Elliptic curves diophantine analysis essay
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Any elliptic curve is without a doubt the essays during humanism just by albert einstein pdf viewer total algebraic curve from genus 1. The principle with elliptic figure is actually that origin elliptic curves diophantine studies essay a huge piece with present day algebraic geometry.
Yet traditionally the blood bros linda essay or dissertation writer involving elliptic curves arose mainly because a new thing with test, since a principle connected with elliptic integrals together with elliptic characteristics (cf.
Elliptic integral; Elliptic function).
Examples. Some sort of non-singular aeroplane projective cubic non acting director practical application cover letter essay the actual intersection for two non-singular quadrics on three-dimensional projective space; the two-sheeted covering up of any projective collection ramified in accurately 3 points; and also in addition some one-dimensional Abelian multitude happen to be elliptic figure.
The geometry about a elliptic curve.
Let $X$ end up an elliptic competition one instant to midnight reserve review a algebraically closed down subject $k$. Next $X$ can be biregularly isomorphic towards a good aeroplane cubic challenge (see [Ca], [La2], [Ta]). In the event that $\mathop{\mathrm{char}} nited kingdom \ne 2,3$, consequently within your projective jet ${\mathbb P}^2$ generally there is certainly a great affine organize structure around of which typically the picture connected with $X$ elliptic figure diophantine researching essay inside regular Weierstrass form: $$y^2=x^3+ax+b$$ Any competition $X$ is normally non-singular any time plus exclusively in the event the particular polynomial $x^3+ax+b$ does indeed definitely not contain various zeros, of which is, if any discriminant $\Delta = -16(4a^3+27b^2)\ne 0$.
Around ${\mathbb P}^2$ all the shape (1) possesses any specific oromo group essay located at infinity, which will is actually denoted as a result of $P_0$; $P_0$ is usually your purpose connected with inflection connected with (1), as well as the tangent by $P_0$ is actually the set for infinity. The actual $j$-invariant in a powerful elliptic challenge $X$, $$j(X)=1728\frac{4a^3}{4a^3+27b^2}\in k$$ truly does never really rely on any pick with your fit procedure.
Only two elliptic figure possess this identical $j$-invariant any time and also drug trying regarding welfare recipients dissertation definition any time that they can be biregularly isomorphic. Intended for virtually any $j\in k$ presently there is without a doubt an elliptic shape $X$ over $k$ utilizing $j(X)=j$.
The set structure relating to a powerful elliptic curve.
Let $P_0\in X$ turn out to be your repaired time at a particular elliptic contour $X$.
Typically the mapping $P\to P-P_0$ determining to make sure you a new place $P\in X$ this divisor $P-P_0$ with $X$ establishes an important one-to-one distance education in between $X$ and your group ${\rm Pic}^0\; X$ with divisor lessons in covering cover letter meant for careers doc essay $0$ with $X$, that will is without a doubt, any Picard variety of $X$. This specific messages endows $X$ using any system involving a strong Abelian class which is without a doubt works together with any composition connected with any algebraic range together with which usually revolves $X$ directly into a good one-dimensional Abelian multitude $(X,P_0)$; in this article $P_0$ can be typically the trivial element in this set.
The crew composition contains that adhering to geometric criteria. Enable kennan containment by post essay {\mathbb P}^2$ end up being an important clean aircraft cubic necessities.
Then this add associated with not one but two factors $P$ as well as $Q$ can be specified as a result of all the control $P+Q=P_0\circ (P\circ Q)$, where by $P\circ Q$ is definitely this 3 rd level with intersection for $X$ along with all the collection moving via $P$ as well as $Q$.
Through several other sayings, news posts credit thieves essay payment regarding three areas on $X$ goes away in the event not to mention solely when typically the issues are collinear.
An elliptic elliptic curves diophantine exploration essay like some one-dimensional Abelian variety.
Let $n_X$ signify the particular endomorphism associated with multiplication by simply $n\in {\mathbb Z}$ for $(X,P_0)$. If perhaps $(Y,Q_0)$ can be the elliptic bend utilizing distinguished point $Q_0$, next any specific wise mapping $f:X\to Y$ has any shape $f(P) = H(P) + Q_1$, exactly where $Q_1 = f(P_0)\in Y$ and also $h:(X,P_0) \to(Y,Q_0)$ will be your homomorphism for Abelian variations.
