55 research outputs found
Critical slope p-adic L-functions of CM modular forms
For ordinary modular forms, there are two constructions of a p-adic
L-function attached to the non-unit root of the Hecke polynomial, which are
conjectured but not known to coincide. We prove this conjecture for modular
forms of CM type, by calculating the the critical-slope L-function arising from
Kato's Euler system and comparing this with results of Bellaiche on the
critical-slope L-function defined using overconvergent modular symbols.Comment: 14 page
-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
Let be an odd prime. Given an imaginary quadratic field
where splits with , and a -ordinary
newform such that verifies the Heegner hypothesis
relative to , we prove a -adic Gross-Zagier formula for the critical
slope -stabilization of (assuming that it is non--critical). In
the particular case when is the newform of weight associated to an
elliptic curve that has good ordinary reduction at , this allows us to
verify a conjecture of Perrin-Riou. The -adic Gross-Zagier formula we prove
has applications also towards the Birch and Swinnerton-Dyer formula for
elliptic curves of analytic rank one.Comment: 35 pages, minor updates. Comments most welcome
Plus/minus p-adic L-functions for GL2n
We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automorphic representations of GL2n using the p-adic L-functions constructed in work of Barrera Salazar et al. (On p-adic l-functions for GL2n in finite slope shalika families, 2021). We use these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor
On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of
a modular form f over the cyclotomic Zp-extension of Q under the assumption
that f is non-ordinary at p. In particular, we give upper bounds of these
groups in terms of Iwasawa invariants of Selmer groups defined using p-adic
Hodge Theory. These bounds have the same form as the formulae of Kobayashi,
Kurihara and Sprung for supersingular elliptic curves.Comment: To appear in Canad. J. Mat
Dieudonne crystals and Wach modules for p-divisible fgroups
Let be a perfect field of characteristic and an extension of
contained in some . Using crystalline
Dieudonn\'e theory, we provide a classification of -divisible groups over
in terms of finite height -modules over
. Although such a classification is a consequence of
(a special case of) the theory of Kisin--Ren, our construction gives an
independent proof and allows us to recover the Dieudonn\'e crystal of a
-divisible group from the Wach module associated to its Tate module by
Berger--Breuil or by Kisin--Ren
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