Part III Essay on Serre's Conjecture
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Artin L-Functions
Artin L-functions Charlotte Euvrard Laboratoire de Mathématiques de Besançon January 10, 2014 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 1 / 12 Definition L=K Galois extension of number fields G = Gal(L=K ) its Galois group Let (r;V ) be a representation of G and c the associated character. Let p be a prime ideal in the ring of integers OK of K , we denote by : Ip the inertia group jp the Frobenius automorphism I V p = fv 2 V : 8i 2 Ip; r(i)(v) = vg Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 2 / 12 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Sum and Euler product Proposition Let d and d0 be the degrees of the representations (r;V ) and (r~;V Ip ). -
Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory. -
Roots of L-Functions of Characters Over Function Fields, Generic
Roots of L-functions of characters over function fields, generic linear independence and biases Corentin Perret-Gentil Abstract. We first show joint uniform distribution of values of Kloost- erman sums or Birch sums among all extensions of a finite field Fq, for ˆ almost all couples of arguments in Fq , as well as lower bounds on dif- ferences. Using similar ideas, we then study the biases in the distribu- tion of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick–Waxman and Keating–Rudnick, building on cohomological in- terpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenval- ues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties. Contents 1. Introduction and statement of the results 1 2. Kloosterman sums and Birch sums 12 3. Angles of Gaussian primes 14 4. Prime polynomials in short intervals 20 5. An extension of the large sieve for Frobenius 22 6. Generic maximality of splitting fields and linear independence 33 7. Proof of the generic linear independence theorems 36 References 37 1. Introduction and statement of the results arXiv:1903.05491v2 [math.NT] 17 Dec 2019 Throughout, p will denote a prime larger than 5 and q a power of p. ˆ 1.1. Kloosterman and Birch sums. -
Motives, Volume 55.2
http://dx.doi.org/10.1090/pspum/055.2 Recent Titles in This Series 55 Uwe Jannsen, Steven Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert £. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. Browder, editor, Nonlinear functional analysis and its applications (University of California, Berkeley, July 1983) 44 William K. Allard and Frederick J. Almgren, Jr., editors, Geometric measure theory and the calculus of variations (Humboldt State University, Areata, California, July/August 1984) 43 Francois Treves, editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984) 42 Anil Nerode and Richard A. -
Surveys in Differential Geometry
Surveys in Differential Geometry Vol. 1: Lectures given in 1990 edited by S.-T. Yau and H. Blaine Lawson Vol. 2: Lectures given in 1993 edited by C.C. Hsiung and S.-T. Yau Vol. 3: Lectures given in 1996 edited by C.C. Hsiung and S.-T. Yau Vol. 4: Integrable systems edited by Chuu Lian Terng and Karen Uhlenbeck Vol. 5: Differential geometry inspired by string theory edited by S.-T. Yau Vol. 6: Essay on Einstein manifolds edited by Claude LeBrun and McKenzie Wang Vol. 7: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer edited by S.-T. Yau Vol. 8: Papers in honor of Calabi, Lawson, Siu, and Uhlenbeck edited by S.-T. Yau Vol. 9: Eigenvalues of Laplacians and other geometric operators edited by A. Grigor’yan and S-T. Yau Vol. 10: Essays in geometry in memory of S.-S. Chern edited by S.-T. Yau Vol. 11: Metric and comparison geometry edited by Jeffrey Cheeger and Karsten Grove Vol. 12: Geometric flows edited by Huai-Dong Cao and S.-T. Yau Vol. 13: Geometry, analysis, and algebraic geometry edited by Huai-Dong Cao and S.-T.Yau Vol. 14: Geometry of Riemann surfaces and their moduli spaces edited by Lizhen Ji, Scott A. Wolpert, and S.-T. Yau Volume XIV Surveys in Differential Geometry Geometry of Riemann surfaces and their moduli spaces edited by Lizhen Ji, Scott A. Wolpert, and Shing-Tung Yau International Press www.intlpress.com Series Editor: Shing-Tung Yau Surveys in Differential Geometry, Volume 14 Geometry of Riemann surfaces and their moduli spaces Volume Editors: Lizhen Ji, University of Michigan Scott A. -
Arxiv:0906.3146V1 [Math.NT] 17 Jun 2009
Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann– Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumer- ation formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geome- try over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, follow- ing Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ⊗F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes. -
