59 research outputs found
Ordinary deformations are unobstructed in the cyclotomic limit
The deformation theory of ordinary representations of the absolute Galois
groups of totally real number fields (over a finite field ) has been studied
for a long time, starting with the work of Hida, Mazur and Tilouine, and
continued by Wiles and others. Hida has studied the behaviour of these
deformations when one considers the -cyclotomic tower of extensions of the
field. In the limit, one obtains a deformation ring classifying the ordinary
deformations of the (Galois group of) the -cyclotomic extension. We show
that if this ring in Noetherian (a natural assumption considered by Hida) it is
free over the ring of Witt vectors of . This however imposes natural
conditions on certain -invariants
Horizontal non-vanishing of Heegner points and toric periods
Let be a totally real field and a modular \GL_2-type
abelian variety over . Let be a CM quadratic extension. Let be
a class group character over such that the Rankin-Selberg convolution
is self-dual with root number . We show that the number of
class group characters with bounded ramification such that increases with the absolute value of the discriminant of .
We also consider a rather general rank zero situation. Let be a
cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let
be a Hecke character over such that the Rankin-Selberg convolution
is self-dual with root number . We show that the number of
Hecke characters with fixed -type and bounded ramification such
that increases with the absolute value of the
discriminant of .
The Gross-Zagier formula and the Waldspurger formula relate the question to
horizontal non-vanishing of Heegner points and toric periods, respectively. For
both situations, the strategy is geometric relying on the Zariski density of CM
points on self-products of a quaternionic Shimura variety. The recent result
\cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental
to the approach.Comment: Adv. Math., to appear. arXiv admin note: text overlap with
arXiv:1712.0214
Horizontal variation of Tate--Shafarevich groups
Let be an elliptic curve over . Let be an odd prime and
an embedding. Let
be an imaginary quadratic field and the corresponding Hilbert class
field. For a class group character over , let be
the field generated by the image of and the prime
of above determined via . Under mild
hypotheses, we show that the number of class group characters such that
the -isotypic Tate--Shafarevich group of over is finite with
trivial -part increases with the absolute value of the
discriminant of
The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication
Let be an elliptic curve over a number field with complex
multiplication by the ring of integers in an imaginary quadratic field . We
give a complete proof of the conjecture of Birch and Swinnerton-Dyer for ,
as well as its equivariant refinement formulated by Gross, under the assumption
that and that is abelian. We also prove
analogous results for CM abelian varieties .Comment: To appear in Cambridge Journal of Mat
On the non-triviality of the -adic Abel-Jacobi image of generalised Heegner cycles modulo , I: modular curves
Generalised Heegner cycles are associated to a pair of an elliptic Hecke
eigenform and a Hecke character over an imaginary quadratic extension K/\Q.
Let be an odd prime split in K/\Q and an odd unramified prime.
We prove the non-triviality of the -adic Abel-Jacobi image of generalised
Heegner cycles modulo over the -anticylotomic extension of . The
result is an evidence for the refined Bloch-Beilinson and the Bloch-Kato
conjecture. In the case of two, it provides a refinement of the results of
Cornut and Vatsal on the non-triviality of Heegner points over the
-anticylotomic extension of .Comment: J. Alg. Geom., to appea
On the non-vanishing of -adic heights on CM abelian varieties, and the arithmetic of Katz -adic -functions
Let be a simple CM abelian variety over a CM field , a rational
prime. Suppose that has potentially ordinary reduction above and is
self-dual with root number . Under some further conditions, we prove the
generic non-vanishing of (cyclotomic) -adic heights on along
anticyclotomic -extensions of . This provides evidence towards
Schneider's conjecture on the non-vanishing of -adic heights. For CM
elliptic curves over \Q, the result was previously known as a consequence of
work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based
on non-vanishing results for Katz -adic -functions and a Gross--Zagier
formula relating the latter to families of rational points on .Comment: Ann. Inst. Fourier, to appea
Anticyclotomic Iwasawa theory of abelian varieties of -type at non-ordinary primes
Let be a prime number, an elliptic curve with good
supersingular reduction at and an imaginary quadratic field such that
the root number of over is . When is split in , Darmon and
Iovita formulated the plus and minus Iwasawa main conjectures for over the
anticyclotomic -extension of , and proved one-sided inclusion:
an upper bound for plus and minus Selmer groups in terms of the associated
-adic -functions. We generalize their results to two new settings:
1. Under the assumption that is split in but without assuming
, we study Sprung-type Iwasawa main conjectures for abelian varieties
of -type, and prove an analogous inclusion.
2. We formulate, relying on the recent work of the first named author with
Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves
when is inert in , and prove an analogous inclusion.Comment: 43 page
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