sd23: Torsion Points on Elliptic Curves
system:sage


<h1 style="text-align: center;"><strong>Torsion Points on Elliptic Curves over Quartic Fields</strong></h1>
<h2 style="text-align: center;"><span style="color: #333333;">William Stein, Univ of Washington</span></h2>
<h3 style="text-align: center;"><span style="color: #333333;">(this is joint work with Sheldon Kamienny and Michael Stoll)</span></h3>
<h2 style="text-align: center;"><span style="color: #333333;">Sage Days 23, July 2010 at the&nbsp;</span></h2>
<p>&nbsp;</p>
<p style="text-align: center;"><img src="lorentz.png" alt="" width="250" /></p>

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<h1 style="text-align: center;">New Contribution</h1>
<p>We prove that there are no elliptic curves over a&nbsp;number field $K$ of degree 4 that have a $K$-rational point of prime order $&gt;17$.</p>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;">We complete the proof that there are no elliptic curves over a</div>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 0px; width: 1px; height: 1px; overflow-x: hidden; overflow-y: hidden;">number eld K of degree  4 that have a K-rational point of prime order &gt; 17.</div>
<p>&nbsp;</p>
<p>&nbsp;</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><strong>Corollary:</strong> $E(K)_{\rm tor}$ 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>"Theorem":</strong>&nbsp;$E(K)_{\rm tor}$&nbsp;is cyclic or a product of two cyclic groups</p>
<p>(Proof:&nbsp;$E(K)_{\rm tor}$ is a finite subgroup of $\CC/\Lambda$.)</p>
<p>&nbsp;</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> (Bepo Levi, 1908 [see Schoof-Schappacher 2004]; remade by Nagel in 1952; then remade by Ogg in 1960's):&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>
<ul>
<li>This is a conjecture about <strong>rational points on</strong> certain <strong>curves of</strong> (possibly) <strong>higher genus</strong>&nbsp;</li>
<li>Levi wrote <strong>three papers</strong> on this problem. &nbsp; &nbsp;He viewed it as classifying the ways in which the chord and tangent method can fail to generate infinitely many points on an elliptic curve. &nbsp; He proved some amazing results! &nbsp; <em>"...&nbsp;the group A = Z/nZ does not occur for n = 14, 16 and n = 20 and A =Z/nZ &times; Z/2Z does not occur for n = 10 and n = 12. In these cases he must study&nbsp;some thorny diophantine equations, defining plane curves of genus 1 or 2. He concludes&nbsp;by infinite descent, very much in the spirit of Fermat. In the jargon of the contemporary&nbsp;arithmetic of elliptic curves the infinite descent involves a 2-descent [Levi 1906&ndash;08, 17]."</em></li>
</ul>

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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>: If $Y_1(p)(K)$ is empty, then there is no elliptic curve $E/K$ with $p \mid \#E(K)_{\rm tor}$.</p>
<p>Also, $Y_0(N)$ is a quotient of $Y_1(N)$, so if $Y_0(N)(K)$ is empty, then so is $Y_1(N)(K)$.&nbsp;</p>

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<h2 style="text-align: center;">Mazur's Theorem (1970s)</h2>
<p><strong>Theorem </strong>(Mazur) We have $Y_1(p)(\QQ)=\emptyset$ for $p&gt;13$. &nbsp;Thus 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 (it turns out that there are infinitely many for each $j$-invariant):</p>

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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()))
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<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).&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;">good or additive reduction</span></em> at $3$, get contradiction by injecting the $p$-torsion mod $3$ since $p&gt;13$. Thus may assume $E$ has multiplicative reduction, and with some work may assume that $x$ reduces (mod 3) 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 contained in a formal group</span></em>, hence torsion free. &nbsp;But $A(\QQ)=A(\QQ)_{\rm tor}$ is finite, so image of $x$ in $A(\QQ)$ 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 $q$-descent, for primes $q$ that divide the numerator of $(p-1)/12$. &nbsp; This is long before much was known toward the Birch and Swinnerton-Dyer conjecture. &nbsp; Now, much more is known. &nbsp;More recently, one can:</p>
<ul>
<li><strong>Merel 1995</strong>: use the bigger <strong><em>winding quotient&nbsp;</em></strong>of $J_0(p)$, which is the <em>maximal&nbsp;analytic&nbsp;rank $0$</em> quotient. &nbsp;This makes the arguments above easier and more generalizable. &nbsp;We know by <em>Kolyvagin-Logachev et al.</em> or by <em>Kato</em> that the winding quotient has rank 0. &nbsp;(For $p=67$ the winding and Eisenstein quotients already differ, since 67a has trivial torsion and rank 0.)<br /><br /></li>
<li><strong>Parent 1999</strong>: use instead 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 (Kolyvagin-Logachev does not apply).</li>
</ul>