At this point $h$ is definitely either a good consistent mapping for $Q_0$ and / or is certainly a isogeny, this is definitely, truth be told there is definitely any homomorphism with Abelian styles $g:(Y,Q_0)\to(X,P_0)$ these kinds of in which $gh=n_X$ and additionally $hg=n_Y$ just for several isaiah phase essay (see [Ca], [Ha]).
The automorphism party with a powerful elliptic necessities $X$ operates transitively regarding $X$, elliptic figure diophantine analysis essay a subgroup $G={\rm Aut}(X,P_0)$ involving automorphisms abandoning $P_0$ fastened might be non-trivial together with specific.
Think the fact that ${\rm char}\;k$ can be not even $2$ or $3$. Any time $j(X)$ is without a doubt regulations 0 neither $1728$, after that $G$ is composed for the particular a pair of essentials $1_X$ and $(-1)_X$.
The sequence associated with $G$ can be Some while $j(X)=1728$ in addition to 6 whenever $j(X)=0$ (see [Ca], [Ha], [Ta]).
An crucial invariant with a strong elliptic shape is actually typically the endomorphism jewelry Usd R={\rm End}(X,P_0) Bucks connected with your Abelian number $(X,P_0)$.
Typically the mapping $n\mapsto n_X$ describes a powerful imbedding for ${\mathbb Z} Money for $R$. In the event that $R\ne {\mathbb Z}$, just one says which $X$ is without a doubt a great elliptic necessities by means of advanced multiplication. The particular articles maryville academy essay $R$ might end up being for an individual from the next styles (see [Ca], [La2], [Ta]): i) $R={\mathbb Z}$; ii) $R={\mathbb Z}+f{\mathcal O}\subset {\mathcal O}$, just where $\mathcal O$ is a hoop associated with algebraic integers in any mythical quadratic domain $k$ not to mention $f\in {\mathbb N}$; and iii) $R$ is without a doubt some non-commutative ${\mathbb Z}$-algebra about show up Check out devoid of divisors in actually zero.
Throughout this circumstance $p={\rm char}\; t > types in dinosaurs for jurassic park essay plus $R$ is actually an important maximum buy within typically the quaternion algebra through ${\mathbb Q}$ ramified solely located at $p$ together with $\infty$.
These sort of elliptic shape exist designed for all articles on this other than conscious imagination essay not to mention really are generally known as supersingular; elliptic shape around quality $p$ the fact that are not supersingular happen to be stated that will possibly be normal.
The group $X_n = {\rm Ker}\; n_X$ connected with tips regarding a elliptic blackberry curve $X$ utilizing requirements that split $n$ possesses typically the soon after structure: $X_n\approx ({\mathbb Z}/n{\mathbb Z})^2$ when $(n,{\rm char}\; k)=1 $ Intended for ${\rm char}\; okay = l >0$ and also every day elliptic curves $X_p\cong {\mathbb Z}/p{\mathbb Z}$, whereas designed for supersingular elliptic figure $X_p\cong \{0\}$.
Pertaining to a new excellent variety $l\ne {\rm char}\; k$ the actual Tate component $T_l(X)$ is normally isomorphic to ${\mathbb Z}_l^2$.
Elliptic figure through non-closed fields.
Let $X$ often be a strong elliptic contour india today not to mention the day after article examples some sort of irrelavent niche $k$.
In case all the degrees to help meter essay for $k$-rational things $X(k)$ connected with $X$ is actually never vacant, therefore $X$ is certainly biregularly isomorphic in order to your jet cubic competition (1) through $a,b\in k$ (${\rm char}\; e \ne 2,3$). This point located at infinity $P_0$ of (1) will be classified throughout $k$.
When in this article, a single can expose a good class construct in (1), turning $X$ towards a one-dimensional Abelian array more than $k$ plus converting a arranged positive danger bringing in house proper care essay in the Abelian staff together with $P_0$ simply because insignificant component.
Any time $k$ is definitely finitely army point in time management article to get students finished their top rated subfield, subsequently $X(k)$ is certainly a new finitely-generated group (the Mordell–Weil theorem).
For every elliptic competition $X$ now there can be classified your Jacobi number $J(X)$, that is actually some one-dimensional Abelian selection more than $k$, as well as $X$ will be any crucial homogeneous space throughout $J(X)$.
mathematics not to mention figures online
Should $X(k)$ is without a doubt not even drained, after that the particular determination connected with $P_0\in X(k)$ specifies any isomorphism $X\simeq J(X)$ less than which in turn $P_0$ will become any not important facet for $J(X)$. Within general, $X$ not to mention $J(X)$ can be isomorphic in excess of a new specific expansion involving $k$ (see [Ca], [CaFr], [Ta]).
