Coleman Integration in Larger Characteristic
|
|
|
- Ferdinand Pearson
- 5 years ago
- Views:
Transcription
1 Coleman Integration in Larger Characteristic ANTS XIII University of Wisconsin, Madison Alex J. Best 17/7/2018 Boston University
2 The big picture Kedlaya Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
3 The big picture CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
4 The big picture BaBrKe Coleman integration CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
5 The big picture Ba-BaTu BaBrKe Coleman integration CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
6 The big picture Ba-BaTu BaBrKe Coleman integration Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
7 The big picture Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
8 The big picture Be Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
9 The big picture Be Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
10 The big picture Be Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more...
11 The big picture Be Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu More curves Kedlaya Õ( p) Arul, Balakrishnan, Best, Bradshaw, Castryk, Costa, Denef, Gaudry, Gurel, Harvey, Kedlaya, Magner, Minzlaff, Shieh, Triantafillou, Tuitman, Vercauteren, and more... There is (at least) one dimension missing: Small p!
12 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions.
13 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions. Longer term goals: Adapt descendants of Kedlaya s algorithm to compute (iterated) Coleman integrals, e.g.:
14 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions. Longer term goals: Adapt descendants of Kedlaya s algorithm to compute (iterated) Coleman integrals, e.g.: Larger characteristic (Harvey)
15 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions. Longer term goals: Adapt descendants of Kedlaya s algorithm to compute (iterated) Coleman integrals, e.g.: Larger characteristic (Harvey) Average polynomial time (Harvey)
16 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions. Longer term goals: Adapt descendants of Kedlaya s algorithm to compute (iterated) Coleman integrals, e.g.: Larger characteristic (Harvey) Average polynomial time (Harvey) Effectiver average polynomial time (Harvey-Sutherland)
17 Motivation (Explicit) Coleman integration is a central tool in (non-abelian) Chabauty and can be applied to find: heights, torsion points and regulators also, but, algorithms lag behind the related ones for zeta functions. Longer term goals: Adapt descendants of Kedlaya s algorithm to compute (iterated) Coleman integrals, e.g.: Larger characteristic (Harvey) Average polynomial time (Harvey) Effectiver average polynomial time (Harvey-Sutherland) Applications to rational points, combining congruence information for many primes, 1-step (Mordell-Weil) sieving.
18 Coleman integration Throughout we take X /Z p a genus g odd degree hyperelliptic curve, and p an odd prime. We pick a lift of the Frobenius map, φ : X X, and write A (resp. A loc (X )) for overconvergent (resp. locally analytic) functions on X.
19 Coleman integration Throughout we take X /Z p a genus g odd degree hyperelliptic curve, and p an odd prime. We pick a lift of the Frobenius map, φ : X X, and write A (resp. A loc (X )) for overconvergent (resp. locally analytic) functions on X. Theorem (Coleman) There is a Q p -linear map x b : Ω1 Q A p A loc (X ) for which: x d = id: Ω 1 A Q p Ω 1 loc FTC b x b x b x φ ω = φ ω d = id: A A loc b Frobenius equivariance
20 Reduction to reduction Balakrishnan-Bradshaw-Kedlaya reduce the problem of computing all Coleman integrals of basis differentials ω i of HdR 1 (X ) between X and a point x X (Q p ), to: 1. Finding tiny integrals between nearby points, 2. Writing φ ω i df i = j a ijω j and evaluating the primitive f i for a point P near x, for each i.
21 Reduction to reduction Balakrishnan-Bradshaw-Kedlaya reduce the problem of computing all Coleman integrals of basis differentials ω i of HdR 1 (X ) between X and a point x X (Q p ), to: 1. Finding tiny integrals between nearby points, 2. Writing φ ω i df i = j a ijω j and evaluating the primitive f i for a point P near x, for each i. Applying φ to the basis x i dx/2y for i = 0,..., 2g 1 gives φ ω i N 1 j=0 (2g+1)j r=0 B j,r x p(i+r+1) 1 p(2j+1)+1 dx y 2y (mod p N ) B j,r Z p are in terms of coefficients of the curve and binomial coefficients.
22 Kedlaya s algorithm Theorem (Kedlaya) The action of φ on HMW 1 (X ) (which determines the zeta function of X ) can be computed in time Õ(p).
