Elliptic curves and p-adic uniformisation
نویسنده
چکیده
The Diophantine theory studies the rational solutions (X, Y, Z) ∈ Q of equation (1). It is convenient to ignore the trivial solution (0, 0, 0) and to identify solutions if they differ by multiplication by a non-zero scalar. Solutions to (1) are thus viewed as points in the projective plane P2(Q). Let E(Q) ⊂ P2(Q) denote this solution set. More generally, if F is any field, let E(F ) ⊂ P2(F ) be the corresponding set of solutions with values in F . It is identified with the set of (x, y) ∈ F 2 satisfying the associated affine equation
منابع مشابه
ON THE DE RHAM AND p-ADIC REALIZATIONS OF THE ELLIPTIC POLYLOGARITHM FOR CM ELLIPTIC CURVES
In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...
متن کاملON THE REAL HODGE AND p-ADIC REALIZATIONS OF THE ELLIPTIC POLYLOGARITHM FOR CM ELLIPTIC CURVES
In this paper, we give an explicit description of the complex and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...
متن کاملTeitelbaum’s exceptional zero conjecture in the anticyclotomic setting
In [Tei], Teitelbaum formulates a conjecture relating first derivatives of the Mazur– Swinnerton-Dyer p-adic L-functions attached to a modular forms of even weight k ≥ 2 to certain L-invariants arising from Shimura curve parametrisations. This article formulates an analogue of Teitelbaum’s conjecture in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic Zp-extension of an ...
متن کاملA p-adic Height Function Of Cryptanalytic Significance
It is noted that an efficient algorithm for calculating a p-adic height could have cryptanalytic applications. Elliptic curves and their generalizations are an active research topic with practical applications in cryptography [1], [2], [3]. If E is an elliptic curve defined over a finite field Fp, where p is prime, and if P and Q are points on the curve E such that Q = nP , then the elliptic cu...
متن کاملExplicit p-adic methods for elliptic and hyperelliptic curves
We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007