Formal groups and zeta-functions of elliptic curves

1971 ◽  
Vol 12 (4) ◽  
pp. 321-336 ◽  
Author(s):  
Walter L. Hill
Keyword(s):  
Elliptic Curves ◽  
Zeta Functions ◽  
Formal Groups

scholarly journals FORMAL GROUPS AND INVARIANT DIFFERENTIALS OF ELLIPTIC CURVES

2015 ◽  
Vol 92 (1) ◽  
pp. 44-51
Author(s):  
MOHAMMAD SADEK
Keyword(s):  
Power Series ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Series Expansion ◽  
Power Series Expansion ◽  
Formal Groups ◽  
Frobenius Endomorphism ◽  
Congruence Relations ◽  
Invariant Differential ◽  
Weierstrass Equations

In this paper, we find a power series expansion of the invariant differential ${\it\omega}_{E}$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.

scholarly journals Higher-rank zeta functions for elliptic curves

2020 ◽  
Vol 117 (9) ◽  
pp. 4546-4558 ◽  
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Vector Bundles ◽  
Zeta Functions ◽  
Smooth Curve ◽  
Isomorphism Classes ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

scholarly journals Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

2018 ◽  
Vol 14 (08) ◽  
pp. 2317-2331
Author(s):  
Marcus du Sautoy
Keyword(s):  
Elliptic Curves ◽  
Zeta Functions ◽  
Symmetric Power ◽  
Meromorphic Continuation ◽  
Natural Boundary ◽  
Integer Polynomials ◽  
Analytic Behavior ◽  
Euler Products ◽  
Natural Boundaries

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

FORMAL GROUPS OF JACOBIAN VARIETIES OF HYPERELLIPTIC CURVES

2010 ◽  
Vol 06 (07) ◽  
pp. 1701-1716
Author(s):  
FUMIO SAIRAIJI
Keyword(s):  
Elliptic Curves ◽  
Characteristic Zero ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Formal Groups ◽  
Jacobian Varieties ◽  
Explicit Constructions ◽  
Modulo P

Let k be a field of characteristic zero. In this paper, we discuss two explicit constructions of the formal groups Ĵ of the Jacobian varieties J of hyperelliptic curves C over k. Our results are generalizations of the classical constructions of formal groups of elliptic curves. As an application of our results, we may decide the type of bad reduction of J modulo p when C is a hyperelliptic curve over ℚ.

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