23 Kedlaya s algorithm Theorem (Kedlaya) The action of φ on HMW 1 (X ) (which determines the zeta function of X ) can be computed in time Õ(p). Theorem (Harvey) If p > (2g + 1)(2N 1) the action of φ on H 1 MW (X ) can be computed in time Õ( p).
24 Kedlaya s algorithm Theorem (Kedlaya) The action of φ on HMW 1 (X ) (which determines the zeta function of X ) can be computed in time Õ(p). Theorem (Harvey) If p > (2g + 1)(2N 1) the action of φ on HMW 1 (X ) can be computed in time Õ( p). The problem solved here is almost the same: determining a ij s.t. φ ω i j a ij ω j image(d).
25 Primitive technology Revised problem Computing f along with ω df when reducing degree. For vanilla Kedlaya this is easy, the reduction procedure is transparent, whenever we subtract dg to reduce, add g onto f. For faster variants, this is not so simple!
26 The reduction process Harvey uses horizontal and vertical reductions to find the action of Frobenius on cohomology,
27 The reduction process Harvey uses horizontal and vertical reductions to find the action of Frobenius on cohomology, abstractly we have: Spaces of differentials W t, indexed by degree, each of dimension 2g. Goal Reduce all differentials from W t to a cohomologous one in W 0, write in terms of fixed basis of W 0.
28 The reduction process Harvey uses horizontal and vertical reductions to find the action of Frobenius on cohomology, abstractly we have: Spaces of differentials W t, indexed by degree, each of dimension 2g. Goal Reduce all differentials from W t to a cohomologous one in W 0, write in terms of fixed basis of W 0. Relations in the de Rham cohomology linear maps R(t): W t W t 1 t, with R(t)ω ω.
29 The reduction process Harvey uses horizontal and vertical reductions to find the action of Frobenius on cohomology, abstractly we have: Spaces of differentials W t, indexed by degree, each of dimension 2g. Goal Reduce all differentials from W t to a cohomologous one in W 0, write in terms of fixed basis of W 0. Relations in the de Rham cohomology linear maps R(t): W t W t 1 t, with R(t)ω ω. Want to evaluate W t ω R(1)R(2) R(t 1)R(t)ω W 0
30 O( p) Key fact Entries of R(t) are fractions of linear functions of t, with Z p coefficients; work of Bostan-Gaudry-Schost (& Harvey) = products can be interpolated R(a, b) = R(a + 1) R(b) R(a t, b + t) This is what gives a Õ( p) algorithm.
31 O( p) Key fact Entries of R(t) are fractions of linear functions of t, with Z p coefficients; work of Bostan-Gaudry-Schost (& Harvey) = products can be interpolated R(a, b) = R(a + 1) R(b) R(a t, b + t) This is what gives a Õ( p) algorithm. We also want an evaluation of the primitive f ω for which ω df ω = R(t)ω. We can keep this extra data throughout the recurrence as f ω is linear in ω.
32 O( p) Key fact Entries of R(t) are fractions of linear functions of t, with Z p coefficients; work of Bostan-Gaudry-Schost (& Harvey) = products can be interpolated R(a, b) = R(a + 1) R(b) R(a t, b + t) This is what gives a Õ( p) algorithm. We also want an evaluation of the primitive f ω for which ω df ω = R(t)ω. We can keep this extra data throughout the recurrence as f ω is linear in ω. Vital remark We must use evaluations of primitives here, instead of trying to compute f as a power series.
33 A problem and a solution Stumbling block This is no longer linear in the index t! You cannot apply BGS to evaluate this recurrence faster.
34 A problem and a solution Stumbling block This is no longer linear in the index t! You cannot apply BGS to evaluate this recurrence faster. Horner to the rescue! Instead of computing a series N i=0 a ix i by computing sequentially ( N ) a i x i we can instead compute i=t t=n,n 1,...,0 (( ((a N )x + a N 1 )x + )x + a 0 ) from the inside to the out. This is an iterated composition of linear functions, each of which is linear in the index t.
35 Explicit recurrence In matrix form we augment the (numerators of) the reduction matrices: y 2t dx/2y x 2g 1 y 2t dx/2y f (P) y 2(t 1) dx/2y (2t 1)r 0,0 + 2s 0,0 (2t 1)r 2g 1,0 + 2s 2g 1, x 2g 1 y 2(t 1) dx/2y (2t 1)r 0,2g 1 + 2s 0,2g 1 (2t 1)r 2g 1,2g 1 + 2s 2g 1,2g 1 f (P) S 0 (x) S 2g 1 (x) y 2 D V (t) so that we keep in memory a vector v W t Q p which gives the evaluation at the end.
