We give a local deformation theoretic proof of Farkas’ conjecture that a principally polarized complex abelian variety of dimension 4 whose theta divisor has an isolated double point of rank 3 at a point of order two is a Jacobian. The argument yields an explicit local normal form for the theta function near such a point. The proof depends only on the facts that the theta function is even, a ge...

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...

We analyze the convergence of the sequential quadratic programming (SQP) method for nonlinear programming for the case in which the Jacobian of the active constraints is rank deecient at the solution and/or strict complementarity does not hold for some or any feasible Lagrange multipliers. We use a nondiieren-tiable exact penalty function, and we prove that the sequence generated by the SQP is ...

1. Complex multiplication. We say an abelian variety A of dimension g over a field K admits sufficiently many Complex Multiplications (smCM) if D = End(A) contains a commutative semi-simple algebra Λ ⊂ D of rank dimQ(Λ) = [Λ : Q] = 2g. In this case we say A is a CM abelian variety. A point z ∈ Ag(k) is called a CM point if z is the moduli point of a polarized abelian variety z = [(A,μ)] such th...

We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K, and denote its Jacobian by J . Let d ≥ 1 be an integer and denote the d-th symmetric power of C by C(d). In this paper we adapt the classic Chabauty–Coleman method to study the K-rational points of C(d). Suppose that J(K) has Mordell–Weil rank at most g− d. We give an explicit and practical criterion...

We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus > 1, it is impractical to apply Hilbert’s Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrar...

This paper explores unsupervised learning of parsing models along two directions.First, which models are identifiable from infinite data? We use a general tech-nique for numerically checking identifiability based on the rank of a Jacobian ma-trix, and apply it to several standard constituency and dependency parsing models.Second, for identifiable models, how do we estimate the p...

in this work are studied the jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.