This thesis is a thorough revision of the Ph.D thesis entitled "The Zeros of Elliptic Curve L- functions" [Spi15] by Simon Spicer. First, we provide basic knowledge and facts about elliptic curves and in Section 8 we explain its algorithm. Let E be an elliptic curve with the global minimal Weierstrass equation E:y2+a1xy+a3y= x3+a2x2+a4x+a6 with the conductorNE.
This procedure stops if L(m)E (1) is not zero to two-bit precision, and then outputs the analytic rank of the elliptic curve rE =m. For elliptic curves with large conductors, we give an upper bound and a lower bound on the rank. In the last section, we have compiled a list of minor errors and typos in his thesis.
Basic Denitions and Backgrounds
For elliptic curveL functions, the points = 1 is called the critical point, and the line <(s) = 1 is called the critical line. The analytic rank of E, denoted byran(E), is defined by the first non-zero coefficient of the Taylor series of L(E, s) at s= 1. According to the Mordell-Weil theorem, the group of rational points becomes onE isE(Q)∼=Etor(Q)×Zr and the algebraic rank of an elliptic curve isral(E) =r, which is the exponent of the non-torque part.
According to the BSD conjecture, the analytic rank and the algebraic rank are exactly equal. Also, the value of the first non-zero coecient of the L series at the central point is calculated by .
The Gamma function
Now through the change of variable, let nt = τ and repeating the integration of parts, we can get. The first and second conditions will be obtained directly by the resulting formula, and the third condition - convexity of log(Γ(s)) can be obtained by denition of convexity and some calculations. Using Theorem III.4, we can make the recursive relation of the Gamma function wider to the whole real line.
The simple poles of the gamma function can also be found by the recursive relation.
A relation to the Riemann zeta function
The Digamma function
L-function
Like the Digamma function, the logarithmic derivative of the augmented L-function will be useful later. If we are not on the critical strip, we can roughly estimate the series L and its logarithmic derivative.
Elliptic curve and Torus
Weierstrass ℘ - function
So ℘0(s) has a triple pole at 0, and by further calculation we can delete all reciprocal terms as follows. First, we want to show that for a point of the elliptic curve Φ(si) = Pi = (xi, yi) the given Φ is a group homomorphism. The homothety classes of networks are in one-to-one correspondence with the C-isomorphism classes of elliptic curves.
The latticeΛ has a basis(ω1, ω2) proper (ie ω1 ∈R) and the real part of ω2 is either 0 or ω1/2 corresponding to the case that the discriminant DE >0 or DE <0 respectively.
The Real period
It can be useful to add the zeros in a specific way and analyze its behavior to get some information about the distribution of the zeros. The lower half plane is not our concern, since conjugates of zeros are also zeros of the zeta function. By Theorem 5.2, the reciprocal sum of non-trivial zeros converges, so the summation of each power of the reciprocals will also converge.
Naturally, we extend this idea to non-trivial zeros of the L-series for an elliptic curve. Under the same philosophy of the Hadamard product, we can obtain the representation of the completed L-function as a product over its zeros, and the logarithmic derivative of the completed L-function as a sum over its zeros. This quantity is deeply related to the completed L-function and the rank of an elliptic curve.
And the bite can be a way to represent the exact value of the analytical rank. Mordell's theorem tells us that the elliptic curve over Q is isomorphic to the product of the torsion points part and the free part. For a non-zero point P on E, x(P) = pq denotes the reduced fraction of the first coordinate of P, assuming q >0.
Then the left side is simply calculated by the Double-and-Add numerical method, and the right side is more simple. And the remaining thing is only a shorthand classification (for example, whether h(P) = log|p|or not). Through the pairing made by the N´eron-Tate height function, the quotient of the Mordell-Weil group E(Q)/Etor(Q) is embedded in Rr where is the algebraic rank of an elliptic curve.
Since we started with a Mordell-Weil group coefficient basis, the value of the regulator does not depend on the choice of generators. Since P is a generator and ˆh is a quadratic form, the regulator is the smallest positive height of the curve. Note that the regulator is defined as the lattice covolume under the N´eron-Tate coupling map.
Note that the basis of the first factor is definitely less than 1, and the Gamma function is increasing since NE >350000, so the inequality still holds for the upper order bound. As the conductance increases, the regulator limits decrease as the conductance power is negative and the gamma value of logNE is in the denominator, so it almost obeys a factorial scale.
The analytic algorithm
Some bounds of the rank
Using this theorem, eventually we get some bounds of bite, rank and principal coefficient. With these inequalities, we can approximate the rate of growth of those quantities to the logarithmic rate of the conductor of E. Let CE0 be the leading coefficient of the Taylor series of the completed L-function, then.
If m has the same parity as E, the derivative of the completed L function is at the central point. BS15a] , Binary quartic forms with bounded invariants, and the boundedness of the average rank of elliptic curves, Annals of Mathematics. BS15b], Ternary cubic forms with bounded invariants, and the existence of positive proportion elliptic curves of rank 0, Annals of Mathematics.
Elk06] Noam D Elkies, Points of low altitude on elliptic curves and d i surfaces: Elliptic surfaces overP1 with small d, International Algorithmic Number Theory Symposium, Springer, 2006, p. IJT14] Özlem Imamo glu, Jonas Jermann, and on the zeros of E2, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. LLL+97] Serge Lang, Serge Lang, Serge Lang, France Mathematician, Serge Lang, and France Mathématicien, Survey of Diophantine geometry, vol.
MS13] Barry Mazur and William Stein, How Explicit is the Explicit Formula?, preprint, available at http://www. Spi15] Simon Vernon Bok Spicer, The Zeros of Elliptic Curve l-Functions: Analytical Algorithms with Explicit Time Complexity, Ph.D. My sincere thanks go to my advisor Peter Jae-Hyun Cho, who provided many mathematical teachings, philosophies, and personal mentorships to bring this article to fruition.
I would also like to thank the other committee members for reading my dissertation, Hae-Sang Sun and Chol Park.
