In the last 15 years, many class number formulas and main conjectures have been proven. Here, we ... more In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(f0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(f0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the onevariable case, we let f denote the p-ordinary Galois representation with values ...
Bulletin of the American Mathematical Society, 1972
The problem of p-adically estimating the number of solutions of an algebraic variety over a finit... more The problem of p-adically estimating the number of solutions of an algebraic variety over a finite field of characteristic p was initiated by the classical result of Che valley-Warning [11]. THEOREM. Let G(x 1 ,..., xj be a polynomial with integral coefficients, of degree less than n. The number of solutions of G(x 1? ...,x M) = 0 modp is divisible by p. My aim in this paper, is to explain a conjecture of N. Katz [3], which provides quite sharp information concerning this general problem, and to indicate some theorems I have obtained which affirm his conjecture, under a mild hypothesis. Proofs of these theorems will be contained in [8]. 1. The Zeta-function of a variety over a finite field Let X 0 be a scheme of finite type over k = F r We wish to study N s , the number of rational points of X 0 over F qS , for all s > 1. The Zeta-function of X 0 /k expresses this information for us: Z(T;X 0 /k) = exp(j£-^| Given any "Weil cohomology" in the terminology of Kleiman's survey article [6], the Zeta-function may be expressed as an alternating product of characteristic polynomials, Z(r;X 0 /fe) = ndet(l-/ 1 .T)<-1 »' tl i where f denotes the endomorphism of H'(X 0) induced by/, the qui power endomorphism of the structure sheaf of X 0. Moreover, if X 0 is proper and smooth, the Zeta-function satisfies a functional equation with respect to a change of variables An expanded version of an invited address delivered to the 688th meeting of the Society at the Massachusetts Institute of Technology in Cambridge, Massachusetts on October 30,
Model Theory Algebra and Geometry 2000 Isbn 0 521 78068 3 Pags 199 227, 2000
This is an introductory exposition to background material useful to appreciate various formulatio... more This is an introductory exposition to background material useful to appreciate various formulations of the Mordell-Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (for example, via the method of Chabauty) relevant to Mordell-Lang. The article turns technical at one point (the step in the proof of the Mordell-Lang Conjecture in characteristic zero which passes from number fields to general fields). Two different procedures are sketched for doing this, with more details given than are readily found in the literature. There is also some discussion of issues of effectivity.
Transactions of the American Mathematical Society, 1964
In [8], Milnor exhibited an example of two finite complexes Xy,X2 which are homeomorphic, but com... more In [8], Milnor exhibited an example of two finite complexes Xy,X2 which are homeomorphic, but combinatorially distinct. That was the first example of the disparity between combinatorial notions versus topological ones on finite complexes. Previously the only results were positive in nature. Papakyriakopoulos had proved the Hauptvermutung for 2-dimensional complexes [11], and Moise [10] (and later Bing) had proved the Hauptvermutung for 3-dimensional manifolds. The object of this paper is to provide an example (see §5) of a finite complex K and two simplicial imbeddings, a,ß:K-*Sm which are combinatorially inequivalent, yet topologically equivalent. The nature of the argument is such that the minimum m it yields is m = 23. The construction of this example uses relative forms of the strong and weak stability theorem (see [6]). It also uses recent results of Haefliger, Bott, and Milnor. To show that the two imbeddings a,ß:K->Sm are not combinatorially equivalent, I notice that Sm/0L(K) * Xy, Sm/ß(K) x X2 (the isomorphisms being combinatorial). Since Xy is combinatorially distinct from X2, the result follows that there is no combinatorial homeomorphism K-.Sm £ Sm such that ko a = ß. In the course of the construction, a general theorem is proved (§6) which provides the possibility of producing many pairs of (combinatorially distinct) combinatorial imbeddings which are topologically equivalent. An unsettled question is the following : Are there combinatorial imbeddings /, g :Sk-> Sm which are combinatorially distinct yet topologically equivalent? By recent results of Zeeman [17], all combinatorial imbeddings of Sk in Sm(m ^ k + 3) are equivalent combinatorially. By the results of Smale [12], all combinatorial imbeddings of Sm_1 in Sm which are combinatorially locally trivial are combinatorially equivalent (for m sufficiently large). It is yet possible, however, for there to be two combinatorial imbeddings
We review a number of ways of "visualizing" the elements of the Shafarevich-Tate group of an elli... more We review a number of ways of "visualizing" the elements of the Shafarevich-Tate group of an elliptic curve E over a number field K. We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X 0 (N), where N is the conductor of the elliptic curve). For a given such E with nontrivial Shafarevich-Tate group, we pose the question: Are all the curves of genus one representing elements of the Shafarevich-Tate group of E isomorphic (over the rationals) to curves contained in a (single) abelian surface A, itself defined over the rationals, containing E as a subelliptic curve, and contained in turn in the new part of the jacobian of a modular curve X 0 (N)? At first view, one might imagine that there are few E with nontrivial Shafarevich-Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich-Tate group is large enough. Nonetheless, among all (modular) elliptic curves E as above, with conductors up to 5500 and with no rational point of order 2, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.
