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Abstract LetKbe an imaginary quadratic field and$$p\geq 5$$a rational prime inert inK. For a$$\mathbb {Q}$$-curveEwith complex multiplication by$$\mathcal {O}_K$$and good reduction atp, K. Rubin introduced ap-adicL-function$$\mathscr {L}_{E}$$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$$\mathscr {L}_{E}$$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE. A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic$${\mathbb {Z}}_p$$-extension$$\Psi _\infty $$of the unramified quadratic extension of$${\mathbb {Q}}_p$$. Along the way, we present a theory of local points over$$\Psi _\infty $$of the Lubin–Tate formal group of height$$2$$for the uniformizing parameter$$-p$$.
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This content will become publicly available on January 1, 2026
Asymptotic second moment of Dirichlet L -functions along a thin coset
Abstract We prove an asymptotic formula for the second moment of central values of DirichletL-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulodinside the full group of characters moduloq. Suppose that$$\nu _p(d) \geq \nu _p(q)/2$$for all primespdividingq. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size$$q^{1/2}$$here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smallerd, with$$\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$$, with a power-saving error term fordlarger than$$q^{2/5}$$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughlyd, which is only slightly smaller than the diagonal main term.
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- Award ID(s):
- 2302210
- PAR ID:
- 10632919
- Publisher / Repository:
- Forum of Mathematics, Sigma
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 13
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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