Use of frobenius map of an elliptic curve












0












$begingroup$


I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) mod p$. Isn't it just an identity map by fermat's little theorem? What's the use of this map in elliptic curves?






group-theory elliptic-curves frobenius-groups







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
    $endgroup$
    – Slade
    Dec 1 '18 at 17:33












  • $begingroup$
    @Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
    $endgroup$
    – satya
    Dec 1 '18 at 17:37










  • $begingroup$
    I'd read Silverman to find out more about the Frobenius map.
    $endgroup$
    – Wuestenfux
    Dec 1 '18 at 18:03






  • 2




    $begingroup$
    Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
    $endgroup$
    – reuns
    Dec 1 '18 at 18:15


















0












$begingroup$


I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) mod p$. Isn't it just an identity map by fermat's little theorem? What's the use of this map in elliptic curves?






group-theory elliptic-curves frobenius-groups







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
    $endgroup$
    – Slade
    Dec 1 '18 at 17:33












  • $begingroup$
    @Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
    $endgroup$
    – satya
    Dec 1 '18 at 17:37










  • $begingroup$
    I'd read Silverman to find out more about the Frobenius map.
    $endgroup$
    – Wuestenfux
    Dec 1 '18 at 18:03






  • 2




    $begingroup$
    Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
    $endgroup$
    – reuns
    Dec 1 '18 at 18:15
















0












0








0





$begingroup$


I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) mod p$. Isn't it just an identity map by fermat's little theorem? What's the use of this map in elliptic curves?






group-theory elliptic-curves frobenius-groups







share|cite|improve this question









$endgroup$




I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) mod p$. Isn't it just an identity map by fermat's little theorem? What's the use of this map in elliptic curves?






group-theory elliptic-curves frobenius-groups




group-theory elliptic-curves frobenius-groups






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 1 '18 at 17:24









satyasatya

857




857








  • 1




    $begingroup$
    Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
    $endgroup$
    – Slade
    Dec 1 '18 at 17:33












  • $begingroup$
    @Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
    $endgroup$
    – satya
    Dec 1 '18 at 17:37










  • $begingroup$
    I'd read Silverman to find out more about the Frobenius map.
    $endgroup$
    – Wuestenfux
    Dec 1 '18 at 18:03






  • 2




    $begingroup$
    Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
    $endgroup$
    – reuns
    Dec 1 '18 at 18:15
















  • 1




    $begingroup$
    Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
    $endgroup$
    – Slade
    Dec 1 '18 at 17:33












  • $begingroup$
    @Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
    $endgroup$
    – satya
    Dec 1 '18 at 17:37










  • $begingroup$
    I'd read Silverman to find out more about the Frobenius map.
    $endgroup$
    – Wuestenfux
    Dec 1 '18 at 18:03






  • 2




    $begingroup$
    Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
    $endgroup$
    – reuns
    Dec 1 '18 at 18:15










1




1




$begingroup$
Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
$endgroup$
– Slade
Dec 1 '18 at 17:33






$begingroup$
Fermat's little theorem says that $x^p = x$ in the finite field $mathbb{F}_p = mathbb{Z}/p$. However, this is false for any nontrivial field extension of $mathbb{F}_p$. In this case, we are working in the algebraic closure of $mathbb{F}_p$.
$endgroup$
– Slade
Dec 1 '18 at 17:33














$begingroup$
@Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
$endgroup$
– satya
Dec 1 '18 at 17:37




$begingroup$
@Slade I am new to this area. May I know what is a field extension and algebraic closure of $F_p$?
$endgroup$
– satya
Dec 1 '18 at 17:37












$begingroup$
I'd read Silverman to find out more about the Frobenius map.
$endgroup$
– Wuestenfux
Dec 1 '18 at 18:03




$begingroup$
I'd read Silverman to find out more about the Frobenius map.
$endgroup$
– Wuestenfux
Dec 1 '18 at 18:03




2




2




$begingroup$
Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
$endgroup$
– reuns
Dec 1 '18 at 18:15






$begingroup$
Let $E : y^2 = x^3+x$ then $E(mathbb{F}_{5^k}) = { (x,y) in mathbb{F}_{5^k}, y^2= x^3+x} cup {O }$, there is a group law on it and the full curve is $bigcup_k E(mathbb{F}_{5^k}) = E(overline{mathbb{F}_{5}})$. You are supposed to know how to construct those finite fields with $5^k$ elements (and how the Frobenius act on them) and get some intuitions on the roots of "the polynomial map defining $P mapsto n P, P in E(overline{mathbb{F}_{5}})$"
$endgroup$
– reuns
Dec 1 '18 at 18:15












1 Answer
1






active

oldest

votes


















2












$begingroup$

In the notation of the notes you link to $E(K)$ stands for the points $(x,y)in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(mathbb{F}_p)$ the Frobenius map $(x,y)mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(overline{mathbb{F}_p})$.