Bulletin of the American Mathematical Society, 2004
I wish to thank Curt McMullen and Noam Elkies for helpful comments on an early draft, and Stephan... more I wish to thank Curt McMullen and Noam Elkies for helpful comments on an early draft, and Stephanie Yang and Ivan Petrakiev for providing the illustrations.
Bulletin of the American Mathematical Society, 2008
Part I of this article was presented in the Current Events Bulletin section of the winter meeting... more Part I of this article was presented in the Current Events Bulletin section of the winter meeting of the AMS on January 7, 2007, in New Orleans. The title of the talk was "The structure of error terms in number theory and an introduction to the Sato-Tate Conjecture". Part I and some of Part II were published in the Current Events Bulletin of the AMS that was distributed at the meeting.
This paper is about computational questions regarding p-adic height pairings on elliptic curves o... more This paper is about computational questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions K v at the places v of K dividing p, a single quantity: the value of the p-adic modular form E 2 associated to the elliptic curve. Thanks to the work of Kedlaya et al., we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E 2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E 2 on the Hasse domain.
This paper is about computational and theoretical questions regarding p-adic height pairings on e... more This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent. 2000 Mathematics Subject Classification: 11F33, 11Y40, 11G50
In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below)... more In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.
In the last 15 years, many class number formulas and main conjectures have been proven. Here, we ... more In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(f0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(f0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the onevariable case, we let f denote the p-ordinary Galois representation with values ...
Bulletin of the American Mathematical Society, 1972
The problem of p-adically estimating the number of solutions of an algebraic variety over a finit... more The problem of p-adically estimating the number of solutions of an algebraic variety over a finite field of characteristic p was initiated by the classical result of Che valley-Warning [11]. THEOREM. Let G(x 1 ,..., xj be a polynomial with integral coefficients, of degree less than n. The number of solutions of G(x 1? ...,x M) = 0 modp is divisible by p. My aim in this paper, is to explain a conjecture of N. Katz [3], which provides quite sharp information concerning this general problem, and to indicate some theorems I have obtained which affirm his conjecture, under a mild hypothesis. Proofs of these theorems will be contained in [8]. 1. The Zeta-function of a variety over a finite field Let X 0 be a scheme of finite type over k = F r We wish to study N s , the number of rational points of X 0 over F qS , for all s > 1. The Zeta-function of X 0 /k expresses this information for us: Z(T;X 0 /k) = exp(j£-^| Given any "Weil cohomology" in the terminology of Kleiman's survey article [6], the Zeta-function may be expressed as an alternating product of characteristic polynomials, Z(r;X 0 /fe) = ndet(l-/ 1 .T)<-1 »' tl i where f denotes the endomorphism of H'(X 0) induced by/, the qui power endomorphism of the structure sheaf of X 0. Moreover, if X 0 is proper and smooth, the Zeta-function satisfies a functional equation with respect to a change of variables An expanded version of an invited address delivered to the 688th meeting of the Society at the Massachusetts Institute of Technology in Cambridge, Massachusetts on October 30,
Model Theory Algebra and Geometry 2000 Isbn 0 521 78068 3 Pags 199 227, 2000
This is an introductory exposition to background material useful to appreciate various formulatio... more This is an introductory exposition to background material useful to appreciate various formulations of the Mordell-Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (for example, via the method of Chabauty) relevant to Mordell-Lang. The article turns technical at one point (the step in the proof of the Mordell-Lang Conjecture in characteristic zero which passes from number fields to general fields). Two different procedures are sketched for doing this, with more details given than are readily found in the literature. There is also some discussion of issues of effectivity.
