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| J_0(N) | |
|---|---|
| Name | J_0(N) |
| Type | Abelian variety |
| Dimension | genus of X_0(N) |
| Related | X_0(N), Hecke algebra, modular forms |
J_0(N).
J_0(N) is the Jacobian variety of the modular curve X_0(N) defined over Q and constructed from the congruence subgroup Γ_0(N) of SL_2(Z). As an abelian variety over Q it carries a natural action of the Hecke algebra and decomposes up to isogeny into simple factors associated with newforms from spaces of cusp forms. The variety relates to the geometry of X_0(N), the arithmetic of elliptic curves over Q, and the theory of modular abelian varieties studied by Eichler, Shimura, and Deligne.
As the Jacobian of X_0(N), the variety parametrizes degree-zero divisor classes on the coarse moduli scheme classifying isogenies of elliptic curves with cyclic subgroups of order N. Its points are related to pairs of elliptic curves with Γ_0(N)-level structure and map to line bundles on X_0(N) via the Abel–Jacobi map. The construction uses the moduli stacks of elliptic curves and the work of Deligne–Rapoport on integral models of modular curves and compactifications by cusps.
The tangent space of J_0(N) at the origin is dual to the space S_2(Γ_0(N)) of weight-two cusp forms; the Eichler–Shimura relation connects Hecke operators T_p to Frobenius and Verschiebung on the l-adic Tate module. Newform theory and Atkin–Lehner–Li theory produce an isotypical decomposition of J_0(N) indexed by Galois conjugacy classes of normalized newforms, giving simple abelian varieties A_f associated to eigenforms f in S_2(Γ_0(N)). Work of Ribet, Mazur, and Ohta analyzes congruences between eigenforms and the resulting isogeny factors, and the Hecke algebra acts as endomorphisms studied in the context of the Langlands program and the Fontaine–Mazur conjecture.
Atkin–Lehner involutions W_Q for Q||N act on X_0(N) and induce automorphisms of J_0(N) whose fixed subvarieties reflect plus/minus decompositions important for level-lowering and level-raising results of Ribet and Wiles. Degeneracy maps between X_0(M) and X_0(N) for M|N induce pullback and pushforward morphisms between corresponding Jacobians, yielding old and new subvarieties; these maps underpin control theorems in the study of congruences and the construction of Eisenstein ideals developed by Mazur and Sharifi.
The Mordell–Weil group J_0(N)(Q) and its torsion subgroup play central roles in the arithmetic of modular curves, in Mazur's study of rational isogenies of elliptic curves, and in the proof of Fermat's Last Theorem via level-lowering and modularity lifting results by Wiles, Taylor, and Ribet. Torsion in J_0(N)(Q) connects to cuspidal subgroups and results of Manin, Drinfeld, and Ogg on component groups; computations by Cremona, Stein, and Merel give explicit classifications in low level. The structure of J_0(N)(Q) informs descent arguments for elliptic curves, visibility phenomena studied by Agashe and Stein, and arithmetic of Heegner points introduced by Gross and Zagier.
For N=1 the Jacobian is trivial while for small levels X_0(N) often has genus zero or one producing elliptic curve factors classified in tables by Cremona and LMFDB contributors. Famous examples include the factor associated to the modular form of conductor 11 giving the elliptic curve studied by Fricke and Atkin, and the optimal quotient A_f corresponding to the newform attached to the Taniyama–Shimura–Weil theorem used by Wiles in the proof of modularity for semistable elliptic curves. Higher-level phenomena involving Eisenstein quotients, cuspidal subgroups of Ogg, and exceptional isogenies appear in explicit computations by Kenku, Momose, and Ligozat.
Each simple isogeny factor A_f of J_0(N) carries an L-function equal to the product of L(f,s) over Galois conjugates of the associated newform, tying J_0(N) to the Birch and Swinnerton-Dyer conjecture for modular abelian varieties studied by Kolyvagin, Gross, Zagier, and Nekovář. Special value formulas relate heights of Heegner points on J_0(N) to derivatives of L-functions, and Iwasawa theory for cyclotomic towers uses p-adic L-functions attached to eigenforms controlling Selmer groups of factors of J_0(N), as in the work of Perrin-Riou, Skinner–Urban, and Kato.
Category:Abelian varieties Category:Modular curves Category:Modular forms Category:Number theory Category:Arithmetic geometry