Torsion Points on Elliptic Curves
system:sage


<h1 style="text-align: center;">Torsion Points on Elliptic Curves over Quartic Fields</h1>
<h2 style="text-align: center;">William Stein</h2>
<h3 style="text-align: center;">(this is joint work with Sheldon Kamienny)</h3>
<h2 style="text-align: center;">University of Washington</h2>
<h2 style="text-align: center;">May 2010</h2>
<p>&nbsp;</p>
<p style="text-align: center;"><img src="uw.png" alt="" width="290" height="210" /></p>

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<h2 style="text-align: center;">Motivating Problem</h2>
<p>Let $K$&nbsp;be a number field. &nbsp;</p>
<p><strong>Theorem</strong> (Mordell-Weil): If $E$ is an elliptic curve over $K$, then $E(K)$ is a finitely generated abelian group.</p>
<p>Thus $E(K)_{\rm tor}$ is a finite group.&nbsp;</p>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">PROBLEMLet K be a number field. &nbsp;Which finite abelian groups</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">E(K)_{tor} occur, as we vary over all elliptic curves E/K?</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">There are a *LOT* of papers on this problem.</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">OBSERVATION: E(K)_{tor} is a finite subgroup of Q^2/Z^2, so E(K)_{tor}</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">is cyclic or a product of two cyclic groups.Theo</div>
<p><strong>Problem: </strong>&nbsp;Which finite abelian groups $E(K)_{\rm tor}$&nbsp;occur, as we vary over all elliptic curves $E/K$?</p>
<p style="text-align: center;">&nbsp;</p>
<p><strong>Observation:</strong>&nbsp;$E(K)_{\rm tor}$ is a finite subgroup of $\CC/\Lambda$, so $E(K)_{\rm tor}$&nbsp;is cyclic or a product of two cyclic groups.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>

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<h2 style="text-align: center;">An Old Conjecture</h2>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">CONJECTURE (LEVI around 1908; OGG in 1960s):&nbsp;</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">&nbsp;&nbsp;When K=Q, the groups E(Q)_{tor} are the 15 groups:</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">&nbsp;&nbsp; &nbsp;Z/mZ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for m&lt;=10 or m=12</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow: hidden;">
<p>&nbsp;&nbsp; (Z/2Z) x (Z/2vZ) &nbsp; &nbsp;for v&lt;=4.</p>
<p>&nbsp;</p>
</div>
<p>&nbsp;</p>
<p><strong>Conjecture</strong> (Levi around 1908; re-made by Ogg in 1960s):&nbsp;</p>
<p>&nbsp;&nbsp;When $K=\QQ$, the groups $E(\QQ)_{\rm tor}$, as we vary over all $E/\QQ$, are the following 15 groups:</p>
<p>&nbsp;&nbsp; &nbsp;$\ZZ/m\ZZ$ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;for $m\leq 10$ or $m=12$</p>
<p>&nbsp;&nbsp; &nbsp;$(\ZZ/2\ZZ) \times (\ZZ/2v\ZZ)$ &nbsp; &nbsp;for $v\leq 4$.</p>
<p>&nbsp;</p>
<p><strong>Note:</strong></p>
<ol>
<li>This is really a conjecture about <strong>rational points on</strong> certain <strong>curves of</strong> (possibly) <strong>higher genus</strong> (title of Michael Stoll's talk today)...</li>
<li>Or, it's a conjecture in <strong>arithmetic dynamics</strong> about <strong>periodic points</strong>.</li>
</ol>
<p>&nbsp;</p>

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<h2 style="text-align: center;">Modular Curves</h2>
<p>The modular curves $Y_0(N)$ and $Y_1(N)$:</p>
<ul>
<li>Let $Y_0(N)$ be the affine&nbsp;<strong>modular curve</strong>&nbsp;over $\QQ$ whose points parameterize isomorphism classes of pairs $(E,C)$, where $C \subset E$ is a <em>cyclic subgroup</em> of order $N$.</li>
<li>Let $Y_1(N)$ be ... &nbsp;of pairs $(E,P)$, where $P\in E(\overline{\QQ})$ is a <em>point</em> of order $N$.</li>
</ul>
<p>Let $X_0(N)$ and $X_1(N)$ be the compactifications of the above affine curves.</p>
<p><strong>Observation</strong>: There is an elliptic curve $E/K$ with $p \mid \#E(K)$ if and only if $Y_1(p)(K)$ is nonempty.</p>
<p>Also, $Y_0(N)$ is a quotient of $Y_1(N)$, so if $Y_0(N)(K)$ is empty, then so is $Y_0(N)$.&nbsp;</p>

