}}{{\text{t}}_{\text{K}}}({\text{V)}}$, so long as there is no ambiguity about the field K. Recall that an element A in GL(V) is called a pseudoreflection if its space of fixed points, Ker(A–1), has codimension one in V, or, equivalently, if the quotient spaceV/Ker(A–1) has dimension one.

      (1.0.2) Given an integer${\text{r}} \geqslant \;{\text{0}}$, and an element A in GL(V), we say that A has drop r if Ker(A–1) has...

    • Appendix to Chapter 1: A Result of Zalesskii
      (pp. 43-50)
    • Chapter 2: Lefschetz Pencils, Especially on Curves
      (pp. 51-70)

      (2.0.1) We work over an algebraically closed field k. Let X/k be a proper smooth connected kscheme of dimension${\text{n}} \geqslant {\text{1}}$, andLon X a very ample invertible${O_X}{\kern 1pt} - {\kern 1pt} {\text{module}}$-module. We embed X in${\text{P(}}{{\text{H}}^0}({\text{X,}}\;{\text{L)}}$, the projective space of hyperplanes in${{\text{H}}^0}({\text{X,}}\;{\text{L)}}$, in the usual way: x in X(k) is mapped to the hyperplane in${{\text{H}}^0}({\text{X,}}\;{\text{L)}}$consisting of those global sections ofLwhich vanish at x. Equivalently, we give ourselves X as a closed subscheme of a projective space${\text{p}}$in such a way that both the following conditions are satisfied:

      ($\mathcal{L}\;{\text{is}}\;{O_X}(1): = \;{\text{the pullback to X}}\;{\text{of}}\;{O_{\text{P}}}(1)$,

      ( the restriction map induces an isomorphism...

    • Chapter 3: Induction
      (pp. 71-78)

      (3.0.1) Let G be a group,${\text{H}} \subset {\text{G}}$a subgroup, R a commutative ring, and V a left R[H]–module. There are two standard notions of the induction of V from H to G. The first, which we call “standard” induction, is

      (${\text{In}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V)}}\;{\text{:}} = \;{\text{R[G]}}{ \otimes _{{\text{R[H]}}}}{\kern 1pt} {\text{V}}$

      with its structure of left R[G]–module through the first factor.

      The second, which we call Mackey induction, is

      (${\text{MaIn}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V)}}\;{\text{:}} = \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[H]}} - {\text{mod}}}}({\text{R[G],}}\;{\text{V)}} = \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{H}} - {\text{sets}}}}({\text{G,}}\;{\text{V)}}$,

      which becomes a left R[G]-module by defining

      $({{\text{L}}_{\text{g}}}\varphi ){\text{(x)}}: = \;\varphi ({\text{xg)}}$.

      (3.0.2) For standard induction, we get, for any left R[G]-module W, one version of Frobenius reciprocity:

      (${\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[H]}} - {\text{mod}}}}({\text{V,}}\;{\text{W|H)}} \cong \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[G]}} - {\text{mod}}}}({\text{In}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V),}}\;{\text{W)}}$,

      the isomorphism being$\psi \mapsto \;({\text{the}}\;{\text{map}}\;{{\text{g}}^ \otimes }{\text{v}} \mapsto {\text{g}}\psi {\text{(v)}}$. Taking for W the trivial...

    • Chapter 4: Middle Convolution
      (pp. 79-84)

      (4.0.1) We fix a prime number$\ell $. We work on${{\text{A}}^1}$over an algebraically closed field k in which$\ell $is invertible. We wish to define a certain class${P_{{\text{conv}}}}$of irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaves}}\;F$on${{\text{A}}^1}$. Given an irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on${{\text{A}}^1}$(or, equivalently, a nonpunctual irreducible perverse sheaf${\text{K}} = F[1]\;{\text{on}}\;{{\text{A}}^1}$), denote by

      (${\text{S:}}\, = \;{\text{Sing(}}F{)_{{\text{finite}}}}$

      finite the finite set of points in${{\text{A}}^1}$at which$F$is not lisse.

      (4.0.2) We say that$F$lies in${P_{{\text{conv}}}}$if

      (${\text{rank(}}F)\; + \;\# {\text{S}}\; + \;{\Sigma _{{\text{t}}\;{\text{in}}\;{\text{S}} \cup {\text{\{ }}\infty {\text{\} }}}}\;{\text{Swa}}{{\text{n}}_{\text{t}}}(F)\; \geqslant \;3$.

      (4.0.3) If k has characteristic zero, then among all irreducible middle extensions$F$, only the constant sheaf${{{\text{\bar Q}}}_\ell }\,$...

  5. Part II: Twist Sheaves, over an Algebraically Closed Field

    • Chapter 5: Twist Sheaves and Their Monodromy
      (pp. 85-116)

      (5.0.1) In this section, we make explicit the “families of twists” we will be concerned with. We fix an algebraically closed field k, a proper smooth connected curve C/k whose genus is denoted g, and a prime number$\ell $invertible in k. We also fix an integer${\text{r}} \geqslant {\text{1}}$, and an irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C of generic rank r. This means that for some dense open set U in C, with${\text{j}}\;{\text{:}}\;{\text{U}} \to {\text{C}}$the inclusion,$F\,|\,U$is a lisse sheaf of rank r on U which is irreducible in the sense that the corresponding r–dimensional${{{\text{\bar Q}}}_\ell }$–representation of${\pi _1}({\text{U)}}$...

