# 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: New Old Natural Maps (ex-8ef092)` ```magma > 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 `NaturalMaps` command still only returns 2 natural maps, one for each divisor of the quotient of the levels. ```magma > 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 $A$. ### `NewQuotient(A): ModAbVar -> ModAbVar, MapModAbVar` The new quotient of the abelian variety $A$. ### `Example: New Old New Subvarieties And Quotients (ex-1eeb69)` ```magma > 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 $A$. ### `OldQuotient(A): ModAbVar -> ModAbVar, MapModAbVar` The old quotient of the abelian variety $A$. ### `Example: New Old Old Subvarieties And Quotients (ex-00e897)` We compute the old subvariety and old quotient of $J_0(100)$, both of which have dimension $6$. ```magma > 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 ${\mathbb{Z}}/12{\mathbb{Z}}\times {\mathbb{Z}}/12{\mathbb{Z}}$. ```magma > 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 ```