L-INVARIANT OF p-ADIC L-FUNCTIONS
Haruzo Hida2006
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Related papers
p-adicL-functions andp-adic periods of modular forms
Inventiones Mathematicae, 1993
Non-vanishing modulo p of Hecke L–values and application
L-Functions and Galois Representations
Big Image of Galois Representations Associated with Finite Slope p-adic Families of Modular Forms
Springer Proceedings in Mathematics & Statistics, 2016
Two p-adic L-functions and rational points on elliptic curves with supersingular reduction
L-Functions and Galois Representations
p -adic Rankin L -series and rational points on CM elliptic curves
Pacific Journal of Mathematics, 2012
Non-vanishing modulo p of Hecke L-values
Geometric Aspects of Dwork Theory, 2004
Image of Λ-adic Galois representations modulo p
Inventiones mathematicae, 2012
Let p ≥ 5 be a prime. If an irreducible component of the spectrum of the 'big' ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of SL 2 (Fp[[T ]]) for the canonical "weight" variable T . This fact appears to be deep, as it is almost equivalent to the vanishing of the µ-invariant of the Kubota-Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function. Another key ingredient of the proof is the anticylotomic main conjecture proven by Rubin/Mazur-Tilouine. Fix a prime p ≥ 5, field embeddings and a positive integer N prime to p. We sometimes identify C p and C by a fixed field isomorphism compatible with the above embeddings. Consider the space of modular forms M k+1 (Γ 0 (N p r+1 ), χ) with (p N, r ≥ 0) and cusp forms S k+1 (Γ 0 (N p r+1 ), χ). Let the ring Z[χ] ⊂ C and Z p [χ] ⊂ Q p be generated by the values χ over Z and Z p , respectively. The Hecke algebra for a p-adically closed subring A ⊂ C p containing Z p [χ] (with uniformizer ). Fix a complete discrete valuation ring W finite over Z p inside Q p (though nothing changes even if W is just a complete discrete valuation ring over Z p inside C p ). Our T (p) is often written as U (p) as the level is divisible by p. The ordinary part H k+1,χ/W ⊂ H k+1,χ/W is then the maximal ring direct summand on which U (p) is invertible. We write e for the idempotent of H k+1,χ/W ; so, e is the p-adic limit in H k+1,χ/W of U (p) n! as n → ∞. Via the fixed isomorphism C p ∼ = C, the idempotent e not only acts on the space of modular forms with coefficients in W but also on the classical space M k+1 (Γ 0 (N p r+1 ), χ). We write the image of the idempotent as M ord k+1 for modular forms and S ord k+1 for cusp forms. Let χ 1 = χ N × the tame p-part of χ. Then, by [H86a] and [H86b], we have a unique 'big' Hecke algebra H = H χ1/W such that (H1) H is free of finite rank over Λ : where ω is the Teichmüller character. Here W [ε] ⊂ Q p is the W -subalgebra generated by the values of ε. The corresponding objects for cusp forms are denoted by the corresponding lower case characters; so, h = Z[χ][T (n)|0 < n ∈ Z] ⊂ End(S k+1 (Γ 0 (N p r+1 ), χ)), h k+1,χ/A = h k+1 (Γ 0 (N p r+1 ), χ; A) := h⊗ Z[χ] A, the ordinary part h k+1,χ/W ⊂ h k+1,χ/W and the big cuspidal Hecke algebra h χ1/W . Replacing modular forms by cusp forms (and upper case symbols by
Related topics
Related papers
Local theta-regulators of an algebraic number -- p-adic Conjectures
2017
Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta_p^theta(eta) in F_p for the Q_p-irreducible characters theta of G=Gal(K/Q). Let V_theta be the theta-irreducible representation. A linear representation L^theta=delta.V_theta is associated withDelta_p^theta(eta) whose nullity is equivalent to delta>1 (Theorem 3.9). Each Delta_p^theta(eta) yields Reg_p^theta(eta) modulo p in the factorization ∏_theta (Reg_p^theta(eta))^phi(1) of Reg_p^G(eta) := Reg_p(eta)/p^[K : Q] (normalized p-adic regulator), where phi divides theta is absolutely irreducible.From the probability Prob(Delta_p^theta(eta) = 0 & L^theta=delta.V_theta)<p^(-f.delta^2) (f= residue degree of p in the field of values of phi) and the Borel--Cantelli heuristic, we conjecture that, for p large enough, Reg_p^G(eta) is a p-adic unit or that p^phi(1) divides exactly Reg_p^G(eta) (existence of a single theta with f=delta=...
On the μ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication
Journal of Number Theory, 1987
The main result of this paper proves that the p-invariant is zero for the lwasawa module which arises naturally in the study of p-power descent on an elliptic curve with complex multiplication and good ordinary reduction at the prime p.
A note on p-adic L-functions
Journal of Number Theory, 1976
An elementary proof is given for the existence of the Kubota-Leopoldt padic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical L-functions at positive integers is discussed.
On p-adic L-functions and cyclotomic fields. II
Nagoya Mathematical Journal, 1977
Letpbe a prime. If one adjoins toQallpn-th roots of unity forn= 1,2,3, …, then the resulting field will contain a unique subfieldQ∞such thatQ∞is a Galois extension ofQwith Gal (Q∞/Q)Zp, the additive group ofp-adic integers. We will denote Gal (Q∞/Q) byΓ. In a previous paper [6], we discussed a conjecture relatingp-adicL-functions to certain arithmetically defined representation spaces forΓ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces forΓ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.
The p-adic L-functions of modular elliptic curves
… Unlimited–2001 and Beyond, Engquist, Schmid, …, 2001
Remarks on the λ p -invariants of cyclic fields of degree p
Acta Arithmetica, 2005
The Iwasawaμ-invariant ofp-adic HeckeL-functions
Annals of Mathematics, 2010
For an odd prime p, we compute the -invariant of the anticyclotomic Katz p-adic L-function of a p-ordinary CM field if the conductor of the branch character is a product of primes split over the maximal real subfield. Except for rare cases where the root number of the p-adic functional equation is congruent to 1 modulo p, the invariant vanishes.
On p-adic L-functions and cyclotomic fields
Nagoya Mathematical Journal, 1975
Let p be a prime. If one adjoins to Q all pn -th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q ∞ such that Q ∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q ∞/Q ) by Γ.
Two <i>p</i>-adic <i>L</i>-functions and rational points on elliptic curves with supersingular reduction
Cambridge University Press eBooks, 2007
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.