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Moduli stack of elliptic curves
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<h2 id="properties">Properties</h2> <h3>Smooth Deligne-Mumford stack</h3> <p>The moduli stack of elliptic curves is a smooth separated <a href="/facts/Deligne%25E2%2580%2593Mumford_stack/kXlptywa">Deligne–Mumford stack</a> of finite type over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb {Z} )} , but is not a scheme as elliptic curves have non-trivial automorphisms. </p> <h3>j-invariant</h3> <p>There is a proper morphism of M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} to the affine line, the coarse moduli space of elliptic curves, given by the <a href="/facts/J-invariant/H1a38AJF"><i>j</i>-invariant</a> of an elliptic curve. </p> <h2 id="construction-over-the-complex-numbers">Construction over the complex numbers</h2> <p>It is a classical observation that every elliptic curve over C {\displaystyle \mathbb {C} } is classified by its <a href="/facts/Period_matrix/9alkmCtU">periods</a>. Given a basis for its integral homology α , β ∈ H 1 ( E , Z ) {\displaystyle \alpha ,\beta \in H_{1}(E,\mathbb {Z} )} and a global holomorphic differential form ω ∈ Γ ( E , Ω E 1 ) {\displaystyle \omega \in \Gamma (E,\Omega _{E}^{1})} (which exists since it is smooth and the dimension of the space of such differentials is equal to the <a href="/facts/Arithmetic_genus/hYdDY8Su">genus</a>, 1), the integrals [ ∫ α ω ∫ β ω ] = [ ω 1 ω 2 ] {\displaystyle {\begin{bmatrix}\int _{\alpha }\omega &\int _{\beta }\omega \end{bmatrix}}={\begin{bmatrix}\omega _{1}&\omega _{2}\end{bmatrix}}} give the generators for a Z {\displaystyle \mathbb {Z} } -lattice of rank 2 inside of C {\displaystyle \mathbb {C} } <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> pg 158. Conversely, given an integral lattice Λ {\displaystyle \Lambda } of rank 2 {\displaystyle 2} inside of C {\displaystyle \mathbb {C} } , there is an embedding of the complex torus E Λ = C / Λ {\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda } into P 2 {\displaystyle \mathbb {P} ^{2}} from the <a href="/facts/Weierstrass_P_function/GHRXDknk">Weierstrass P function</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> pg 165. This isomorphic correspondence ϕ : C / Λ → E ( C ) {\displaystyle \phi :\mathbb {C} /\Lambda \to E(\mathbb {C} )} is given by z ↦ [ ℘ ( z , Λ ) , ℘ ′ ( z , Λ ) , 1 ] ∈ P 2 ( C ) {\displaystyle z\mapsto [\wp (z,\Lambda ),\wp '(z,\Lambda ),1]\in \mathbb {P} ^{2}(\mathbb {C} )} and holds up to <a href="/facts/Homothetic_transformation/WUmuRuzC">homothety</a> of the lattice Λ {\displaystyle \Lambda } , which is the equivalence relation z Λ ∼ Λ for z ∈ C ∖ { 0 } {\displaystyle z\Lambda \sim \Lambda ~{\text{for}}~z\in \mathbb {C} \setminus \{0\}} It is standard to then write the lattice in the form Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } for τ ∈ h {\displaystyle \tau \in {\mathfrak {h}}} , an element of the <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a>, since the lattice Λ {\displaystyle \Lambda } could be multiplied by ω 1 − 1 {\displaystyle \omega _{1}^{-1}} , and τ , − τ {\displaystyle \tau ,-\tau } both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over C {\displaystyle \mathbb {C} } . There is an additional equivalence of curves given by the action of the SL 2 ( Z ) = { ( a b c d ) ∈ Mat 2 , 2 ( Z ) : a d − b c = 1 } {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{Mat}}_{2,2}(\mathbb {Z} ):ad-bc=1\right\}} where an elliptic curve defined by the lattice Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } is isomorphic to curves defined by the lattice Z ⊕ Z ⋅ τ ′ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau '} given by the <a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">modular action</a> ( a b c d ) ⋅ τ = a τ + b c τ + d = τ ′ {\displaystyle {\begin{aligned}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot \tau &={\frac {a\tau +b}{c\tau +d}}\\&=\tau '\end{aligned}}} Then, the moduli stack of elliptic curves over C {\displaystyle \mathbb {C} } is given by the stack quotient M 1 , 1 ≅ [ SL 2 ( Z ) ∖ h ] {\displaystyle {\mathcal {M}}_{1,1}\cong [{\text{SL}}_{2}(\mathbb {Z} )\backslash {\mathfrak {h}}]} Note some authors construct this moduli space by instead using the action of the <a href="/facts/Modular_group/wL6l14Hk">Modular group</a> PSL 2 ( Z ) = SL 2 ( Z ) / { ± I } {\displaystyle {\text{PSL}}_{2}(\mathbb {Z} )={\text{SL}}_{2}(\mathbb {Z} )/\{\pm I\}} . In this case, the points in M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} having only trivial stabilizers are dense. </p> <p> {\displaystyle \qquad } </p> <h3>Stacky/Orbifold points</h3> <p>Generically, the points in M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} are isomorphic to the classifying stack B ( Z / 2 ) {\displaystyle B(\mathbb {Z} /2)} since every elliptic curve corresponds to a double cover of P 1 {\displaystyle \mathbb {P} ^{1}} , so the Z / 2 {\displaystyle \mathbb {Z} /2} -action on the point corresponds to the involution of these two branches of the covering. There are a few special points<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> pg 10-11 corresponding to elliptic curves with <a href="/facts/J-invariant/H1a38AJF"> j {\displaystyle j} -invariant</a> equal to 1728 {\displaystyle 1728} and 0 {\displaystyle 0} where the automorphism groups are of order 4, 6, respectively<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> pg 170. One point in the <a href="/facts/Fundamental_domain/e68KDYWu">Fundamental domain</a> with stabilizer of order 4 {\displaystyle 4} corresponds to τ = i {\displaystyle \tau =i} , and the points corresponding to the stabilizer of order 6 {\displaystyle 6} correspond to τ = e 2 π i / 3 , e π i / 3 {\displaystyle \tau =e^{2\pi i/3},e^{\pi i/3}} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>pg 78. </p> <h4>Representing involutions of plane curves</h4> <p>Given a plane curve by its <a href="/facts/Weierstrass_equation/60jvaHSA">Weierstrass equation</a> y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} and a solution ( t , s ) {\displaystyle (t,s)} , generically for <a href="/facts/J-invariant/H1a38AJF">j-invariant</a> j ≠ 0 , 1728 {\displaystyle j\neq 0,1728} , there is the Z / 2 {\displaystyle \mathbb {Z} /2} -involution sending ( t , s ) ↦ ( t , − s ) {\displaystyle (t,s)\mapsto (t,-s)} . In the special case of a curve with <a href="/facts/Complex_multiplication/YQNcbNXP">complex multiplication</a> y 2 = x 3 + a x {\displaystyle y^{2}=x^{3}+ax} there the Z / 4 {\displaystyle \mathbb {Z} /4} -involution sending ( t , s ) ↦ ( − t , − 1 ⋅ s ) {\displaystyle (t,s)\mapsto (-t,{\sqrt {-1}}\cdot s)} . The other special case is when a = 0 {\displaystyle a=0} , so a curve of the form y 2 = x 3 + b {\displaystyle y^{2}=x^{3}+b} there is the Z / 6 {\displaystyle \mathbb {Z} /6} -involution sending ( t , s ) ↦ ( ζ 3 t , − s ) {\displaystyle (t,s)\mapsto (\zeta _{3}t,-s)} where ζ 3 {\displaystyle \zeta _{3}} is the third <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a> e 2 π i / 3 {\displaystyle e^{2\pi i/3}} . </p> <h3>Fundamental domain and visualization</h3> <p>There is a subset of the upper-half plane called the <a href="/facts/Fundamental_domain/e68KDYWu">Fundamental domain</a> which contains every isomorphism class of elliptic curves. It is the subset D = { z ∈ h : | z | ≥ 1 and Re ( z ) ≤ 1 / 2 } {\displaystyle D=\{z\in {\mathfrak {h}}:|z|\geq 1{\text{ and }}{\text{Re}}(z)\leq 1/2\}} It is useful to consider this space because it helps visualize the stack M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} . From the quotient map h → SL 2 ( Z ) ∖ h {\displaystyle {\mathfrak {h}}\to {\text{SL}}_{2}(\mathbb {Z} )\backslash {\mathfrak {h}}} the image of D {\displaystyle D} is surjective and its interior is injective<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>pg 78. Also, the points on the boundary can be identified with their mirror image under the involution sending Re ( z ) ↦ − Re ( z ) {\displaystyle {\text{Re}}(z)\mapsto -{\text{Re}}(z)} , so M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} can be visualized as the projective curve P 1 {\displaystyle \mathbb {P} ^{1}} with a point removed at infinity<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>pg 52. </p> <h3>Line bundles and modular functions</h3> <p>There are