On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N, divisible by an even number of primes, and let Γ′ be a congruence subgroup of level M, where M is prime to N. For each D dividing N and divisible by an even number of primes, the Shimura curve X ^D (Γ₀(N/D) ∩ Γ′) associated to the indefinite quaternion algebra of discriminant D and Γ₀(N/D) ∩ Γ′-level structure is well defined, and we can consider its Jacobian J ^D (Γ₀(N/D) ∩ Γ′). Let J ^D denote the N/D-new subvariety of this Jacobian. By the Jacquet-Langlands correspondence [J-L] and Faltings’ isogeny theorem [Fa], there are Hecke-equivariant isogenies among the various varieties J ^D defined above. However, since the isomorphism of Jacquet-Langlands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties J ^D in terms of the maps they induce on the character groups of the tori corresponding to the mod p reductions of these varieties for p dividing N. Our characterization of such maps in these terms allows us to classify the possible kernels of maps from J ^D to J ^D′, for D dividing D′, up to support on a small finite set of maximal ideals of the Hecke algebra. This allows us to compute the Tate modules J ^D of J ^D at all non-Eisenstein of residue characteristic l > 3. These computations have implications for the multiplicities of irreducible Galois representations in the torsion of Jacobians of Shimura curves; one such consequence is a “multiplicity one” result for Jacobians of Shimura curves.     

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

Compactified Jacobians of curves with spine decompositions

A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri’s moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Here we give a sufficient condition for when these two types of moduli spaces are equal. Eduardo Esteves is Supported by CNPq, Processos 301117/04-7 and 470761/06-7, by CNPq/FAPERJ, Processo E-26/171.174/2003, and by the Institut Mittag–Leffler (Djursholm, Sweden).     

Endomorphism algebras of Jacobians of certain superelliptic curves

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Let p be a prime, and q a power of p. Using Galois theory, we show that over a field K of characteristic zero, the endomorphism algebras of the Jacobians of certain superelliptic curves yq=f(x) are products of cyclotomic fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=z5ZzOy1K_Ec.     

Topology of the compactified Jacobians of singular curves

We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p, q), (4, 2q, s), (6, 8, s), or (6, 10, s). Further, we derive a combinatorial method to compute the Betti numbers of the compactified Jacobian of an unibranched rational curve with singularities like above. Some of the Betti numbers can be stated explicitly.     

Universal algebraic equivalences between tautological cycles on Jacobians of curves

We present a collection of algebraic equivalences between tautological cycles on the Jacobian J of a curve, i.e., cycles in the subring of the Chow ring of J generated by the classes of certain standard subvarieties of J. These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly. Supported in part by NSF grant DMS-0302215.     

Singular pencils of quadrics and compactified Jacobians of curves

LetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.     

Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation , where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant. Received April 29, 1997; in final form September 22, 1997     

On component groups of Jacobians of quaternionic modular curves

Let \({D}\) be a division ring with center \({F}\). The aim of the paper is to show that if \({F}\) is uncountable or \({D}\) is finite dimensional over \({F}\), then every subnormal subgroup of the multiplicative group \({D^*}\) of \({D}\) satisfying a nontrivial generalized power central group identity is contained in \({F}\). As a corollary, Conjecture 2 in (Herstein, Israel J Math 31:180–188, 1978) holds in case \({D}\) is finite dimensional.     

On the size of the Jacobians of curves over finite fields

Abdract  Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.      

Realization of modular Galois representations in the Jacobians of modular curves

The Ramanujan Journal - In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod $$ell $$ Galois representations...     

Polarizations of isotypical components of Jacobians with group action

We outline a method to compute the type of the induced polarization of an abelian subvariety of a canonically polarized Jacobian of a smooth projective curve. The method works for curves of not too big genus admitting a “large” group of automorphisms. Several examples are given.     

Fundamental S-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves

In 2010, Platonov proposed a fundamentally new approach to the torsion problem in Jacobi varieties of hyperelliptic curves over the field of rational numbers. This new approach is based on the calculation of fundamental units in hyperelliptic fields. It was applied to prove the existence of torsion points of new orders. In the paper, the notion of the degree of an S-unit is introduced and a relationship between the degree of an S-unit and the order of the corresponding torsion point of the Jacobian of a hyperelliptic curve is established. A complete exposition of the new method and results obtained on the basis of this method is contained in [2].