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On the $$p$$-adic $$L$$-function of a modular form at a supersingular prime. (English) Zbl 1074.11061
In this important paper the author shows what the “arithmetically significant” part of the $$p$$-adic $$L$$-function attached to a modular form $$f$$ of weight $$k$$ should be in the supersingular case. Actually one has to deal with two functions indexed by + and $$-$$. For instance when $$f$$ comes from a (modular) elliptic curve $$E$$ with $$a_p=0$$ (which is, for $$p>3$$, tantamount to supersingularity at $$p$$), and $$\alpha$$ is a root of $$x^2+p$$, then $$L_p(E,\alpha,T) = G^+(T) + G^-(T)\alpha$$ where $$G^\pm(T)$$ has $$p$$-adic rational coefficients. From the interpolation approach to $$p$$-adic $$L$$-functions it follows that $$G^\pm(T)$$ has an infinite series of trivial zeros (in the supersingular case which we assume from now on), and this actually implies that $$L_p(E,\alpha,T)$$ has infinitely many zeros as well, even though we don’t see them immediately. (These observations are due to B. Perrin-Riou.) A similar result holds for $$L$$-functions involving twists. The task is to eliminate these predictable zeros.
The paper carefully reviews the construction of $$p$$-adic $$L$$-functions, following Vishik and others, in §2. Then the author proceeds to modify $$G^\pm(T)$$ so as to eliminate the trivial zeros. To make this beautiful idea work, he defines two power series (of necessity, with unbounded denominators in their coefficients) $$\log^+_p$$ and $$\log^-_p$$, the so-called half-logarithms, whose zeros are exactly the trivial zeros of $$G^\pm(T)$$. For $$k=2$$, these half-logs live in $${\mathbb Q}_p[[T]]$$. In general, they involve an extra integral parameter $$j=0,\ldots,k-2$$. The definition of the half-logs is quite elegant: one takes the product of all $$p^n$$-th cyclotomic polynomials with $$n$$ of a fixed parity, each polynomial divided by $$p$$. Recall that these cyclotomic polynomials are Eisenstein in $$T$$ with constant coefficient exactly $$p$$; so the division by $$p$$ changes this into 1, and one sees at least that the constant term of the infinite product makes sense. It is then shown that $$L^\pm_p(f,T)=G^\pm(T)/\log_p^\pm(T)$$ has bounded coefficients (we over-simplify notation), and hence only finitely many zeros, by Weierstraß preparation. This is the main result (§5). It is shown that if $$f$$ comes from the elliptic curve $$E$$, then $$L^\pm_p(f,T)=L^\pm_p(E,T)$$ has no denominators at all. In §6 this is applied in order to determine the growth of the analytical order of the $$p$$-part of the Tate-Shafarevich groups in a cyclotomic tower. (Under the B-SD conjecture, the analytical order is just the order.) This is Proposition 6.10. The long sums of $$p$$-powers correspond to the trivial zeros of the $$L$$-functions and are responsible for the guaranteed doubly exponential growth, and the $$\lambda$$ and $$\mu$$ terms are the usual Iwasawa invariants of $$L^\pm_p(E,T)$$. In the particular case that the quotient $$L(E,1)/\Omega_E$$ is a $$p$$-unit, these invariants all vanish and one recaptures (part of) a result of Kurihara. Here, remarkably enough, the growth no longer depends on the curve $$E$$! Actually Kurihara shows more in this situation: he proves the relevant case of B-SD, equating analytic and algebraic order.
The paper under review concludes with some very interesting numerical results obtained with the system MAGMA. They describe nontrivial roots and $$\mu$$-invariants of the $$3$$-adic $$L$$-functions attached to twists of the curve $$X_0(32)$$.

##### MSC:
 11R23 Iwasawa theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
##### Keywords:
L-functions; elliptic curves; trivial zeros; Iwasawa invariants
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##### References:
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