36 Many integrals simultaneously We may wish to do this with multiple points in several residue disks. Instead of repeating the whole procedure (repeating computing the Frobenius matrix), augment with many points. y 2t dx/2y x 2g 1 y 2t dx/2y f (P1) f (PL) y 2(t 1) dx/y (2t 1)r 0,0 + 2s 0,0 (2t 1)r 2g 1,0 + 2s 2g 1, x 2g 1 y 2(t 1) dx/y (2t 1)r 0,2g 1 + 2s 0,2g 1 (2t 1)r 2g 1,2g 1 + 2s 2g 1,2g 1 f (P1) S 0(x(P 1)) S 2g 1(x(P 1)) y 2 (P 1)D V (t) f (PL) S 0(x(P L)) S 2g 1(x(P L)) 0 y(p L) 2 D V (t)
37 Many integrals simultaneously We may wish to do this with multiple points in several residue disks. Instead of repeating the whole procedure (repeating computing the Frobenius matrix), augment with many points. y 2t dx/2y x 2g 1 y 2t dx/2y f (P1) f (PL) y 2(t 1) dx/y (2t 1)r 0,0 + 2s 0,0 (2t 1)r 2g 1,0 + 2s 2g 1, x 2g 1 y 2(t 1) dx/y (2t 1)r 0,2g 1 + 2s 0,2g 1 (2t 1)r 2g 1,2g 1 + 2s 2g 1,2g 1 f (P1) S 0(x(P 1)) S 2g 1(x(P 1)) y 2 (P 1)D V (t) f (PL) S 0(x(P L)) S 2g 1(x(P L)) 0 y(p L) 2 D V (t) Note This matrix and its iterates have the same fixed form, when running BGS don t try and interpolate entries that are always 0 better run time.
38 Thanks for listening! Questions/comments?
39 Some amusing factoids Core algorithm is potentially more general? Evaluation of primitives is integration, but maybe of a different type? It is faster to evaluate the power series this way than to evaluate the power series you get from vanilla Kedlaya, even when multiple points are needed computing the power series is no use.
Coleman Integration in Larger Characteristic
Coleman Integration in Larger Characteristic ANTS XIII University of Wisconsin, Madison Alex J. Best 17/7/2018 Boston University The big picture Be Ba-BaTu BaBrKe Coleman integration Mi-ArBeCoMaTr Ha CaDeVe-Go-GaGu-Sh-Tu
Counting points on curves: the general case
Counting points on curves: the general case Jan Tuitman, KU Leuven October 14, 2015 Jan Tuitman, KU Leuven Counting points on curves: the general case October 14, 2015 1 / 26 Introduction Algebraic curves
Computations with Coleman integrals
Computations with Coleman integrals Jennifer Balakrishnan Harvard University, Department of Mathematics AWM 40 Years and Counting (Number Theory Session) Saturday, September 17, 2011 Outline 1 Introduction
Counting points on smooth plane quartics
Counting points on smooth plane quartics David Harvey University of New South Wales Number Theory Down Under, University of Newcastle 25th October 2014 (joint work with Andrew V. Sutherland, MIT) 1 / 36
Computing zeta functions of certain varieties in larger characteristic
Computing zeta functions of certain varieties in larger characteristic New York University 16th March 2010 Effective methods in p-adic cohomology Mathematical Institute, University of Oxford Notation q
Computing L-series of geometrically hyperelliptic curves of genus three. David Harvey, Maike Massierer, Andrew V. Sutherland
Computing L-series of geometrically hyperelliptic curves of genus three David Harvey, Maike Massierer, Andrew V. Sutherland The zeta function Let C/Q be a smooth projective curve of genus 3 p be a prime
Counting Points on Curves using Monsky-Washnitzer Cohomology
Counting Points on Curves using Monsky-Washnitzer Cohomology Frederik Vercauteren frederik@cs.bris.ac.uk Jan Denef jan.denef@wis.kuleuven.ac.be University of Leuven http://www.arehcc.com University of
p-adic Integration on Curves of Bad Reduction
p-adic Integration on Curves of Bad Reduction Eric Katz (University of Waterloo) joint with David Zureick-Brown (Emory University) January 16, 2014 Eric Katz (Waterloo) p-adic Integration January 16, 2014
A survey of p-adic approaches to zeta functions (plus a new approach)
