New and Old Subvarieties and Natural Maps
Natural Maps
Suppose \(M\) and \(N\) are positive integers and \(M\) divides \(N\). There are natural maps in both directions between \(J_0(N)\) and \(J_0(M)\) (and likewise for \(J_1\), etc.), for each divisor of \(t = N/M\), which correspond to maps of the form \(f(q) \mapsto f(q^t)\) and their duals. Since any modular abelian variety \(A\) in Magma is equipped with a map \(A\to J_e\) and \(J_p\to A\), where \(J_e\) and \(J_p\) are attached to modular symbols, the problem of defining natural maps between \(A\) and \(B\) is reduced to defining natural maps between modular abelian varieties attached to modular symbols.
- NaturalMap(A, B, d): ModAbVar, ModAbVar, RngIntElt -> MapModAbVar
Given abelian varieties \(A\) and \(B\) and an integer \(d\) return the natural map from \(A\) to \(B\) induced, in a potentially complicated way, from the map \(f(q)\mapsto\) \(f(q^d)\) on modular forms. In situations where the modular forms associated to \(A\) and \(B\) have nothing to do with each other, then we define this map to be the zero map.
- NaturalMap(A, B): ModAbVar, ModAbVar -> MapModAbVar
The natural map from the abelian variety \(A\) to the abelian variety \(B\) induced by the identity on modular forms, or the zero map if there is none.
- NaturalMaps(A, B): ModAbVar, ModAbVar -> SeqEnum
Given abelian varieties \(A\) and \(B\) return a sequence of the natural maps from \(A\) to \(B\) for all divisors \(d\) of the level of \(A\) over the level \(B\), or the level of \(B\) over the level \(A\).
- Example: Natural Maps Between Products (ex-8ef092)
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> A := JZero(11)*JZero(22); > B := JZero(11)*JZero(33); > phi := NaturalMap(A,B); > phi; Homomorphism N(1) from JZero(11) x JZero(22) to JZero(11) x JZero(33) (not printing 6x8 matrix) > Nullity(phi); 1 > f := NaturalMap(A,B,3); f; Homomorphism N(3) from JZero(11) x JZero(22) to JZero(11) x JZero(33) (not printing 6x8 matrix) > Nullity(f); 2 > NaturalMaps(JZero(11),JZero(33)); [ Homomorphism N(1) from JZero(11) to JZero(33) given on integral homology by: [ 1 0 -2 2 -3 0] [ 1 -1 0 1 -2 1], Homomorphism N(3) from JZero(11) to JZero(33) given on integral homology by: [ 0 -2 1 2 0 -1] [-1 0 1 1 -1 -1] ]
If we take a product of several copies of \(J_0(11)\) and of several copies of \(J_0(22)\), the
NaturalMapscommand still only returns 2 natural maps, one for each divisor of the quotient of the levels.> A := JZero(11)^2; > B := JZero(22)^3; > NaturalMaps(A,B); [ Homomorphism N(1) from JZero(11) x JZero(11) to JZero(22) x JZero(22) x JZero(22) given on integral homology by: [ 0 1 -2 3 0 1 -2 3 0 1 -2 3] [ 1 -1 1 0 1 -1 1 0 1 -1 1 0] [ 0 1 -2 3 0 1 -2 3 0 1 -2 3] [ 1 -1 1 0 1 -1 1 0 1 -1 1 0], Homomorphism N(2) from JZero(11) x JZero(11) to JZero(22) x JZero(22) x JZero(22) given on integral homology by: [-1 0 2 -2 -1 0 2 -2 -1 0 2 -2] [-1 2 -1 0 -1 2 -1 0 -1 2 -1 0] [-1 0 2 -2 -1 0 2 -2 -1 0 2 -2] [-1 2 -1 0 -1 2 -1 0 -1 2 -1 0] ]
New Subvarieties and Quotients
These commands compute the new and \(r\)-new subvarieties and quotients of an abelian variety \(A\) of level \(N\). The \(r\)-new subvariety of \(A\) is the intersection of the kernels of all natural maps from \(A\) to modular abelian varieties of level \(N/r\). The new subvariety is the intersection of the \(r\)-new subvarieties over all prime divisors \(r\) of \(N\). The \(r\)-new quotient of \(A\) is the quotient of \(A\) by the sum of all images in \(A\) under all natural maps of abelian varieties of level \(N/r\).
- NewSubvariety(A, r): ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
The \(r\)-new subvariety of the abelian variety \(A\).
- NewSubvariety(A): ModAbVar -> ModAbVar, MapModAbVar
The new subvariety of the abelian variety \(A\).
- NewQuotient(A, r): ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
The \(r\)-new quotient of the abelian variety \(A\).
- Example: New Subvarieties and Quotients of J0(33) (ex-1eeb69)
Run in calculator (opens in a new tab)
> J := JZero(33); > Dimension(J); 3 > Dimension(NewSubvariety(J,3)); 1 > Dimension(NewSubvariety(J)); 1 > Dimension(NewSubvariety(J,11)); 3 > Dimension(NewQuotient(J)); 1 > Dimension(OldSubvariety(J)); 2 > Dimension(OldSubvariety(J,3)); 2
Old Subvarieties and Quotients
These commands compute the old and \(r\)-old subvarieties and quotients of an abelian variety \(A\) of level \(N\). The \(r\)-old subvariety of \(A\) is the sum of the images of all natural maps from modular abelian varieties of level \(N/r\) to \(A\). The old subvariety is the sum of the \(r\)-old subvarieties as \(r\) varies over the divisors of \(N\). The \(r\)-old quotient of \(A\) is the quotient of \(A\) by its \(r\)-new subvariety.
- OldSubvariety(A, r): ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
The \(r\)-old subvariety of the abelian variety \(A\).
- OldSubvariety(A): ModAbVar -> ModAbVar, MapModAbVar
The old subvariety of the abelian variety \(A\).
- OldQuotient(A, r): ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
The \(r\)-old quotient of the abelian variety \(A\).
- Example: Old Subvariety and Quotient of J0(100) (ex-00e897)
Run in calculator (opens in a new tab)
We compute the old subvariety and old quotient of \(J_0(100)\), both of which have dimension \(6\).
> J := JZero(100); J; Modular abelian variety JZero(100) of dimension 7 and level 2^2*5^2 over Q > J_old := OldSubvariety(J); J_old; Modular abelian variety JZero(100)_old of dimension 6 and level 2^2*5^2 over Q > phi := Embeddings(J_old)[1]; > Codomain(phi); Modular abelian variety JZero(100) of dimension 7 and level 2^2*5^2 over Q > Jold := OldQuotient(J); Jold; Modular abelian variety JZero(100)^old of dimension 6 and level 2^2*5^2 over Q
The new subvariety and new quotient of \(J_0(100)\) intersect in a finite subgroup isomorphic to \({\mathbb{Z}}/12{\mathbb{Z}}\times {\mathbb{Z}}/12{\mathbb{Z}}\).
> J_new := NewSubvariety(J); J_new; Modular abelian variety JZero(100)_new of dimension 1 and level 2^2*5^2 over Q > G, A := J_new meet J_old; G; Finitely generated subgroup of abelian variety with invariants [ 12, 12 ] > Dimension(A); 0