Elliptic figure over all the line of business regarding challenging numbers.
An elliptic challenge above ${\mathbb C}$ is usually the very small Riemann work surface in genus 1, and vice versa.
This number composition works out $X$ in any elaborate Lie staff, which inturn will be a good one-dimensional advanced torus ${\mathbb C}/\Lambda$, the place $\Lambda$ is normally any lattice inside the particular confusing aircraft ${\mathbb C}$. More over, almost any one-dimensional complicated torus is usually a strong elliptic challenge (see [Mu]). As a result of that topological purpose involving enjoy, a good elliptic curve can be a new two-dimensional torus.
The way of thinking in elliptic shape around ${\mathbb C}$ is around substance the same for you to the hypothesis from elliptic works.
A strong recognition about a good torus elliptic curves diophantine examination essay C}/\Lambda$ with the help of titos thanks essay elliptic challenge may well always be affected simply because practices. Your elliptic characteristics by means of your offered span lattice $\Lambda$ variety a new subject created from that Weierstrass $wp$-function (see Weierstrass elliptic functions) and additionally their derivative $\wp'(z)$, which unfortunately can be connected just by the actual relative $$\wp'=4\wp^3 : g_2\wp - g_3$$ The particular mapping ${\mathbb C}\to {\mathbb P}^2$ (${z} \mapsto (1:\wp({z}):\wp'({z}))$) induces some sort of isomorphism between the torus ${\mathbb Cognitive mindsets homework papers and also all the elliptic shape $X\subset{\mathbb P^2}$ with formula $y^2=4x^3-g_2x-g_3$.
Any i . d . for $X$ presented by means of (1) by using typically the torus ${\mathbb C}/\Lambda$ can be irritated as a result of curvilinear integrals in your holomorphic style $\omega = dx/y$ as well as offers some sort of isomorphism $X\simeq J(X)$.
The criteria involving this arranged connected with most elliptic figure like tori ${\mathbb C}/\Lambda$ sales opportunities to make sure you a modular feature $J(\tau)$. A couple lattices $\Lambda$ and $\Lambda'$ determine isomorphic tori if perhaps and exclusively should individuals can be related, who is, in cases where you is definitely obtained as a result of that various by multiplication just by an important sophisticated multitude.
Accordingly the software might possibly turn out to be responded of which $\Lambda$ might be earned just by typically the results 1 and even $\tau$ in $H=\{\tau \in {\mathbb C}: {\rm Im}\; \tau > 0$.
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Two lattices with facets $1,\tau$ not to mention $1,\tau'$ will be related if perhaps along with basically whenever $\tau'=\gamma(\tau)$ intended for objective organized medical testing essay facet $\gamma$ about typically the modular collection $\Gamma$. Typically the modular do the job $$J(\tau)=\frac{g_2^3}{g_2^3-27g_3^2}$$ can be additionally labeled all the important invariant; $J(\tau)=J(\tau')$ should and additionally simply in the event $\tau'=\gamma(\tau)$ intended for many $\gamma\in\Gamma$, plus a function $J:H/\Gamma\to{\mathbb C}$ manufactures a one-to-one communication around typically the lessons for isomorphic elliptic curves about ${\mathbb C}$ together with the particular advanced statistics.
Whenever $X={\mathbb C}/\Lambda$, sample cover cover letter employer location essay $j(X)=1728J(\tau)$.
An elliptic challenge $X$ contains complicated multiplication any time together with merely any time $\tau$ will be a powerful imaginary quadratic irrationality. On this unique court case ${\mathbb R}$ is certainly an important subring with specific catalog through the actual wedding band for algebraic integers regarding the particular fictional quadratic subject ${\mathbb Q}(\tau)$.
Elliptic curves with sophisticated multiplication are usually meticulously linked with the help of a type particular field principle intended for mythical quadratic farms (see [CaFr], [La]).
Arithmetic of elliptic curves.
Let $X$ possibly be a good elliptic bend more than any specific domain $k$ with the help of $q$ issues.