Transactions of the American Mathematical Society, 1964
In [8], Milnor exhibited an example of two finite complexes Xy,X2 which are homeomorphic, but com... more In [8], Milnor exhibited an example of two finite complexes Xy,X2 which are homeomorphic, but combinatorially distinct. That was the first example of the disparity between combinatorial notions versus topological ones on finite complexes. Previously the only results were positive in nature. Papakyriakopoulos had proved the Hauptvermutung for 2-dimensional complexes [11], and Moise [10] (and later Bing) had proved the Hauptvermutung for 3-dimensional manifolds. The object of this paper is to provide an example (see §5) of a finite complex K and two simplicial imbeddings, a,ß:K-*Sm which are combinatorially inequivalent, yet topologically equivalent. The nature of the argument is such that the minimum m it yields is m = 23. The construction of this example uses relative forms of the strong and weak stability theorem (see [6]). It also uses recent results of Haefliger, Bott, and Milnor. To show that the two imbeddings a,ß:K->Sm are not combinatorially equivalent, I notice that Sm/0L(K) * Xy, Sm/ß(K) x X2 (the isomorphisms being combinatorial). Since Xy is combinatorially distinct from X2, the result follows that there is no combinatorial homeomorphism K-.Sm £ Sm such that ko a = ß. In the course of the construction, a general theorem is proved (§6) which provides the possibility of producing many pairs of (combinatorially distinct) combinatorial imbeddings which are topologically equivalent. An unsettled question is the following : Are there combinatorial imbeddings /, g :Sk-> Sm which are combinatorially distinct yet topologically equivalent? By recent results of Zeeman [17], all combinatorial imbeddings of Sk in Sm(m ^ k + 3) are equivalent combinatorially. By the results of Smale [12], all combinatorial imbeddings of Sm_1 in Sm which are combinatorially locally trivial are combinatorially equivalent (for m sufficiently large). It is yet possible, however, for there to be two combinatorial imbeddings
We review a number of ways of "visualizing" the elements of the Shafarevich-Tate group of an elli... more We review a number of ways of "visualizing" the elements of the Shafarevich-Tate group of an elliptic curve E over a number field K. We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X 0 (N), where N is the conductor of the elliptic curve). For a given such E with nontrivial Shafarevich-Tate group, we pose the question: Are all the curves of genus one representing elements of the Shafarevich-Tate group of E isomorphic (over the rationals) to curves contained in a (single) abelian surface A, itself defined over the rationals, containing E as a subelliptic curve, and contained in turn in the new part of the jacobian of a modular curve X 0 (N)? At first view, one might imagine that there are few E with nontrivial Shafarevich-Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich-Tate group is large enough. Nonetheless, among all (modular) elliptic curves E as above, with conductors up to 5500 and with no rational point of order 2, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.
Bulletin of the American Mathematical Society, 2004
I wish to thank Curt McMullen and Noam Elkies for helpful comments on an early draft, and Stephan... more I wish to thank Curt McMullen and Noam Elkies for helpful comments on an early draft, and Stephanie Yang and Ivan Petrakiev for providing the illustrations.
Bulletin of the American Mathematical Society, 2008
Part I of this article was presented in the Current Events Bulletin section of the winter meeting... more Part I of this article was presented in the Current Events Bulletin section of the winter meeting of the AMS on January 7, 2007, in New Orleans. The title of the talk was "The structure of error terms in number theory and an introduction to the Sato-Tate Conjecture". Part I and some of Part II were published in the Current Events Bulletin of the AMS that was distributed at the meeting.
This paper is about computational questions regarding p-adic height pairings on elliptic curves o... more This paper is about computational questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions K v at the places v of K dividing p, a single quantity: the value of the p-adic modular form E 2 associated to the elliptic curve. Thanks to the work of Kedlaya et al., we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E 2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E 2 on the Hasse domain.
This paper is about computational and theoretical questions regarding p-adic height pairings on e... more This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent. 2000 Mathematics Subject Classification: 11F33, 11Y40, 11G50
In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below)... more In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.
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