scholarly journals Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

2017 ◽  
Vol 4 (1) ◽  
pp. 7-15 ◽  
Author(s):  
Robert Xin Dong
Keyword(s):  
Elliptic Curves ◽  
Bergman Kernel ◽  
Complex Structure ◽  
Elliptic Functions ◽  
Hyperelliptic Curves ◽  
Asymptotic Formulas ◽  
Holomorphic Family ◽  
Structure Information ◽  
Taylor Expansions ◽  
Bergman Kernel Asymptotics

Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

scholarly journals A direct approach to Bergman kernel asymptotics for positive line bundles

2008 ◽  
Vol 46 (2) ◽  
pp. 197-217 ◽  
Author(s):  
Robert Berman ◽  
Bo Berndtsson ◽  
Johannes Sjöstrand
Keyword(s):  
Bergman Kernel ◽  
Direct Approach ◽  
Line Bundles ◽  
Bergman Kernel Asymptotics ◽  
Positive Line

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Further Contributions to the Theory of Peripheral Jets in Ground Proximity

1992 ◽  
Vol 36 (01) ◽  
pp. 88-90
Author(s):  
David S. Tselnik
Keyword(s):  
General Solution ◽  
Infinite Series ◽  
Elliptic Functions ◽  
Jet Flow ◽  
New Method ◽  
Asymptotic Formulas ◽  
Flow Problems ◽  
Complicated Task

A number of plane inviscid jet flow problems of interest in hydrodynamics require the use of elliptic functions theory. Generally speaking, finding the general solution to a problem in terms of elliptic functions is not a complicated task. However, finding solutions as rapidly convergent infinite series or as sound asymptotic formulas is often not as easy, and special ways of treatment may prove to be necessary. In parallel with solving the problem of peripheral jets, the author's earlier paper (1985) proposed some such ways of treatment. In the present paper, a new method of treatment is proposed (and used);this approach may be of help in studies where the methods of elliptic functions theory have to be used.

On a class of unirational varieties

1951 ◽  
Vol 47 (3) ◽  
pp. 496-503 ◽  
Author(s):  
L. Roth
Keyword(s):  
General Theory ◽  
Elliptic Curves ◽  
Hyperelliptic Curves ◽  
Considerable Part ◽  
First Case

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

Determinantal representations of elliptic curves via Weierstrass elliptic functions

2018 ◽  
Vol 34 ◽  
pp. 125-136 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curves ◽  
Algebraic Curve ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Riemann Theta Functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

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