A survey of p-adic approaches to zeta functions (plus a new approach) Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://math.ucsd.edu/~kedlaya/slides/
Computing L-series coefficients of hyperelliptic curves
Computing L-series coefficients of hyperelliptic curves Kiran S. Kedlaya and Andrew V. Sutherland Massachusetts Institute of Technology May 19, 2008 Demonstration The distribution of Frobenius traces Let
Explicit p-adic methods for elliptic and hyperelliptic curves
Explicit p-adic methods for elliptic and hyperelliptic curves Jennifer S Balakrishnan Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK e-mail: balakrishnan@mathsoxacuk Abstract
A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties
A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties Steffen Müller Universität Hamburg joint with Jennifer Balakrishnan and William Stein Rational points on curves: A p-adic and
A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties
A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties Steffen Müller Universität Hamburg joint with Jennifer Balakrishnan (Harvard) and William Stein (U Washington) Forschungsseminar
Counting points on hyperelliptic curves
University of New South Wales 9th November 202, CARMA, University of Newcastle Elliptic curves Let p be a prime. Let X be an elliptic curve over F p. Want to compute #X (F p ), the number of F p -rational
ITERATED COLEMAN INTEGRATION FOR HYPERELLIPTIC CURVES
ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE JENNIFER BALAKRIHNAN Astract The Coleman integral is a p-adic line integral Doule Coleman integrals on elliptic curves appear in Kim s nonaelian Chaauty
Rational Points on Curves in Practice. Michael Stoll Universität Bayreuth Journées Algophantiennes Bordelaises Université de Bordeaux June 8, 2017
Rational Points on Curves in Practice Michael Stoll Universität Bayreuth Journées Algophantiennes Bordelaises Université de Bordeaux June 8, 2017 The Problem Let C be a smooth projective and geometrically
Alternative Models: Infrastructure
Alternative Models: Infrastructure Michael J. Jacobson, Jr. jacobs@cpsc.ucalgary.ca UNCG Summer School in Computational Number Theory 2016: Function Fields Mike Jacobson (University of Calgary) Alternative
Computing zeta functions of nondegenerate toric hypersurfaces
Computing zeta functions of nondegenerate toric hypersurfaces Edgar Costa (Dartmouth College) December 4th, 2017 Presented at Boston University Number Theory Seminar Joint work with David Harvey (UNSW)
FINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016
FINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016 PREPARED BY SHABNAM AKHTARI Introduction and Notations The problems in Part I are related to Andrew Sutherland
Hecke Operators for Arithmetic Groups via Cell Complexes. Mark McConnell. Center for Communications Research, Princeton
Hecke Operators for Arithmetic Groups via Cell Complexes 1 Hecke Operators for Arithmetic Groups via Cell Complexes Mark McConnell Center for Communications Research, Princeton Hecke Operators for Arithmetic
E 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?
![E 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?](/thumbs/84/90664534.jpg "E 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?")
. Characteristic Classes from the viewpoint of Operator Theory. Introduction Overarching Question: How can you tell if two vector bundles over a manifold are isomorphic? Let X be a compact Hausdorff space.
Towards a precise Sato-Tate conjecture in genus 2
Towards a precise Sato-Tate conjecture in genus 2 Kiran S. Kedlaya Department of Mathematics, Massachusetts Institute of Technology Department of Mathematics, University of California, San Diego kedlaya@mit.edu,
this to include the explicit maps, please do so!
