1 Background

In [3], Mazur and Swinnerton-Dyer constructed a p-adic L-series attached to a modular elliptic curve E/Q for each prime p of good, ordinary reduction. This L-series is a power series in Z_{p}[[T]] that interpolates the algebraic part of special values of the complex L-series of E twisted by various characters. As this power series has bounded coefficients, by the p-adic Weierstrauss preparation theorem, one can associated to it a well-defined polynomial part that encodes all of the zeroes of the L-series. The Main Conjecture for elliptic curves relates this polynomial to the characteristic polynomial of the dual of the Selmer group of E. This is a very deep conjecture as it relates the analytic data of the p-adic L-series to a polynomial built up out of the arithmetic data of E.

In [1] and [5], the construction of these p-adic L-series was generalized both to higher weight modular forms and to primes of supersingular reduction. The cost of allowing non-ordinary primes was that the resulting L-series is no longer guaranteed to have bounded coefficients and hence could have infinitely many zeroes. Also, in the supersingular case, the construction yields two p-adic Lseries, one for each root of x^{2} − a_{p}x + p^{k−1}. Here a_{p} is the eigenvalue of the p-th Hecke operator and k is the weight of the modular form.

In the case where the slopes of the two roots of this equation are distinct, Mazur has shown that one of the two L-series does indeed have infinitely many zeroes. When a_{p} = 0 and hence necessarily the two slopes are the same, Greenberg has shown that both of the L-series have infinitely many zeroes. The arithmetic nature of these zeroes are not yet well understood especially in the context of a main conjecture.

2 Results

The main result I have shown in my thesis explains the infinitude of zeroes in the case a_{p} = 0 by describing them as arising from an infinite family of trivial zeroes and finitely many more arithmetically interesting zeroes. I have worked this out for arbitrary weight, but for ease of exposition, I will explain this only for elliptic curves.

Let E be an elliptic curve over Q with a_{p} = 0. Denote by α and \overline{α} the two roots of x^{2} + p = 0. We then have two p-adic L-series, L_{α}(T), L\overline{α}(T) ∈ Q_{p}(α)[[T]]. Write,

L_{α}(T) = G(T) + H(T) · α with G, H ∈ Q_{p}[[T]].

The interpolation property defining L_{α} and L\overline{α} forces G to vanish at ζ_{p^{2n}} − 1 and H to vanish at ζ_{p^{2n−1}} − 1 for all n ≥ 1 where ζ_{p^{k}} is a p^{k}-th root of unity in \overline{Q}_{p}. Hence, the power series G and H have an infinite set of “trivial” zeroes. Greenberg uses this observation to prove that the two L-series have infinitely many zeroes.

Building on this argument, I am able to construct power series Φ^{+} and Φ^{−} that vanish precisely at the forced zeroes of G and H respectively. These are constructed as an infinite product of cyclotomic polynomials. The next step is to examine the functions g = G/Φ^{+} and h = H/Φ^{−}. The relation of Φ^{+} and Φ^{−} to L_{α}(T) is somewhat analogous to the relation of the gamma function to the Riemann zeta function. The gamma function forces the zeta function to vanish at all of the negative even integers and the interesting zeroes of the zeta function are discovered after these zeroes are removed from consideration. In our setting, we divide G and H by Φ^{+} and Φ^{−} respectively in hoping to uncover their more interesting zeroes. For this reason, I refer to Φ^{+} and Φ^{−} as gamma-like functions.

By studying the rate of growth of G,H,Φ^{+} and Φ^{−}, one sees that g and h are bounded functions and actually have integral coefficients. Hence

L_{α}(T) = g(T) · Φ^{+}(T) + h(T) · Φ^{−}(T) · α.

with g, h ∈ Z_{p}[[T]]. With this description of the p-adic L-series, we see that its zeroes are controlled by the infinite family of zeroes of Φ^{+} and Φ^{−} along with the finite collection of zeroes of g and h.