So to understand it you first need to understand what $overline{mathbb{F}_p}$ is. It is an algebraic closure of $mathbb{F}_p$, i.e. an algebraic extension of $mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $mathbb{F}_p$.



In fact $mathbb{F}_p$ is the subset of elements in $overline{mathbb{F}_p}$ that are fixed by the map $xmapsto x^p$. Hence $E(mathbb{F}_p)$ is the subset of elements of $E(overline{mathbb{F}_p})$ that are fixed by the Frobenius map.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021589%2fuse-of-frobenius-map-of-an-elliptic-curve%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    In the notation of the notes you link to $E(K)$ stands for the points $(x,y)in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(mathbb{F}_p)$ the Frobenius map $(x,y)mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(overline{mathbb{F}_p})$.



    So to understand it you first need to understand what $overline{mathbb{F}_p}$ is. It is an algebraic closure of $mathbb{F}_p$, i.e. an algebraic extension of $mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $mathbb{F}_p$.



    In fact $mathbb{F}_p$ is the subset of elements in $overline{mathbb{F}_p}$ that are fixed by the map $xmapsto x^p$. Hence $E(mathbb{F}_p)$ is the subset of elements of $E(overline{mathbb{F}_p})$ that are fixed by the Frobenius map.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      In the notation of the notes you link to $E(K)$ stands for the points $(x,y)in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(mathbb{F}_p)$ the Frobenius map $(x,y)mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(overline{mathbb{F}_p})$.



      So to understand it you first need to understand what $overline{mathbb{F}_p}$ is. It is an algebraic closure of $mathbb{F}_p$, i.e. an algebraic extension of $mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $mathbb{F}_p$.



      In fact $mathbb{F}_p$ is the subset of elements in $overline{mathbb{F}_p}$ that are fixed by the map $xmapsto x^p$. Hence $E(mathbb{F}_p)$ is the subset of elements of $E(overline{mathbb{F}_p})$ that are fixed by the Frobenius map.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        In the notation of the notes you link to $E(K)$ stands for the points $(x,y)in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(mathbb{F}_p)$ the Frobenius map $(x,y)mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(overline{mathbb{F}_p})$.



        So to understand it you first need to understand what $overline{mathbb{F}_p}$ is. It is an algebraic closure of $mathbb{F}_p$, i.e. an algebraic extension of $mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $mathbb{F}_p$.



        In fact $mathbb{F}_p$ is the subset of elements in $overline{mathbb{F}_p}$ that are fixed by the map $xmapsto x^p$. Hence $E(mathbb{F}_p)$ is the subset of elements of $E(overline{mathbb{F}_p})$ that are fixed by the Frobenius map.






        share|cite|improve this answer









        $endgroup$



        In the notation of the notes you link to $E(K)$ stands for the points $(x,y)in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(mathbb{F}_p)$ the Frobenius map $(x,y)mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(overline{mathbb{F}_p})$.



        So to understand it you first need to understand what $overline{mathbb{F}_p}$ is. It is an algebraic closure of $mathbb{F}_p$, i.e. an algebraic extension of $mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $mathbb{F}_p$.



        In fact $mathbb{F}_p$ is the subset of elements in $overline{mathbb{F}_p}$ that are fixed by the map $xmapsto x^p$. Hence $E(mathbb{F}_p)$ is the subset of elements of $E(overline{mathbb{F}_p})$ that are fixed by the Frobenius map.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 1 '18 at 18:12









        MichalisNMichalisN

        4,5771227




        4,5771227






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021589%2fuse-of-frobenius-map-of-an-elliptic-curve%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Plaza Victoria

            Image
            Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
            Read more

            Puebla de Zaragoza

            Image
            Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
            Read more

            Musa

            Image
            Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
            Read more