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<h2 style="text-align: center;">Mazur's Theorem (1970s)</h2>
<p><strong>Theorem </strong>(Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p\leq 13$.</p>
<p>Combined with previous work of Kubert and Ogg, one sees that Mazur's theorem implies Levi's conjecture, i.e., a complete classification of the finite groups $E(\QQ)_{\rm tor}$.</p>
<p>Here are representative curves by the way (there are infinitely many for each $j$-invariant):</p>

{{{id=14|
for ainvs in ([0,-2],[0,8],[0,4],[4,0],[0,-1,-1,0,0],[0,1],
        [1, -1, 1, -3, 3],[7,0,0,16,0], [1,-1,1,-14,29],
        [1,0,0,-45,81], [1, -1, 1, -122, 1721], [-4,0],
        [1,-5,-5,0,0], [5,-3,-6,0,0], [17,-60,-120,0,0]  ):
    E = EllipticCurve(ainvs)
    view((E.torsion_subgroup().invariants(), E))
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[\right], y^2 = x^3 - 2 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[2\right], y^2 = x^3 + 8 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[3\right], y^2 = x^3 + 4 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[4\right], y^2 = x^3 + 4x \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[5\right], y^2 - y = x^3 - x^2 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[6\right], y^2 = x^3 + 1 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[7\right], y^2 + xy + y = x^3 - x^2 - 3x + 3 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[8\right], y^2 + 7xy = x^3 + 16x \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[9\right], y^2 + xy + y = x^3 - x^2 - 14x + 29 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[10\right], y^2 + xy = x^3 - 45x + 81 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[12\right], y^2 + xy + y = x^3 - x^2 - 122x + 1721 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[2, 2\right], y^2 = x^3 - 4x \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[4, 2\right], y^2 + xy - 5y = x^3 - 5x^2 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[6, 2\right], y^2 + 5xy - 6y = x^3 - 3x^2 \right)</span></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left[8, 2\right], y^2 + 17xy - 120y = x^3 - 60x^2 \right)</span></html>
}}}

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<h2 style="text-align: center;">Mazur's Method</h2>
<p><strong>Theorem&nbsp;</strong>(Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p\leq 13$.</p>
<p>Basic idea of the proof: &nbsp;</p>
<ol>
<li>Find a <em><span style="text-decoration: underline;">rank zero quotient</span></em> $A$ of $J_0(p)$ such that...</li>
<li>... the induced map $f:X_0(p) \to A$ is a <span style="text-decoration: underline;">f</span><em><span style="text-decoration: underline;">ormal immersion</span></em> at infinity (this means that the induced map on complete local rings is surjective, or equivalently, that the induced map on cotangent spaces is surjective).&nbsp;</li>
<li>Then consider the <span style="text-decoration: underline;"><em>point</em></span> $x \in Y_0(p)$ corresponding to a pair $(E,\langle P \rangle)$, where $P$ has order $p$. &nbsp;</li>
<li>If $E$ has <em><span style="text-decoration: underline;">potentially good reduction</span></em> at $3$, get contradiction by injecting $p$-torsion mod $3$ since $p&gt;13$, so $E$ has multiplicative reduction, hence we may assume $x$ reduces to the cusp $\infty$.&nbsp;</li>
<li>The image of $x$ in $A(\QQ)$ is thus in the kernel of the reduction map mod $3$. &nbsp; &nbsp; But this <em><span style="text-decoration: underline;">kernel of reduction is a formal group</span></em>, hence torsion free. &nbsp;But $A(\QQ)=A(\QQ)_{\rm tor}$ is finite, so image of $x$ is 0.&nbsp;</li>
<li><em><span style="text-decoration: underline;">Use that $f$ is a formal immersion</span></em> at infinity along with step 5, to show that $x=\infty$, which is a contradiction since $x\in Y_0(p).$</li>
</ol> 
<ul>
</ul>

<p>Mazur uses for $A$ the&nbsp;<em>Eisenstein quotient</em>&nbsp;of $J_0(p)$ because he is able to prove -- way back in the 1970s! -- that this quotient has rank $0$ by doing a $p$-descent. &nbsp; This is long before much was known toward the BSD conjecture. &nbsp;More recently one can:</p>
<ul>
<li><strong>Merel 1995</strong>: use the&nbsp;<strong><em>winding quotient&nbsp;</em></strong>of $J_0(p)$, which is the maximal&nbsp;<em>analytic&nbsp;</em>rank $0$ quotient. &nbsp;This makes the arguments easier, and we know by Kolyvagin-Logachev et al. or by Kato that the winding quotient has rank 0.<br /><br /></li>
<li><strong>Parent 1999</strong>: use the winding quotient of $J_1(p)$, which leads to a similar argument as above. &nbsp;This quotient has rank 0 by Kato's theorem. &nbsp;</li>
</ul>