Let K_{∞} be the cyclotomic Z_{p} extension of Q. The above result can be used to study the analytic invariants of an elliptic curve E along this extension. For example, it is a long standing conjecture that E(K_{∞}) is a finitely generated group. By the Birch and Swinnerton-Dyer conjecture, this should translate into E(K_{∞}) having finite analytic rank. The above result implies that its analytic rank is bounded by the sum of the lambda invariants of g and h.

Also, I can compute asymptotic formulas for the the analytic size of the Tate-Shafarevich group along K_{∞}. Here analytic size means the size as predicted by the Birch and Swinnerton-Dyer conjecture. These sizes grow without bound along K_{∞} and depend upon the Iwasawa invariants of g and h.

Recently, Kurihara [2] has produced formulas for the algebraic size of the Tate-Shafarevich group at a supersingular prime by using Kato’s Euler system. He does this under some strong hypotheses that for instance force my g and h to be invertible and hence to have all of their Iwasawa invariants equal to zero. In this case, his algebraic formulas and my analytic formulas agree perfectly as predicted by the Birch and Swinnerton-Dyer conjecture.

3 Proposed Research

The above research was done in the case where a_{p} = 0 which is a special subcase of the “most” supersingular case (i.e. when the slopes of the two roots of x^{2} − a_{p}x + p^{k−1} are equal). The hypothesis a_{p} = 0 was the key input that caused the trivial zeroes to take on a very simple form. It is reasonable to expect in the most supersingular case that even when a_{p} does not vanish, the functions G and H will still have an an infinite family of trivial zeroes. I hope to be able to understand these zeroes so that I can build the corresponding Φ^{+} and Φ^{−}. With these more general gamma-like functions in hand, I hope to be able to prove a result as in the case where a_{p} = 0.

I have collected some numerical data that strongly suggests that the above is possible. In the elliptic curve case, the only time when p is supersingular and a_{p} ≠ 0 is when p = 2 or 3. I have calculated several examples of these G and H functions when p = 3 and a_{3} ≠ 0. In computing the newton polygons of G (resp. H), it appears that for every root of Φ^{+} (resp. Φ^{−}) there is a corresponding root of G (resp. H) with the same valuation. Furthermore, it appears that G and H have only finitely many additional zeroes beyond this family of zeroes described above. So numerically this situation seems quite similar to the a_{p} = 0 case.

In the unequal slope case, as mentioned earlier, Mazur has proved that one of the two p-adic L-series has infinitely many zeroes. However, his proof does not shed any light on which of the two L-series has this property. I have done some calculations in this higher weight case and it appears that both L-series have infinitely many zeroes. Furthermore, the newton polygons of the L-series carry a very regular pattern which is also true in the a_{p} = 0 case. For this reason, I hope to find some corresponding theory in terms of Φ^{+} and Φ^{−} in this setting. Barry Mazur has expressed great hope that this kind of approach to studying L-series constructed from unbounded distributions will work in a very general setting.

I also hope to study more closely the g and h functions that appear in the elliptic curve case. The Iwasawa invariants of these functions control the growth of the Tate-Shafarevich group as discussed earlier. For this reason, they should have some interpretation on the algebraic side. If I am able to work at the University of Washington, I hope to be able to discuss these ideas with Prof. Greenberg as he is an expert on Iwasawa theory of elliptic curves. One large goal would be to understand in some way the algebraic nature of these two analytically defined functions.

References

[1] Y. Amice and J. Vélu, Distributions p-adiques associées aux séries de Hecke. (French), in Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), 119–131. Astérisque, Nos. 24-25, Soc. Math. France, Paris, 1975.

[2] M. Kurihara, On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, preprint.

[3] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil Curves, Invent. Math. 25 (1974), 1–61.

[4] B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1–48.

[5] M. M. Višik, Nonarchimedean measures associated with Dirichlet series. (Russian) Mat. Sb. (N.S.) 99(141) (1976), no. 2, 248–260.