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J = J0(67); J
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Abelian variety J0(67) of dimension 5
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for A in J.decomposition():
    print A
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Simple abelian subvariety 67a(1,67) of dimension 1 of J0(67)
Simple abelian subvariety 67b(1,67) of dimension 2 of J0(67)
Simple abelian subvariety 67c(1,67) of dimension 2 of J0(67)
}}}

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E = EllipticCurve('67a1')
E.torsion_order()
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1
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E.rank()
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0
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E.isogeny_class()   # so all mod-p reps are surjective, so no Eisenstein
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([Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x - 21 over Rational Field], [1])
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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 $p \mid \#E(K)_{\rm tor}.$</p>
<p>Let $S(d) = \{ \text{torsion primes for degree } \leq d \}$. &nbsp;</p>
<p>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; (Not effective.)</p>
<p><strong>Idea of 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 linear independence of Hecke operators (or $q$-expansions of modular forms) 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>Abromovich</strong> soon proved that $S(d)$ is finite for $d\leq 14$.&nbsp;</p>
<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 <strong>quadratic fields</strong> 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)   for v = 1,2.
</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 in all degrees, 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>$\max(S(d)) \leq (3^{d/2}+1)^2$.</p>

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for d in [2..10]:
    print '%2s%10s    %s'%(d, floor((3^(d/2)+1)^2), d^(3*d^2))
///
 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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<p style="text-align: center;"><strong><span style="font-size: x-large;">Remark (Merel, personal communication, 2010-05-10)</span></strong></p>
<ol>
<li>The known bounds for $S(d)$ are exponential in $d$. &nbsp;However, <em>a polynomial bound on $S(d)$ in $d$ is expected</em>. Therefore, one can not expect to computationally determine the exact list of torsion primes in degree for many more $d$'s.&nbsp;</li>
<li>The bound is obtained by considering (essentially) two cases (according to the type of reduction modulo $\ell$ of your elliptic curve) : in one case it is easily seen to be exponential in $d$, the hard case finally yields a bound which is polynomial in $d$ (something like $O(d^8)$ in my paper, $O(d^6)$ after Oesterl&eacute;, I suspect one can lower it to $O(d^2)$). Unsatisfying!</li>
<li>If you want a bound depending on the field $K$ (instead of just the degree of $K$), you can obtain a bound like O(size of the residue field of $K$ of smallest order).</li>
</ol>
<p><strong>&nbsp;</strong></p>

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<h2 style="text-align: center;">Parent's Kamienny Method: Nailing Down S(3)</h2>
<p>By Oesterle, we know that $\max(S(3)) \leq 37$. &nbsp;</p>
<p>In 1999, <strong>Parent </strong>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 $17$ (using an involved argument with formal groups) 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, 2004] showed that the groups that appear for <strong>infinitely many $j$-invariants</strong> are:</p>
<pre>    Z/mZ           for m&lt;=16, 18, 20
    Z/2Z x Z/2vZ   for v&lt;=7
</pre>
<pre><span style="font-family: 'times new roman', times;"><strong>Remark</strong>: </span><span style="font-family: 'times new roman', times;">Parent</span><span style="font-family: 'times new roman', times;"> also gave an explicit bound for the torsion of order powers of prime numbers in his thesis...</span></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:</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 $S(4)$, I made Parent's (1999) approach very explicit in case $d=4$ and $\ell=2$ (he gives a general criterion for any $d$ and $\ell$, which he calls "Kamienny's Criterion"...). &nbsp;One arrives that the following (where $t$ is a certain explicitly computable element of the Hecke algebra). &nbsp; &nbsp;With $\ell=2, d=4$, we have $(1+\ell^{d/2})^2=25$.</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 <strong>several hours,</strong><strong><em><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;">This is of course a "theorem that relies on a big computer calculation" that involves substantial code. &nbsp;</span></span></em></p>