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<h2 style="text-align: center;">Kamienny-Mazur</h2>
<p>A prime $p$ is a <strong>torsion prime for degree $d$</strong> if there is a number field $K$ of degree $d$ and an elliptic curve $E/K$ such that $p \mid \#E(K)_{\rm tor}$.&nbsp;</p>
<p>Let $S(d) = \{ \text{torsion primes for degree } \leq d \}$. &nbsp;For example, $S(1) = \{2,3,5,7\}$.&nbsp;</p>
<p>Finding all possible torsion structure over all fields of degree $\leq d$ <em>often involves </em>determining $S(d)$<em>,</em> then doing some additional work (which we won't go into). &nbsp;E.g.,</p>
<p><strong>Theorem </strong>(Frey, Faltings): If $S(d)$ is finite, then the set of groups $E(K)_{\rm tor}$, as $E$ varies over all elliptic curves over all number fields $K$ of degree $\leq d$, is finite.&nbsp;</p>
<p><strong>Kamienny and Mazur: </strong>Replace&nbsp;$X_0(p)$ by the <em>symmetric power</em><strong>&nbsp;</strong>$X_0(p)^{(d)}$ and gave an explicit criterion in terms of independence of Hecke operators for $f_d: X_0(p)^{(d)} \to J_0(p)$ to be a formal immersion at $(\infty, \infty,\ldots,\infty)$. &nbsp; A point $y\in X_0(p)(K)$, where $K$ has degree $d$, then defines a point $\tilde{y} \in X_0(p)^{(d)}(\QQ)$, etc.</p>
<p><strong>Theorem (Kamienny and Mazur):</strong></p>
<ul>
<li><span>$S(2) = \{2,3,5,7,11,13\}$,</span></li>
<li><span>$S(d)$ is finite for $d\leq 8$,</span></li>
<li><span>$S(d)$ has density 0 for all $d$.</span></li>
</ul>
<p><strong>Corollary (Uniform Boundedness): </strong>There is a fixed constant $B$ such that if $E/K$ is an elliptic curve over a number field of degree $\leq 8$, then $\# E(K)_{\rm tor} \leq B$.</p>
<p>(Very surprising!)</p>

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<h2 style="text-align: center;">Torsion Structures over Quadratic Fields</h2>
<p><strong>Theorem</strong> (Kenku, Momose,&nbsp;Kamienny, Mazur): The complete list of subgroups that appear over quadratic fields is:</p>
<pre>            Z/mZ            for m&lt;=16 or m=18
           (Z/2Z) x (Z/2vZ) for v&lt;=6.
           (Z/3Z) x (Z/3vZ) for v=1,2
           (Z/4Z) x (Z/4vZ)
</pre>
<p>and each occurs for infinitely many $j$-invariants.</p>

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<h2 style="text-align: center;">What is $S(d)$?</h2>
<p>Kamienny, Mazur: "We expect that $max(S(3)) \leq 19$, but it would simply be too embarrassing to parade the actual astronomical finite bound that our proof gives."</p>
<p>But soon, Merel in a <em>tour de force,</em>&nbsp;proves (by using the winding quotient and a deep modular symbols argument about independence of Hecke operators):</p>
<p><strong>Theorem (Merel, 1996): &nbsp;</strong>$\max(S(d)) &lt; d^{3 d^2}$, for $d\geq 2$.</p>
<p>thus proving the full Universal Boundedness Conjecture, which is a huge result.</p>
<p>Shortly thereafter Oesterle modifies Merel's argument to get a much better upper bound:</p>
<p><strong>Theorem (Oesterle): </strong>$\max(S(d)) &lt; (3^{d/2}+1)^2$.</p>

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for d in [1..10]:
    print '%2s%10s    %s'%(d, floor((3^(d/2)+1)^2), d^(3*d^2))
///
 1         7    1
 2        16    4096
 3        38    7625597484987
 4       100    79228162514264337593543950336
 5       275    26469779601696885595885078146238811314105987548828125
 6       784    1097324413128695095014498519762948444299315170409742569521688363865669310779664367616
 7      2281    16959454617563682698054005840792102521632243876732771232150341713141856731878591823809299439924812705151100914349041188035543
 8      6724    247330401473104534060502521019647190035131349101211839914063056092897225106531867170316401061243044989597671426016139339351365034306751209967546155101893167916606772148699136
 9     19964    7602033756829688179535612101927342434798006222913345882096671718462026450847558385638399133044640009857513126790996106341658482736771462692522663416083613709397190583473914100243037919870652143046001421207236044960360057945209303129
10     59536    1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
}}}

