scholarly journals Special values of motivic L-functions and zeta-polynomials for symmetric powers of elliptic curves

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Steffen Löbrich ◽  
Wenjun Ma ◽  
Jesse Thorner
Keyword(s):  
Elliptic Curves ◽  
Symmetric Powers ◽  
Special Values

scholarly journals Generating functions of planar polygons from homological mirror symmetry of elliptic curves

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jonas Kaszian ◽  
Jie Zhou
Keyword(s):  
Elliptic Curves ◽  
Theta Function ◽  
Generating Functions ◽  
Mirror Symmetry ◽  
Rational Functions ◽  
Mock Theta Function ◽  
Homological Mirror Symmetry ◽  
Jacobi Theta Function ◽  
Special Values

Abstract We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.

scholarly journals On the potential automorphy of certain odd-dimensional Galois representations

2010 ◽  
Vol 146 (3) ◽  
pp. 607-620 ◽  
Author(s):  
Thomas Barnet-Lamb
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Symmetric Powers

AbstractIn a previous paper, the potential automorphy of certain Galois representations to GLn for n even was established, following the work of Harris, Shepherd–Barron and Taylor and using the lifting theorems of Clozel, Harris and Taylor. In this paper, we extend those results to n=3 and n=5, and conditionally to all other odd n. The key additional tools necessary are results which give the automorphy or potential automorphy of symmetric powers of elliptic curves, most notably those of Gelbert, Jacquet, Kim, Shahidi and Harris.

scholarly journals CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

2012 ◽  
Vol 08 (04) ◽  
pp. 945-961 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curve ◽  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Alternate Proof ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

scholarly journals Quadratic points on non-split Cartan modular curves

2021 ◽  
pp. 1-23
Author(s):  
Philippe Michaud-Rodgers
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Quadratic Fields ◽  
Modular Curves ◽  
Symmetric Powers

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

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