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<h1 style="text-align: center;"><span style="font-size: xx-large;">$p\leq 31$: A complete solution Stoll + Hess + [CES]</span></h1>
<p><span style="font-size: medium;">I talked about this problem in Vancouver recently. &nbsp;Michael Stoll was there, and afterwords we worked on the case of smaller $p$...</span></p>
<p><strong>Theorem (Kamienny, Stein, Stoll): $S(4) = \{2,3,5,7,11,13,17\}$</strong></p>
<p>Proof uses that ${\rm rank}(J_1(p))=0$ for $p\leq 31$ (and more!), (see [Conrad-Edixhoven-Stein] about the arithmetic of $J_1(p)$ for small $p$), so one can use much more direct geometric arguments. &nbsp; &nbsp;This also involves some large computations with Magma on explicit algebraic curves, e.g., Riemann-Roch spaces, enumerating and reducing divisors, etc., built on top of Florian Hess's function fields package. &nbsp;<strong>Stoll:</strong> "Finding the degree 4 points takes about 3&nbsp;hours [...] &nbsp;The other problem is that Magma crashes once in a while when turning a point&nbsp;into a place. This will be fixed in the next release, but for now, one may&nbsp;have to try the actual checking a few times until it runs through." &nbsp;(2010-07-06: Stoll just retried the computation in the absolutely latest version of Magma and the crashes are now fixed.)</p>
<p><strong>Conjecture (Stein):</strong> $J_1(p)(\QQ)_{\rm tor}$ is generated by differences of rational cusps.</p>
<p>See extensive data about this conjecture in Conrad-Edixhoven-Stein. &nbsp;Proof of this for certain $p$ is a key input to the above theorem.&nbsp;</p>
<p><strong>Function fields project in Sage:</strong>&nbsp;I really want to implement enough of Hess's algorithm(s) so that the above computation can be done entirely with Sage. &nbsp;See <a href="http://trac.sagemath.org/sage_trac/ticket/9054" target="_blank">trac 9054</a>. This is challenging, since nobody has every implemented Hess's algorithm, except Hess.&nbsp;</p>
<p>&nbsp;</p>
<p style="text-align: center;"><img src="j1p.png" alt="" width="450" /></p>

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<h2 style="text-align: center;">Future Work</h2>
<ol>
<li>Implement Hess's algorithm, to be able to deal with $p\leq 31$ using Sage.</li>
<li>Determine if $J_1(p)(\QQ)_{\rm tor}$ is cuspidal for $p=29$. (I proved this for $p&lt;29$ and $p=31$.)</li>
<li>Make the algorithm for showing that $\max(S(4)) \leq 31$ much more efficient, then repeat my calculations, but for<strong> $d=5$ </strong>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>
<li><strong>Pie in the sky -- Isogeny degrees?</strong> -- still an open problem even over <em>quadratic fields</em>! &nbsp;              
<ul>
<li><strong>Cremona</strong>: "I'm also very interested in the corresponding question for $X_0(\ell)$, so we know what the possible prime degrees of isogenies are for a given field (or degree). I had some interesting correspondence about this with Parent about 6 months ago; <em>he says that is still wide open for quadratic fields</em>! &nbsp; &nbsp;My student Kimi is implementing isogenies of degree 11, 17, 19 (the genus 1 cases) in Sage (work in progress). But to have a genuine isogeny_class() function over any non-Q number fields we need a bound." &nbsp;&nbsp;</li>
<li><strong>Mazur </strong>(email): "It would be also&nbsp;interesting if you could, say, rule out a few &nbsp;primes $p$ occurring as&nbsp;$p$-isogenies over such fields (for non CM curves)?"</li>
</ul>
</li>
</ol>

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float((1+2^(5/2))^2)
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44.313708498984766
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271
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