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<h2 style="text-align: center;">Parent's Method: Nailing Down S(3)</h2>
<p>By Oesterle, we know that $\max(S(3)) \leq 37$. &nbsp;</p>
<p>In 1999, Parent made Kamienny's method applied to $J_1(p)$ explicit and computable, and used this to bound $S(3)$ explicitly, showing that $\max(S(3)) \leq 17$. &nbsp; This makes crucial use of Kato's theorem toward the Birch and Swinnerton-Dyer conjecture! &nbsp;</p>
<p>In subsequent work, Parent rules out $17$ finally giving the answer:</p>
<p>$$ &nbsp;S(3) = \{2,3,5,7,11,13\} &nbsp;$$</p>
<p>The list of groups $E(K)_{\rm tor}$ that occur for $K$ cubic is still<em> unknown</em>. &nbsp;However, using the notion of <em>trigonality</em> of modular curves (having a degree 3 map to $P^1$), Jeon, Kim, and Schweizer showed that the groups that appear for infinitely many $j$-invariants are:</p>
<pre>    Z/mZ           for m&lt;=16, 18, 20
    Z/2Z x Z/2vZ   for v&lt;=7
</pre>

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<h2 style="text-align: center;">What about Degree 4?</h2>
<p>By Oesterle, we know that $\max(S(4)) \leq 97$.</p>
<p>Recently, Jeon, Kim, and Park (2006), again used gonality (and big computations with Singular), to show that the groups that appear for infinitely many $j$-invariants for curves over quartic fields are:</p>
<pre>    Z/mZ           for m&lt;=18, or m=20, m=21, m=22, m=24
    Z/2Z x Z/2vZ   for v&lt;=9
    Z/3Z x Z/3vZ   for v&lt;=3
    Z/4Z x Z/4vZ   for v&lt;=2
    Z/5Z x Z/5Z 
    Z/6Z x Z/6Z</pre>
<p><strong>Question (Kamienny to me):</strong> Is $S(4) = \{2,3,5,7,11,13,17\}?$</p>

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<h2 style="text-align: center;">Explicit Kamienny-Parent for $d=4$</h2>
<p>To attack the above unsolved problem about $S(4)$, we made Parent's (1999) approach very explicit in case $d=4$ and $\ell=2$ (he gives a general criterion for any $d$...). &nbsp;One arrives that the following (where $t$ is a certain explicitly computed element of the Hecke algebra):</p>
<p><img src="parent4.png" alt="" width="650" /></p>
<p>NOTES:</p>
<ol>
<li>This looks pretty crazy, but this is <em>really just a way of expressing the condition that a certain map is a formal immersion</em>.&nbsp;</li>
<li>As $p$ gets large, there are a <strong>LOT</strong> of 4-tuples of elements of the Hecke algebra to test for independence mod 2.</li>
<li>Here is code that implements this algorithm: <a href="code.sage" target="_blank">code.sage</a></li>
</ol>

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<h2 style="text-align: center;">Running the Algorithm</h2>
<p>After a few <strong><em>days<span style="font-weight: normal;">&nbsp;<span style="font-style: normal;">we find that the criterion is <strong>not satisfied</strong>&nbsp;for $p=29,31$, but it is for $37\leq p \leq 97$.&nbsp;</span></span></em></strong></p>
<p><strong><em><span style="font-weight: normal;"><span style="font-style: normal;">Conclusion:</span></span></em></strong></p>
<p><em><span style="font-weight: normal;"><span style="font-style: normal;"><strong>Theorem (Kamienny, Stein): </strong>&nbsp;$\max(S(4)) \leq 31$.&nbsp;</span></span></em></p>
<p><em><span style="font-weight: normal;"><span style="font-style: normal;">It's unclear to me, but Kamienny seems to also have a proof that rules out $29,31$, which would nearly answer the big question for degree $4$.&nbsp;</span></span></em></p>

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<h2 style="text-align: center;">Future Work</h2>
<ol>
<li>Kamienny (unpublished): "Moreover&nbsp;<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial;"><span style="background-color: #ffffff;">29</span></span>, 31, 41 , and 59 can't occur over any quartic field... &nbsp;I've known an easy geometric proof for a long time, but I simply forgot about it..." &nbsp;</li>
<li>Kamienny (unpublished): "For 19 and 23 we only get the result for fields in which at least one of 2, 3 doesn't remain prime. &nbsp;We can try dealing with 19 and 23 by looking (later) at equations for the modular curves if that's computable."</li>
<li>Alternatively, deal with 19 and 23 in a way similar to how Parent dealt with $p=17$ for $d=3$, which was the one case he couldn't address using his criterion.&nbsp; (His paper on $p=17$ looks very painful though!)</li>
<li>Make the algorithm for showing that $\max(S(4)) \leq 31$ more efficient. &nbsp;Right now it takes way too long.</li>
<li>Given 3, repeat my calculations, but for $d=5$ and hope to replace the Oesterle bound of $\max(S(5)) \leq 271$ by $$\max(S(5)) \leq 43 \quad\text{ &nbsp;(or something close)}$$</li>
</ol>

{{{id=35|
float((1+2^(5/2))^2)
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44.313708498984766
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{{{id=2|
previous_prime(275)
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271
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