Contents 1. Introduction 1 2. Warmup: descent on A 2 + B 3 = N 2 3. A 2 + B 3 = N: enriched descent 3 4. The Faltings height 5 5. Isogeny and heights 6 6. The core of the proof that the height doesn t
Counting Points on Curves of the Form y m 1 = c 1x n 1 + c 2x n 2 y m 2
Counting Points on Curves of the Form y m 1 = c 1x n 1 + c 2x n 2 y m 2 Matthew Hase-Liu Mentor: Nicholas Triantafillou Sixth Annual Primes Conference 21 May 2016 Curves Definition A plane algebraic curve
Elliptic Curve Discrete Logarithm Problem
Elliptic Curve Discrete Logarithm Problem Vanessa VITSE Université de Versailles Saint-Quentin, Laboratoire PRISM October 19, 2009 Vanessa VITSE (UVSQ) Elliptic Curve Discrete Logarithm Problem October
Master s thesis topics Algebraic Geometry and Number Theory
Master s thesis topics Algebraic Geometry and Number Theory 2017-2018 This is the list of possible topics for a master s thesis proposed by the staff members of the research group Algebraic Geometry and
1. Motivation: search for a p-adic cohomology theory
AN INTRODUCTION TO RIGID COHOMOLOGY CHRISTOPHER LAZDA 1. Motivation: search for a p-adic cohomology theory Let k be a perfect field of characteristic p > 0, W = W (k) the ring of Witt vectors of k, and
Denition (Monsky-Washinitzer [4]) In the sequence of homomorphisms 0 ;!^0 d ;!^ ;! d d ;!^n;!0 the cohomology groups of the de Rham complex over A y R
![Denition (Monsky-Washinitzer [4]) In the sequence of homomorphisms 0 ;!^0 d ;!^ ;! d d ;!^n;!0 the cohomology groups of the de Rham complex over A y R](/thumbs/78/77678304.jpg "Denition (Monsky-Washinitzer [4]) In the sequence of homomorphisms 0 ;!^0 d ;!^ ;! d d ;!^n;!0 the cohomology groups of the de Rham complex over A y R")
Generalizing Kedlaya's order counting based on Miura theory Joe Suzuki Abstract K. Kedlaya proposed an method to count the number of F q -rational pointsinahyperelliptic curve, using the Leschetz xed points
Computing zeta functions in families of C a,b curves using deformation
Computing zeta functions in families of C a,b curves using deformation Wouter Castryck 2, Hendrik Hubrechts 1, and Frederik Vercauteren 2 1 Department of Mathematics, University of Leuven, Celestijnenlaan
RATIONAL POINTS ON CURVES
RATIONAL POINTS ON CURVES MICHAEL STOLL In these lectures we will discuss how to determine the set of rational points on a curve of genus at least 2 over Q. We will use hyperelliptic curves throughout
Rational Points on Curves of Genus 2
Rational Points on Curves of Genus 2 Michael Stoll Jacobs University Bremen Warwick, January 25, 2008 Curves of Genus 2 A curve of genus 2 over Q is given by an equation C : y 2 = f 6 x 6 + f 5 x 5 + f
The Sato-Tate conjecture for abelian varieties
The Sato-Tate conjecture for abelian varieties Andrew V. Sutherland Massachusetts Institute of Technology March 5, 2014 Mikio Sato John Tate Joint work with F. Fité, K.S. Kedlaya, and V. Rotger, and also
Rational Points on Curves
Rational Points on Curves 1 Q algebraic closure of Q. k a number field. Ambient variety: Elliptic curve E an elliptic curve defined over Q. E N = E E. In general A an abelian variety defined over Q. For
Galois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a.
Galois fields 1 Fields A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except by zero) can be performed, and satisfy the usual rules. More
Fundamental groups, polylogarithms, and Diophantine
Fundamental groups, polylogarithms, and Diophantine geometry 1 X: smooth variety over Q. So X defined by equations with rational coefficients. Topology Arithmetic of X Geometry 3 Serious aspects of the
NOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
Equidistributions in arithmetic geometry
Equidistributions in arithmetic geometry Edgar Costa Dartmouth College 14th January 2016 Dartmouth College 1 / 29 Edgar Costa Equidistributions in arithmetic geometry Motivation: Randomness Principle Rigidity/Randomness
An Extension of Kedlaya s Algorithm to Hyperelliptic Curves in Characteristic 2
An Extension of Kedlaya s Algorithm to Hyperelliptic Curves in Characteristic 2 Jan Denef 1 and Frederik Vercauteren 2,3 1 Department of Mathematics University of Leuven Celestijnenlaan 200B, B-3001 Leuven-Heverlee,
Factoring Algorithms Pollard s p 1 Method. This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors.
Factoring Algorithms Pollard s p 1 Method This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors. Input: n (to factor) and a limit B Output: a proper factor of
Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
Rational points on curves and tropical geometry.