line bundles L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}} over the moduli stack M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} whose sections correspond to <a href="/facts/Modular_form/PcnbYR1f">modular functions</a> f {\displaystyle f} on the upper-half plane h {\displaystyle {\mathfrak {h}}} . On C × h {\displaystyle \mathbb {C} \times {\mathfrak {h}}} there are SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} -actions compatible with the action on h {\displaystyle {\mathfrak {h}}} given by SL 2 ( Z ) × C × h → C × h {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )\times {\displaystyle \mathbb {C} \times {\mathfrak {h}}}\to {\displaystyle \mathbb {C} \times {\mathfrak {h}}}} The degree k {\displaystyle k} action is given by ( a b c d ) : ( z , τ ) ↦ ( ( c τ + d ) k z , a τ + b c τ + d ) {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}:(z,\tau )\mapsto \left((c\tau +d)^{k}z,{\frac {a\tau +b}{c\tau +d}}\right)} hence the trivial line bundle C × h → h {\displaystyle \mathbb {C} \times {\mathfrak {h}}\to {\mathfrak {h}}} with the degree k {\displaystyle k} action descends to a unique line bundle denoted L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}} . Notice the action on the factor C {\displaystyle \mathbb {C} } is a <a href="/facts/Representation_theory/3ydLtaGH">representation</a> of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} on Z {\displaystyle \mathbb {Z} } hence such representations can be tensored together, showing L ⊗ k ⊗ L ⊗ l ≅ L ⊗ ( k + l ) {\displaystyle {\mathcal {L}}^{\otimes k}\otimes {\mathcal {L}}^{\otimes l}\cong {\mathcal {L}}^{\otimes (k+l)}} . The sections of L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}} are then functions sections f ∈ Γ ( C × h ) {\displaystyle f\in \Gamma (\mathbb {C} \times {\mathfrak {h}})} compatible with the action of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} , or equivalently, functions f : h → C {\displaystyle f:{\mathfrak {h}}\to \mathbb {C} } such that f ( ( a b c d ) ⋅ τ ) = ( c τ + d ) k f ( τ ) {\displaystyle f\left({\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot \tau \right)=(c\tau +d)^{k}f(\tau )} This is exactly the condition for a holomorphic function to be modular. </p> <h4>Modular forms</h4> <p>The modular forms are the modular functions which can be extended to the compactification L ⊗ k ¯ → M ¯ 1 , 1 {\displaystyle {\overline {{\mathcal {L}}^{\otimes k}}}\to {\overline {\mathcal {M}}}_{1,1}} this is because in order to compactify the stack M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} , a point at infinity must be added, which is done through a gluing process by gluing the q {\displaystyle q} -disk (where a modular function has its q {\displaystyle q} -expansion)<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>pgs 29-33. </p> <h3>Universal curves</h3> <p>Constructing the universal curves E → M 1 , 1 {\displaystyle {\mathcal {E}}\to {\mathcal {M}}_{1,1}} is a two step process: (1) construct a versal curve E h → h {\displaystyle {\mathcal {E}}_{\mathfrak {h}}\to {\mathfrak {h}}} and then (2) show this behaves well with respect to the SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} -action on h {\displaystyle {\mathfrak {h}}} . Combining these two actions together yields the quotient stack [ ( SL 2 ( Z ) ⋉ Z 2 ) ∖ C × h ] {\displaystyle [({\text{SL}}_{2}(\mathbb {Z} )\ltimes \mathbb {Z} ^{2})\backslash \mathbb {C} \times {\mathfrak {h}}]} </p> <h4>Versal curve</h4> <p>Every rank 2 Z {\displaystyle \mathbb {Z} } -lattice in C {\displaystyle \mathbb {C} } induces a canonical Z 2 {\displaystyle \mathbb {Z} ^{2}} -action on C {\displaystyle \mathbb {C} } . As before, since every lattice is homothetic to a lattice of the form ( 1 , τ ) {\displaystyle (1,\tau )} then the action ( m , n ) {\displaystyle (m,n)} sends a point z ∈ C {\displaystyle z\in \mathbb {C} } to ( m , n ) ⋅ z ↦ z + m ⋅ 1 + n ⋅ τ {\displaystyle (m,n)\cdot z\mapsto z+m\cdot 1+n\cdot \tau } Because the τ {\displaystyle \tau } in h {\displaystyle {\mathfrak {h}}} can vary in this action, there is an induced Z 2 {\displaystyle \mathbb {Z} ^{2}} -action on C × h {\displaystyle \mathbb {C} \times {\mathfrak {h}}} ( m , n ) ⋅ ( z , τ ) ↦ ( z + m ⋅ 1 + n ⋅ τ , τ ) {\displaystyle (m,n)\cdot (z,\tau )\mapsto (z+m\cdot 1+n\cdot \tau ,\tau )} giving the quotient space E h → h {\displaystyle {\mathcal {E}}_{\mathfrak {h}}\to {\mathfrak {h}}} by projecting onto h {\displaystyle {\mathfrak {h}}} . </p> <h4>SL2-action on Z2</h4> <p>There is a SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} -action on Z 2 {\displaystyle \mathbb {Z} ^{2}} which is compatible with the action on h {\displaystyle {\mathfrak {h}}} , meaning given a point z ∈ h {\displaystyle z\in {\mathfrak {h}}} and a g ∈ SL 2 ( Z ) {\displaystyle g\in {\text{SL}}_{2}(\mathbb {Z} )} , the new lattice g ⋅ z {\displaystyle g\cdot z} and an induced action from Z 2 ⋅ g {\displaystyle \mathbb {Z} ^{2}\cdot g} , which behaves as expected. This action is given by ( a b c d ) : ( m , n ) ↦ ( m , n ) ⋅ ( a b c d ) {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}:(m,n)\mapsto (m,n)\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix}}} which is matrix multiplication on the right, so ( m , n ) ⋅ ( a b c d ) = ( a m + c n , b m + d n ) {\displaystyle (m,n)\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=(am+cn,bm+dn)} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Fundamental_domain/e68KDYWu">Fundamental domain</a></li> <li><a href="/facts/Homothetic_transformation/WUmuRuzC">Homothety</a></li> <li><a href="/facts/Level_structure_(algebraic_geometry)/aJBArxJa">Level structure (algebraic geometry)</a></li> <li><a href="/facts/Moduli_of_abelian_varieties/e7MS1EwR">Moduli of abelian varieties</a></li> <li><a href="/facts/Shimura_variety/varFgmR6">Shimura variety</a></li> <li><a href="/facts/Modular_curve/UE3kT5SW">Modular curve</a></li> <li><a href="/facts/Elliptic_cohomology/Y0nSafzr">Elliptic cohomology</a></li></ul> <ul><li>Hain, Richard (2008), <i>Lectures on Moduli Spaces of Elliptic Curves</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0812.1803">0812.1803</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008arXiv0812.1803H">2008arXiv0812.1803H</a></li> <li>Lurie, Jacob (2009), <a href="http://www.math.harvard.edu/~lurie/papers/survey.pdf"><i>A survey of elliptic cohomology</i></a> (PDF)</li> <li>Olsson, Martin (2016), <i>Algebraic spaces and stacks</i>, Colloquium Publications, vol. 62, American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1470427986</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://ncatlab.org/nlab/show/moduli+stack+of+elliptic+curves">moduli+stack+of+elliptic+curves</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li> <li><a href="http://stacks.math.columbia.edu/tag/072K">"The moduli stack of elliptic curves"</a>, <i>Stacks project</i></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184. <a href="978-0-387-09494-6" target="_blank">978-0-387-09494-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184. <a href="978-0-387-09494-6" target="_blank">978-0-387-09494-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Galbraith, Steven. "Elliptic Curves" (PDF). Mathematics of Public Key Cryptography. Cambridge University Press – via The University of Auckland. <a href="https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf" target="_blank">https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Serre, Jean-Pierre (1973). A Course in Arithmetic. New York: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550. <a href="978-1-4684-9884-4" target="_blank">978-1-4684-9884-4</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Serre, Jean-Pierre (1973). A Course in Arithmetic. New York: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550. <a href="978-1-4684-9884-4" target="_blank">978-1-4684-9884-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Henriques, André G. "The Moduli stack of elliptic curves". In Douglas, Christopher L.; Francis, John; Henriques, André G; Hill, Michael A. (eds.). Topological modular forms (PDF). Providence, Rhode Island. ISBN 978-1-4704-1884-7. OCLC 884782304. Archived from the original (PDF) on 9 June 2020 – via University of California, Los Angeles. <a href="978-1-4704-1884-7" target="_blank">978-1-4704-1884-7</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>