The Work of Pierre Deligne
THE WORK OF PIERRE DELIGNE W.T. GOWERS 1. Introduction Pierre Deligne is indisputably one of the world's greatest mathematicians. He has re- ceived many major awards, including the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, and the Wolf Prize in 2008. While one never knows who will win the Abel Prize in any given year, it was virtually inevitable that Deligne would win it in due course, so today's announcement is about as small a surprise as such announcements can be. This is the third time I have been asked to present to a general audience the work of the winner of the Abel Prize, and my assignment this year is by some way the most difficult of the three. Two years ago, I talked about John Milnor, whose work in geometry could be illustrated by several pictures. Last year, the winner was Endre Szemer´edi,who has solved several problems with statements that are relatively simple to explain (even if the proofs were very hard). But Deligne's work, though it certainly has geometrical aspects, is not geometrical in a sense that lends itself to pictorial explanations, and the statements of his results are far from elementary. So I am forced to be somewhat impressionistic in my description of his work and to spend most of my time discussing the background to it rather than the work itself. 2. The Ramanujan conjecture One of the results that Deligne is famous for is solving a conjecture of Ramanujan. This conjecture is not the obvious logical starting point for an account of Deligne's work, but its solution is the most concrete of his major results and therefore the easiest to explain. -
A Refinement of the Artin Conductor and the Base Change Conductor
A refinement of the Artin conductor and the base change conductor C.-L. Chai and C. Kappen Abstract For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bArK of the classical Artin distribution ArK . It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects ArK in the sense that ArK is equal to the sum of bArK and its conjugate distribution. 1 Compared with 2 ArK , the bisection bArK provides a higher resolution on the level of tame ramification. In the second part of this article, we prove that the base change conductor c(T ) of an analytic K-torus T is equal to the value of bArK on the Qp-rational ∗ ∗ Galois representation X (T ) ⊗Zp Qp that is given by the character module X (T ) of T . We hereby generalize a formula for the base change conductor of an algebraic K-torus, and we obtain a formula for the base change conductor of a semiabelian K-variety with potentially ordinary reduction. Contents 1 Introduction 1 2 Notation 4 3 A bisection of the Artin character 5 4 Analytic tori 15 5 The base change conductor 17 6 A formula for the base change conductor 24 7 Application to semiabelian varieties with potentially ordinary reduc- tion 27 References 30 1. Introduction The base change conductor c(A) of a semiabelian variety A over a local field K is an interesting arithmetic invariant; it measures the growth of the volume of the N´eron model of A under base change, cf. -
Background on Local Fields and Kummer Theory
MATH 776 BACKGROUND ON LOCAL FIELDS AND KUMMER THEORY ANDREW SNOWDEN Our goal at the moment is to prove the Kronecker{Weber theorem. Before getting to this, we review some of the basic theory of local fields and Kummer theory, both of which will be used constantly throughout this course. 1. Structure of local fields Let K=Qp be a finite extension. We denote the ring of integers by OK . It is a DVR. There is a unique maximal ideal m, which is principal; any generator is called a uniformizer. We often write π for a uniformizer. The quotient OK =m is a finite field, called the residue field; it is often denoted k and its cardinality is often denoted q. Fix a uniformizer π. Every non-zero element x of K can be written uniquely in the form n uπ where u is a unit of OK and n 2 Z; we call n the valuation of x, and often denote it S −n v(x). We thus have K = n≥0 π OK . This shows that K is a direct union of the fractional −n ideals π OK , each of which is a free OK -module of rank one. The additive group OK is d isomorphic to Zp, where d = [K : Qp]. n × ∼ × The decomposition x = uπ shows that K = Z × U, where U = OK is the unit group. This decomposition is non-canonical, as it depends on the choice of π. The exact sequence 0 ! U ! K× !v Z ! 0 is canonical. Choosing a uniformizer is equivalent to choosing a splitting of this exact sequence. -