Rational points on curves and tropical geometry. David Zureick-Brown (Emory University) Eric Katz (Waterloo University) Slides available at http://www.mathcs.emory.edu/~dzb/slides/ Specialization of Linear
Genus 2 Curves of p-rank 1 via CM method
School of Mathematical Sciences University College Dublin Ireland and Claude Shannon Institute April 2009, GeoCrypt Joint work with Laura Hitt, Michael Naehrig, Marco Streng Introduction This talk is about
A non-abelian conjecture of Birch and Swinnerton-Dyer type
A non-abelian conjecture of Birch and Swinnerton-Dyer type Minhyong Kim Bordeaux, July, 2012 Dedicated to Martin Taylor on the occasion of his 60th birthday. Diophantine geometry: general remarks Diophantine
Ate Pairing on Hyperelliptic Curves
Ate Pairing on Hyperelliptic Curves R. Granger, F. Hess, R. Oyono, N. Thériault F. Vercauteren EUROCRYPT 2007 - Barcelona Pairings Pairings Let G 1, G 2, G T be groups of prime order l. A pairing is a
Part II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
The 70th William Lowell Putnam Mathematical Competition Saturday, December 5, 2009
The 7th William Lowell Putnam Mathematical Competition Saturday, December 5, 9 A1 Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(a) + f(b) + f(c) + f(d) =.
Diagonals and Walks. Alin Bostan 1 Louis Dumont 1 Bruno Salvy 2. November 4, Introduction Diagonals Walks. 1 INRIA, SpecFun.
and Alin Bostan 1 Louis Dumont 1 Bruno Salvy 2 1 INRIA, SpecFun 2 INRIA, AriC November 4, 2014 1/16 Louis Dumont and 2/16 Louis Dumont and Motivations In combinatorics: Lattice walks w i,j : number of
Hypersurfaces and the Weil conjectures
Hypersurfaces and the Weil conjectures Anthony J Scholl University of Cambridge 13 January 2010 1 / 21 Number theory What do number theorists most like to do? (try to) solve Diophantine equations x n +
Hyperelliptic-curve cryptography. D. J. Bernstein University of Illinois at Chicago
Hyperelliptic-curve cryptography D. J. Bernstein University of Illinois at Chicago Thanks to: NSF DMS 0140542 NSF ITR 0716498 Alfred P. Sloan Foundation Two parts to this talk: 1. Elliptic curves; modern
Applications to the Beilinson-Bloch Conjecture
Applications to the Beilinson-Bloch Conjecture Green June 30, 2010 1 Green 1 - Applications to the Beilinson-Bloch Conjecture California is like Italy without the art. - Oscar Wilde Let X be a smooth projective
Hyperelliptic curves
1/40 Hyperelliptic curves Pierrick Gaudry Caramel LORIA CNRS, Université de Lorraine, Inria ECC Summer School 2013, Leuven 2/40 Plan What? Why? Group law: the Jacobian Cardinalities, torsion Hyperelliptic
Deterministic factorization of sums and differences of powers
arxiv:1512.06401v1 [math.nt] 20 Dec 2015 Deterministic factorization of sums and differences of powers Markus Hittmeir University of Salzburg Mathematics Department Abstract Let a,b N be fixed and coprime
CHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
Bjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006
University of California at Berkeley MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional (organized by Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár,, Alice Silverberg,
Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Spring Semester 2015
Department of Mathematics, University of California, Berkeley YOUR OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Spring Semester 205. Please write your - or 2-digit exam number on this
Diophantine geometry and non-abelian duality
Diophantine geometry and non-abelian duality Minhyong Kim Leiden, May, 2011 Diophantine geometry and abelian duality E: elliptic curve over a number field F. Kummer theory: E(F) Z p H 1 f (G,T p(e)) conjectured
Elliptic Curves Spring 2019 Problem Set #7 Due: 04/08/2019
18.783 Elliptic Curves Spring 2019 Problem Set #7 Due: 04/08/2019 Description These problems are related to the material covered in Lectures 13-14. Instructions: Solve problem 1 and then solve one of Problems
Motivic Periods, Coleman Functions, and the Unit Equation
Motivic Periods, Coleman Functions, and the Unit Equation An Ongoing Project D. Corwin 1 I. Dan-Cohen 2 1 MIT 2 Ben Gurion University of the Negev April 10, 2018 Corwin, Dan-Cohen Motivic Periods, Coleman
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN
VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN Abstract. For any field k and integer g 3, we exhibit a curve X over k of genus g such that X has no non-trivial automorphisms over k. 1. Statement
Period Domains. Carlson. June 24, 2010
Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes
Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016
Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on
TOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology
TOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology DONALD M. DAVIS Abstract. We use ku-cohomology to determine lower bounds for the topological complexity of mod-2 e lens spaces. In the
Galois Theory and Diophantine geometry ±11
Galois Theory and Diophantine geometry ±11 Minhyong Kim Bordeaux, January, 2010 1 1. Some Examples 1.1 A Diophantine finiteness theorem: Let a, b, c, n Z and n 4. Then the equation ax n + by n = c has
Chapter 9 Factorisation and Discrete Logarithms Using a Factor Base
Chapter 9 Factorisation and Discrete Logarithms Using a Factor Base February 15, 2010 9 The two intractable problems which are at the heart of public key cryptosystems, are the infeasibility of factorising
Explicit isogenies and the Discrete Logarithm Problem in genus three
Explicit isogenies and the Discrete Logarithm Problem in genus three Benjamin Smith INRIA Saclay Île-de-France Laboratoire d informatique de l école polytechnique (LIX) EUROCRYPT 2008 : Istanbul, April
Cover and Decomposition Index Calculus on Elliptic Curves made practical
Cover and Decomposition Index Calculus on Elliptic Curves made practical Application to a previously unreachable curve over F p 6 Vanessa VITSE Antoine JOUX Université de Versailles Saint-Quentin, Laboratoire
Non-generic attacks on elliptic curve DLPs
Non-generic attacks on elliptic curve DLPs Benjamin Smith Team GRACE INRIA Saclay Île-de-France Laboratoire d Informatique de l École polytechnique (LIX) ECC Summer School Leuven, September 13 2013 Smith
The Canonical Sheaf. Stefano Filipazzi. September 14, 2015
The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over
Algebraic Geometry: Elliptic Curves and 2 Theorems
Algebraic Geometry: Elliptic Curves and 2 Theorems Chris Zhu Mentor: Chun Hong Lo MIT PRIMES December 7, 2018 Chris Zhu Elliptic Curves and 2 Theorems December 7, 2018 1 / 16 Rational Parametrization Plane
Cofactors and Laplace s expansion theorem
Roberto s Notes on Linear Algebra Chapter 5: Determinants Section 3 Cofactors and Laplace s expansion theorem What you need to know already: What a determinant is. How to use Gauss-Jordan elimination to
LECTURE 2 FRANZ LEMMERMEYER
LECTURE 2 FRANZ LEMMERMEYER Last time we have seen that the proof of Fermat s Last Theorem for the exponent 4 provides us with two elliptic curves (y 2 = x 3 + x and y 2 = x 3 4x) in the guise of the quartic
Congruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
Problème du logarithme discret sur courbes elliptiques
Problème du logarithme discret sur courbes elliptiques Vanessa VITSE Université de Versailles Saint-Quentin, Laboratoire PRISM Groupe de travail équipe ARITH LIRMM Vanessa VITSE (UVSQ) DLP over elliptic
Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field
Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field Alin Bostan 1 Gilles Christol 2 Philippe Dumas 1 1 Inria (France) 2 Institut de mathématiques de Jussieu (France) Specfun
Counting points on elliptic curves over F q
Counting points on elliptic curves over F q Christiane Peters DIAMANT-Summer School on Elliptic and Hyperelliptic Curve Cryptography September 17, 2008 p.2 Motivation Given an elliptic curve E over a finite
Elliptic and Hyperelliptic Curve Cryptography
Elliptic and Hyperelliptic Curve Cryptography Renate Scheidler Research supported in part by NSERC of Canada Comprehensive Source Handbook of Elliptic and Hyperelliptic Curve Cryptography Overview Motivation
Seminar on Motives Standard Conjectures
Seminar on Motives Standard Conjectures Konrad Voelkel, Uni Freiburg 17. January 2013 This talk will briefly remind you of the Weil conjectures and then proceed to talk about the Standard Conjectures on
Lecture 8 - Algebraic Methods for Matching 1
CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanford.edu) February 1, 2018 Lecture 8 - Algebraic Methods for Matching 1 In the last lecture we showed that
