Conjecture For Elliptic Curves Research Articles

The Birch-Swinnerton-Dyer conjecture of elliptic curves over mathbbQwith complex multiplication

The Birch-Swinnerton-Dyer conjecture of elliptic curves over mathbbQwith complex multiplication

Watkins’s conjecture for elliptic curves with non-split multiplicative reduction

Let E E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of E E is bounded by the 2 2 -adic valuation of the modular degree of E E . We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2 2 , provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

Critically separable rational maps in families

AbstractGiven a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case.

Modular abelian varieties of odd modular degree

In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^\alpha$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and Stein, finds necessary condition for elliptic curves to have odd modular degree. In the process we prove Watkins's conjecture for elliptic curves with odd modular degree and a nontrivial rational torsion point.

Fine Selmer groups of elliptic curves over p-adic Lie extensions

The aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic Iwasawa theory. Let F be a finite extension of Q, p an odd prime number, and F cyc the cyclotomic Zp-extension of Q. Let K(F cyc) denote the maximal unramified abelian p-extension of F cyc, in which every prime of F cyc above p splits completely, and define Y (F cyc) to be the Galois group of K(F cyc) over F cyc. Put = G(F cyc/F ), and write ( ) for the Iwasawa algebra of . Iwasawa proved that Y (F cyc) is always a finitely generated torsion ( )-module, and he conjectured [12] that in fact Y (F cyc) is always a finitely generated Zp-module. At present, this conjecture has only been proven when F is abelian over Q (see [5],[26]). Perhaps surprisingly, it does not seem to have been pointed out in the literature that there is a precise analogue of Iwasawa’s conjecture for elliptic curves over F cyc. Let E be an elliptic curve over F , and let

The main conjecture for CM elliptic curves at supersingular primes

At a prime of ordinary reduction, the Iwasawa for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi's conjecture relates modified Selmer groups, which he defined, with modified padic L-functions defined by the first author. In this paper we prove Kobayashi's conjecture for elliptic curves with complex multiplication.

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E D : y 2 = x 3−D 2 x

We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves.

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E D : y 2 = x 3 − D 2 x

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E D : y 2 = x 3 − D 2 x

On the semi-simplicity of the $U_p$ -operator on modular forms

For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several levels or weights at the same time, we will denote this by , or ! . If " is a prime number dividing , our $# is also known under the name %&# . One of our main results can be stated very easily: if ('*) and ",+ does not divide , then the operator $# is semi-simple. We can prove the same result for weight /. , under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in [11, 0 2] that this semi-simplicity is implied by Tate’s conjecture claiming that for 1 projective and smooth over a finite field of characteristic " , and 23 54 , 687:9 =@? AB C1D FEHGJILKNMO equals the order of PQ C1R SB at 2 . Ulmer proved in [17] that $# is semi-simple, for 5'T. and ",U not dividing , under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic " . His method is quite different from ours: assuming that $# is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve. The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform V of weight , prime-to-" level and character W , the polynomial X UZY\[ #]X<^\W8 _" `" !ba,c has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at " in the two-dimensional Galois representations associated to V ; it is also the characteristic polynomial of the crystalline Frobenius asociated to V . We show that this crystalline Frobenius cannot be a scalar. In Sections 2 and 3 we prove the results concerning these Frobenius elements for 'd) and e f) , respectively. Section 2 is quite elementary, whereas in Section 3 we use a lot of the machinary for comparing " -adic etale and crystalline cohomology. In Section 4 we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section 5 gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. g partially supported by the Institut Universitaire de France

Some Remarks on the Main Conjecture for Elliptic Curves with Complex Multiplication

Some Remarks on the Main Conjecture for Elliptic Curves with Complex Multiplication