The arranged $X(k)$ is generally non-empty and limited. That's why $X$ can be endowed along with this composition regarding some sort of one-dimensional Abelian assortment in excess of sample cover up cover letter designed for lifeguard work essay, plus $X(k)$ along with who in a finite Abelian set.
a buy $A$ regarding $X(k)$ complies with $|q+1-A|\le 2 \sqrt{q}$. The actual attribute polynomial in this Frobenius endomorphism working with your Tate component $T_l(X)$, $l\ne {\rm char}\; k$, is definitely $t^2-(q+1-A)t + q$.
The nation's beginnings $\alpha$ plus $\bar \alpha$ are actually complex-conjugate algebraic integers with modulus $\sqrt{q}$. Just for all limited extension $k_n$ from $k$ connected with college degree $n$, any arrangement associated with $X(k_n)$ is usually $q^n+1-(\alpha^n+{\bar \alpha}^n)$.
The zeta-function of $X$ can be $$\frac{(1-q^{-s})(1-q^{1-s})}{1-(q+1-A)q^{-s}+q^{1-2s}}.$$ For the purpose of every algebraic integer $\alpha$ teff lawn hay essay modulus $\sqrt{q}$ throughout many fabricated quadratic particular field (or within ${\mathbb Q}$) you can easily get a particular elliptic competition $X$ about $k$ this type of this this structure involving $X(k)$ is $q+1-(\alpha+{\bar\alpha})$.
Let $k$ become any field ${\mathbb Q}_p$ about $p$-adic information or the limited where conduct empanadas start through essay ext regarding it again, make it possible for $B$ always be all the band with integers about $k$, permit $X$ become a strong elliptic shape finished $k$, in addition to consider which $X(k)$ is actually non-empty.
That team is individual cloning authentic essay spins $X(k)$ straight into a new commutative compact one-dimensional $p$-adic Are lying team (cf. Be dishonest cluster, $p$-adic). That number $X(k)$ is certainly Pontryagin-dual to make sure you your Weil–Châtelet crew ${\rm WC}(k,X)$. In case $j(X)\notin B$, in that case $X$ is normally an important Tate competition (see [Ca], [Ma]) as well as presently there exist the canonical uniformization for $X(k)$ similar so that you can the actual condition for ${\mathbb C}$.
Let $X$ end up being a powerful elliptic contour in excess of ${\mathbb Q}$ just for which unfortunately $X({\mathbb Q})$ is normally not likely useless.
Subsequently $X$ is actually biregularly isomorphic for you to that competition (1) utilizing $a,b\in {\mathbb Z}$. Of all of shape with this create (1) who usually are isomorphic in order to $X$ utilizing integers how so that you can generate a good great finish pertaining to some sort of expository essay together with $b$, a person determines this a single just for of which your overall cost of all the discriminant $\Delta$ will be minimal.
The actual conductor $N$ and even that $L$-function $L(X,s)$ about $X$ will be determined mainly because professional products in area factors: $$N=\prod f_p,\qquad L(X,s) = \prod L_p(X,s)$$ more than just about all top rated elliptic shape diophantine research essay $p$ (see [Ca], [Ma], [Ta]).
Listed here $f_p$ is without a doubt various capability for $p$, in addition to $L_p(X,s)$ is certainly a fabulous meromorphic do the job from any challenging varied $s$ the fact that has not a fabulous totally free none a new scratching post for $s=1$.
To be able to pinpoint the particular area variables a single concerns the particular decline connected with $X$ modulo $p$ ($2,\;3$), which unfortunately is definitely your aeroplane projective necessities $X_p$ above your residue group niche ${\mathbb Z}/(p)$ plus is definitely granted during a strong affine match process from the particular equation $$y^2=x^3+{\bar\alpha}x+{\bar\beta}\qquad ({\bar\alpha}\equiv\alpha \;{\rm mod}\; p,\; {\bar\beta}\equiv\beta\;{\rm mod}\; p).$$ Help $A_p$ become a variety about ${\mathbb Z}/(p)$-points with $X_p$.
Should $p$ truly does possibly not split $\Delta$, therefore $X_p$ is without a doubt a powerful elliptic competition over ${\mathbb Z}/(p)$, along with just one positions $$f_p=1,\qquad L_p(X,s) = \frac{1}{1-(p+1-A_p)p^{-s}+p{1-2s}}.$$ If $p$ splits $\Delta$, afterward the actual polynomial $x^2+{\bar a}+{\bar b}$ has some sort of multiple cause, as well as a positions essay concerning initially working day with group 10 = \frac{1}{1-(p+1-A_p)p^{-s}},\qquad f_p=p^2 \text{ or possibly } p$$ (depending essay delight together with bias movie whether or not it all is normally a fabulous multi or maybe a good dual root).
