Glenn Stevens
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{\bf Title:} $p$-adic Analytic Families and $p$-Adic Periods of Modular
Forms
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{\bf Abstract:} In 1985, Mazur, Tate, and Teitelbaum performed numerical
experiments to test a $p$-adic analog of the conjecture of Birch and
Swinnerton-Dyer. To their initial surprise, they discovered that the
order of vanishing at the center of the critical strip of the $p$-adic
$L$-function of a newform that is ``split-multiplicative at $p$" has a
parity that is opposite to that of the order of vanishing of the complex
$L$-function. Indeed, in this case they conjectured that the
$p$-adic $L$-function has an ``extra zero" at the center of the critical
strip and that this extra zero is accompanied by an ``extra factor" in
the leading Taylor coefficient, which they called the $\cal L$-invariant
of the newform. In case the weight of the newform is two and the complex
$L$-function does not vanish at the center of the critical strip, this
conjecture was proved by Ralph Greenberg and the speaker.
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This lecture will describe the higher weight conjecture of Mazur,
Tate, and Teitelbaum, with Coleman's $\cal L$-invariant and outline a
proof that makes essential use of Coleman's construction of $p$-adic
analytic families of modular forms. More precisely, let $f$ be a
classical newform of weight $k_0+2\ge 2$ and assume that $f$ is split
multiplicative at $p$, i.e., that $f$ has level $Np$ where $p\not|N$ and
that $a_p(f) = p^{k_0/2}$. Using his theory of $p$-adic integration,
Coleman defined an $\cal L$-invariant ${\cal L}(f)$ which he conjectured
should be equal to the Mazur-Tate-Teitelbaum $\cal L$-invariant. The
purpose of this note is to outline a proof of Coleman's conjecture. More
precisely we will prove the following theorem.
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{\bf Theorem.} $L_p^\prime(f, 1+k_0/2) = {\cal L}(f)\cdot
L_\infty(f,1+k_0/2)$.
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As an interesting consequence of the proof we show how monodromy on the
filtered module associated to $f$ can be described in terms of the
derivative of the eigenvalue $a_p$ along a $p$-adic analytic family of
modular forms through $f$.
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Robert F. Coleman
Last modified: Mon Mar 8 10:11:49 PST