On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime
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 \S2. 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 \({\mathbb Q}_pT\). 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 (\S5). 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 \S6 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 \(X_0(32)\).
scientific article; zbMATH DE number 1990939