L-FUNCTIONS Contents Introduction 1 1. Algebraic Number Theory 3 2
L-FUNCTIONS ALEKSANDER HORAWA Contents Introduction1 1. Algebraic Number Theory3 2. Hecke L-functions 10 3. Tate's Thesis: Fourier Analysis in Number Fields and Hecke's Zeta Functions 13 4. Class Field Theory 52 5. Artin L-functions 55 References 79 Introduction These notes, meant as an introduction to the theory of L-functions, form a report from the author's Undergraduate Research Opportunities project at Imperial College London under the supervision of Professor David Helm. The theory of L-functions provides a way to study arithmetic properties of integers (and, more generally, integers of number fields) by first, translating them to analytic properties of certain functions, and then, using the tools and methods of analysis, to study them in this setting. The first indication that the properties of primes could be studied analytically came from Euler, who noticed the factorization (for a real number s > 1): 1 X 1 Y 1 = : ns 1 − p−s n=1 p prime This was later formalized by Riemann, who defined the ζ-function by 1 X 1 ζ(s) = ns n=1 for a complex number s with Re(s) > 1 and proved that it admits an analytic continuation by means of a functional equation. Riemann established a connection between its zeroes and the distribution of prime numbers, and it was also proven that the Prime Number Theorem 1 2 ALEKSANDER HORAWA is equivalent to the fact that no zeroes of ζ lie on the line Re(s) = 1. Innocuous as it may seem, this function is very difficult to study (the Riemann Hypothesis, asserting that the 1 non-trivial zeroes of ζ lie on the line Re(s) = 2 , still remains one of the biggest open problems in mathematics). -
Pierre Deligne
www.abelprize.no Pierre Deligne Pierre Deligne was born on 3 October 1944 as a hobby for his own personal enjoyment. in Etterbeek, Brussels, Belgium. He is Profes- There, as a student of Jacques Tits, Deligne sor Emeritus in the School of Mathematics at was pleased to discover that, as he says, the Institute for Advanced Study in Princeton, “one could earn one’s living by playing, i.e. by New Jersey, USA. Deligne came to Prince- doing research in mathematics.” ton in 1984 from Institut des Hautes Études After a year at École Normal Supériure in Scientifiques (IHÉS) at Bures-sur-Yvette near Paris as auditeur libre, Deligne was concur- Paris, France, where he was appointed its rently a junior scientist at the Belgian National youngest ever permanent member in 1970. Fund for Scientific Research and a guest at When Deligne was around 12 years of the Institut des Hautes Études Scientifiques age, he started to read his brother’s university (IHÉS). Deligne was a visiting member at math books and to demand explanations. IHÉS from 1968-70, at which time he was His interest prompted a high-school math appointed a permanent member. teacher, J. Nijs, to lend him several volumes Concurrently, he was a Member (1972– of “Elements of Mathematics” by Nicolas 73, 1977) and Visitor (1981) in the School of Bourbaki, the pseudonymous grey eminence Mathematics at the Institute for Advanced that called for a renovation of French mathe- Study. He was appointed to a faculty position matics. This was not the kind of reading mat- there in 1984. -
Multiplicative Reduction and the Cyclotomic Main Conjecture for Gl2
Pacific Journal of Mathematics MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Volume 283 No. 1 July 2016 PACIFIC JOURNAL OF MATHEMATICS Vol. 283, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.283.171 MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER We show that the cyclotomic Iwasawa–Greenberg main conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg main conjecture was established in[Skinner and Urban 2014], in combination with work of Kato[2004], for a large class of newforms f 2 Sk.00.N// that are ordinary at an odd prime p - N, subject to k ≡ 2 .mod p − 1/ and certain conditions on the mod p Galois representation associated with f . The purpose of this note is to extend this result to the case where p j N (in which case k is necessarily equal to 2). P1 n Recall that the coefficients an of the q-expansion f D nD1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f ) are algebraic integers that generate a finite extension Q. f / ⊂ C of Q. Let p be an odd prime and let L be a finite extension of the completion of Q. f / at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q.