Class groups and Galois representations
and Galois representations UC Berkeley ENS February 15, 2008 For the J. Herbrand centennaire, I will revisit a subject that I studied when I first came to Paris as a mathematician, in 1975 1976. At the
Introduction to Arithmetic Geometry Fall 2013 Problem Set #10 Due: 12/3/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Problem Set #10 Due: 12/3/2013 These problems are related to the material covered in Lectures 21-22. I have made every effort to proof-read them, but
Quantum wires, orthogonal polynomials and Diophantine approximation
Quantum wires, orthogonal polynomials and Diophantine approximation Introduction Quantum Mechanics (QM) is a linear theory Main idea behind Quantum Information (QI): use the superposition principle of
Algebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
On the evaluation of modular polynomials
On the evaluation of modular polynomials Andrew V. Sutherland Massachusetts Institute of Technology ANTS X July 10, 2012 http://math.mit.edu:/ drew 1 / 16 Introduction Let l be a prime and let F q be a
arxiv:math/ v1 [math.nt] 21 Sep 2004
![arxiv:math/ v1 [math.nt] 21 Sep 2004](/thumbs/96/126761909.jpg "arxiv:math/ v1 [math.nt] 21 Sep 2004")
arxiv:math/0409377v1 [math.nt] 21 Sep 2004 ON THE GCD OF AN INFINITE NUMBER OF INTEGERS T. N. VENKATARAMANA Introduction In this paper, we consider the greatest common divisor (to be abbreviated gcd in
What we have learned What is algorithm Why study algorithm The time and space efficiency of algorithm The analysis framework of time efficiency Asympt
Lecture 3 The Analysis of Recursive Algorithm Efficiency What we have learned What is algorithm Why study algorithm The time and space efficiency of algorithm The analysis framework of time efficiency
Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography
Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography Naomi Benger and Michael Scott, 1 School of Computing, Dublin City University, Ireland nbenger@computing.dcu.ie
A talk given at University of Wisconsin at Madison (April 6, 2006). Zhi-Wei Sun
A tal given at University of Wisconsin at Madison April 6, 2006. RECENT PROGRESS ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093,
Problem 1A. Find the volume of the solid given by x 2 + z 2 1, y 2 + z 2 1. (Hint: 1. Solution: The volume is 1. Problem 2A.
Problem 1A Find the volume of the solid given by x 2 + z 2 1, y 2 + z 2 1 (Hint: 1 1 (something)dz) Solution: The volume is 1 1 4xydz where x = y = 1 z 2 This integral has value 16/3 Problem 2A Let f(x)
arxiv:hep-th/ v1 29 Nov 2000
BRS-CHERN-SIMONS FORMS AND CYCLIC HOMOLOGY Denis PERROT 1 arxiv:hep-th/11267v1 29 Nov 2 Centre de Physique Théorique, CNRS-Luminy, Case 97, F-13288 Marseille cedex 9, France perrot@cpt.univ-mrs.fr Abstract
A Generalization of Wilson s Theorem
A Generalization of Wilson s Theorem R. Andrew Ohana June 3, 2009 Contents 1 Introduction 2 2 Background Algebra 2 2.1 Groups................................. 2 2.2 Rings.................................
ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009)
ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009) These are errata for the Third Edition of the book. Errata from previous editions have been fixed
MATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
l-adic Representations
l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
Basic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
Basic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
A Classical Introduction to Modern Number Theory
Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory Second Edition Springer Contents Preface to the Second Edition Preface v vii CHAPTER 1 Unique Factorization 1 1 Unique Factorization
arxiv: v1 [math.nt] 31 Dec 2011
![arxiv: v1 [math.nt] 31 Dec 2011](/thumbs/93/112099542.jpg "arxiv: v1 [math.nt] 31 Dec 2011")
arxiv:1201.0266v1 [math.nt] 31 Dec 2011 Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization Andrej Dujella and Filip Najman Abstract In this paper,
CSE 548: Analysis of Algorithms. Lecture 4 ( Divide-and-Conquer Algorithms: Polynomial Multiplication )
CSE 548: Analysis of Algorithms Lecture 4 ( Divide-and-Conquer Algorithms: Polynomial Multiplication ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Spring 2015 Coefficient Representation