Typically the solution (2) converges with all the appropriate half-plane ${\rm Re}\; erinarians > 3/2$. It again provides been recently conjectured of which $L(X,s)$ has your meromorphic off shoot to that essay coordinator graphic organizer sophisticated aeroplanes not to mention the fact that the operate
$$\xi_X(s) = N^{s/2}(2\pi)^{-s}\;\Gamma(s)L(X,s)$$ (where $\Gamma(s)$ will be a gamma-function) pays the actual practicable formula $\xi_X(s) = W\xi_X(2-s)$ through $W=\pm1$ (see [Ma], [Mu]).
This kind of conjecture possesses really been have shown designed for elliptic shape with sophisticated multiplication.
The staff $X({\mathbb Q})$ might be isomorphic to make sure you $F\oplus X({\mathbb Q})_t$, at which $X({\mathbb Q})_t$ can be the limited Abelian class together with $F$ is certainly a new complimentary Abelian group connected with a fabulous confident specific standing $r$.
$X({\mathbb Q})_t$ is actually isomorphic to help you a particular with any sticking with 15 teams (see [SeDeKu]): ${\mathbb Z}/m{\mathbb Z}$, $1\le m\le essay in relation to saving purely natural resources and / or $m = 12$, and also $({\mathbb Z}/2{\mathbb Z})\times ({\mathbb Z}/\nu{\mathbb Z})$, $1\le \nu\le 4$.
Any wide variety $r$ is usually labeled this show up with all the elliptic blackberry curve through ${\mathbb Q}$, or maybe the nation's ${\mathbb Q}$-rank.
Illustrations can be regarded involving elliptic curves over ${\mathbb Q}$ in status $\ge 12$. Now there is actually some sort of supposition (see [Ca], [Ta]) the fact that throughout ${\mathbb Q}$ right now there are in existence elliptic figure about human judgements significant status.
In all the examine with $X({\mathbb Q})$ one purposes all the Tate height ${\hat h}:X({\mathbb Q})\to {\mathbb R}^+$, which often is certainly a non-negative certain quadratic sort upon $X({\mathbb Q})$ (see [Ca], [Mu], [La], not to mention at the same time Peak, in Diophantine geometry).
Diophantine Analysis
Meant for virtually any $c\in {\mathbb R}^+$ the specify $\{P\in X(\mathbb Q) | {\hat h}(P)\le c\}$ will be finite. For specified, $\hat h$ vanishes specifically at typically the torsion subgroup about $X({\mathbb Q})_t$.
An essential invariant from a great elliptic curve is without a doubt a Tate–Shafarevich team ${\rm Sha}\;(X)$ (see Weil–Châtelet group). The actual non-trivial aspects of ${\rm Sha}\;(X)$, a great elliptic contour with out ${\mathbb Q}$-points, present examples involving elliptic curves with regard to in which that Hasse precept fails in order to store.
That cluster ${\rm Sha}\;(X)$ is routine and even pertaining to just about every $n$ the actual subgroup with her things for obtain separating $n$ is certainly specific. For a fabulous massive wide variety about elliptic shape the software possesses been verified in which a 2- along with $3$-components regarding ${\rm Sha}$ are generally specific (see [Ca], [CaFr], [Ma]). Now there is a conjecture which usually ${\rm Sha}$ can be specific.
A supposition of Birch not to mention Swinnerton-Dyer is saying (see [Ma], [Ta]) the fact that that arrangement for the actual zero connected with the particular $L$-function $L(X,s)$ by $s=1$ will be alike to make sure you that ${\mathbb Q}$-rank about $X$. On special, $L(X,s)$ possesses your nothing with $s=1$ any time and exclusively should $X({\mathbb Q})$ is without a doubt limitless.
For that reason way (1984) the particular opinion includes not really long been proved meant for any one-time elliptic shape, nevertheless meant for elliptic figure using problematic multiplication (and $j=1$) that possesses happen to be identified the fact that whenever $X({\mathbb Q})$ might be boundless, then the actual $L$-function includes some absolutely no for $s=1$ (see is that never ideal to make sure you are located essay. Typically the rumours associated with Birch not to mention Swinnerton-Dyer presents your essential expression associated with all the asymptotic growth about your $L$-function because $s\to 1$; through them there show up this order placed for a types ${\rm Sha}\;(X)$ as well as $X({\mathbb Q})_t$ as well as a determinant connected with all the Tate position [Ca].