  6. Part III: Twist Sheaves, over a Finite Field

    • Chapter 6: Dependence on Parameters
      (pp. 117-124)

      (6.0.1). The following lemma is standard. We include it for ease of reference.

      Lemma 6. 0. 2 Let T be an arbitrary scheme, X/T a proper smooth T–scheme with geometrically connected fibres everywhere of dimension N,Lan invertible${O_{\text{X}}}$–module, and L in${{\text{H}}^0}({\text{X,}}\;L)$a global section. Suppose L is nonzero on each geometric fibre of X/T, i. e., for every geometric point t of T, the image${{\text{L}}_{\text{t}}}$of L in${{\text{H}}^0}({{\text{X}}_{\text{t}}},\;{L_{\text{t}}})$is nonzero. Then the locus “L = 0 as section ofL”, call it Z, is a Cartier divisor in X, which is flat over...

    • Chapter 7: Diophantine Applications over a Finite Field
      (pp. 125-146)

      (7.0.1) In this section, we work over a finite field k, of cardinality q and characteristic p. We fix a proper, smooth, geometrically connected curve C/k of genus g, an effective divisor D on C of degree${\text{d}} \geqslant {\text{2g}} + 1$, a prime number$\ell $invertible in k, an integer${\text{r}} \geqslant {\text{1}}$, and a geometrically irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C of generic rank r. We denote by${\text{Sing(}}F)\; \subset \;{\text{C}}$the finite set of closed points of C at whichFis not lisse, and by${\text{Sing(}}F{)_{{\text{finite}}}}$the intersection${\text{Sing(}}F)\; \cap ({\text{C}}\, - \,{\text{D)}}$.

      The space

      (${\text{X}}\;{\text{:}} = \;Fct({\text{C,}}\;{\text{d,}}\;{\text{D,}}\;{\text{Sing(}}F{)_{{\text{finite}}}})$

      has a natural structure of scheme over k, cf. Proposition 6.1.10....

    • Chapter 8: Average Order of Zero in Twist Families
      (pp. 147-178)

      (8.0.1) In this section, we work over a finite field k of odd characteristic. We give ourselves data${\text{(C/k, D, }}\ell {\text{, r, }}F{\text{, }}\chi {\text{, }}\iota {\text{, w)}}$as in 7.0. We suppose that after extension of scalars from k to${{\text{\bar k}}}$, our data${\text{(C/k, D, }}\ell {\text{, r, }}F{\text{, }}\chi {\text{)}}$satisfies all the hypotheses of Theorem 5.5.1.

      (8.0.2) We further suppose that$F({\text{w}}/2)$is symplectically self–dual on C/k, and that$\chi $has order 2.

      Then, by Poincaré duality,$G(({\text{w}} + 1)\,/\,2)$is orthogonally self–dual as a lisse sheaf on

      ${\text{X}}\;{\text{:}}\, = Fct({\text{C,}}\;{\text{d,}}\;{\text{D,}}\;{\text{Sing(}}F{)_{{\text{finite}}}})$.

      By Theorem 5.5.1,$G$has${{\text{G}}_{{\text{geom}}}}$either SO(N) or O(N).

      (8.1.1) Given a finite extension E/k, and f in X(E), we define the analytic...

  7. Part IV: Twist Sheaves, over Schemes of Finite Type over $\mathbb{Z}$

    • Chapter 9: Twisting by “Primes”, and Working over $\mathbb{Z}$
      (pp. 179-206)

      (9.0.1) In this section, we work over an arbitrary scheme T, which will play the role of a parameter scheme in what follows. We fix a proper, smooth, geometrically connected curve C/T of genus g, and an integer${\text{d}} \geqslant {\text{2g}} + 1$. We denote by${\text{Ja}}{{\text{c}}^{\text{d}}}({\text{C/T)}}$, or simply${\text{Ja}}{{\text{c}}^{\text{d}}}$, the open and closed subscheme of${\text{Pi}}{{\text{c}}_{{\text{C/T}}}}$formed by divisor classes of degree d. We denote by${\text{Di}}{{\text{v}}^{\text{d}}}({\text{C/T)}}$the space of effective divisors in C of degree d. Thus for any T–scheme Y, a Y–valued point of${\text{Di}}{{\text{v}}^{\text{d}}}({\text{C/T)}}$is a closed subscheme of${\text{C}}{ \times _{\text{T}}}\,{\text{Y}}$which is finite and locally free over Y...

    • Chapter 10: Horizontal Results
      (pp. 207-234)

      (10.0.1) We fix a prime number$\ell $, an integer${\text{n}} \geqslant {\text{2}}$, and a character

      $\chi :\;{\mu _{\text{n}}}({\text{Z[1/}}\ell {\text{n,}}\;{\zeta _{\text{n}}}{\text{])}}\; \to \;{({{{\text{\bar Q}}}_\ell })^ \times }$

      of order n. We fix a nonempty connected normal${\text{Z[1/}}\ell {\text{n,}}\;{\zeta _{\text{n}}}]\,$–scheme T of finite type. We fix a proper, smooth, geometrically connected curve C/T of genus g. We suppose given an effective Cartier divisor S in C which is finite etale over T of degree${\text{s}} \geqslant {\text{0}}$(with the convention that S is empty if e = 0). We suppose given a lisse${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C – S of rank${\text{r}} \geqslant {\text{1}}$. If n is 4 or 6, we suppose that${\text{r}} \leqslant 2$. We suppose given an integer w, and...

  8. References
    (pp. 235-240)
  9. Index
    (pp. 241-249)