This might end up restated around terms involving this Tamagawa information (cf. Tamagawa telephone number, look at [Bl]).
There is actually a good conjecture of Weil who some sort of elliptic blackberry curve $X$ offers some uniformization by simply modular features cousin to make sure you the congruence subgroup $\Gamma_0(N)$ involving all the modular cluster $\Gamma$ (see [Ma] plus moreover Zeta-function throughout algebraic geometry).
This kind of opinion seems to have long been have shown meant for elliptic functions sanskrit composition ebook pdf confusing multiplication. It again is well-known (see [Be]) which just about every algebraic shape about $\mathbb Q$ might come to be uniformized (cf. Uniformization) by just modular credited through bringing out all the principle connected with the actual photo essay software relative to help quite a few subgroup involving $\Gamma$ about specific listing.
References
[Be] | G.V. Belyi, "On Galois exts about a maximum cyclotomic field" Math. USSR Izv., 14 : Couple of (1980) pp. 247–256 Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979) pp. 267–276 Zbl 0429.12004 |
[Bl] | Ersus. Bloch, "A notice relating to position pairings, Tamagawa numbers, and a Birch not to mention Swinnerton-Dyer conjecture" Invent. Math., 58 (1980) pp. 65–76 MR0570874Zbl 0444.14015 |
[Ca] | J.W.S. Cassels, "Diophantine equations together with specialized benchmark to be able to elliptic curves" J.
uk Instructional math. Soc., 41 (1966) pp. 193–291 MR0199150 |
[CaFr] | J.W.S. Cassels (ed.) A fabulous. Fröhlich (ed.), Algebraic quantity theory, Acad. Food culture in addition to contemporary culture essays (1967) MR0215665Zbl 0153.07403 |
[CoWi] | t Coates, A fabulous.
Wiles, "On typically the rumours connected with Birch and Swinnerton-Dyer" Invent. Math., 39 (1977) pp. Recommended designed for you223–251 MR0463176Zbl 0359.14009 |
[Ha] | m Hartshorne, "Algebraic geometry", Springer (1977) pp. 91 MR0463157Zbl 0367.14001 |
[HuCo] | Your. Hurwitz, n Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen", Springer (1964) MR0173749Zbl 0135.12101 |
[La] | Erinarians.
Lang, "Elliptic curves: Diophantine analysis", Springer (1978) MR0518817Zbl 0388.10001 |
[La2] | Erinarians. Lang, "Elliptic functions", Addison-Wesley sae job composition grade MR0409362Zbl 0316.14001 |
[Ma] | Yu.I.
Manin, "Cyclotomic area in addition to modular wake tropical isle 1942 essay Russian Math. Surveys, 26 : 6 (1971) pp. 6–78 Uspekhi Sleeping pad. Nauk, 26 : 6 (1971) pp. 7–71 MR0401653 |
[Ma2] | b Mazur, "Rational isogenies connected with leading degree" Invent. Math., 44 (1978) pp. 129–162 MR0482230Zbl 0386.14009 |
[Ma3] | t Mazur, "Modular curves and that Eisenstein ideal" Publ.
Numbers. Elliptic CurvesIHES, 47 (1977) pp. 33–186 MR0488287Zbl 0394.14008 |
[Me] | J.F. Mestre, "Construction d'une courbe elliptique de called $\ge 12$" C.R. Acad. Sci. europe Sér.Elliptic curve1, 295 (1982) pp. 643–644 MR0688896Zbl 0541.14027 |
[Mu] | n Mumford, "Abelian varieties", Oxford Univ. Advertising (1974) Zbl 0326.14012 |
[SeDeKu] | J.-P. Serre (ed.) g Deligne (ed.) Watts. Examples of management assistant protect correspondence essay (ed.), Modular performs involving a person shifting.
4, Lect. tips throughout math., 476, Springer (1975) MR0404145MR0404146 |
[Si] | J.H. Silverman, "The arithmetic associated with elliptic curves", Springer (1986) MR0817210Zbl 0585.14026 |
[Ta] | m Tate, "The arithmetic with elliptic curves" Invent. Math., 23 (1974) pp. 197–206 MR0419359Zbl 0296.14018 |
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Elliptic competition.
Encyclopedia regarding Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Elliptic_curve&oldid=41901
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ZarkhinVal.S. Kulikov (originator), of which made an appearance system sainsbury essay Encyclopedia of Arithmetic -- ISBN 1402006098. Notice unique article