Non-abelian congruences between L -values of elliptic curves

Daniel Delbourgo[1]; Tom Ward[2]

  • [1] Monash University School of Mathematical Sciences Victoria 3800 (Australia)
  • [2] University of Nottingham School of Mathematical Sciences Nottingham NG7 2RD (United Kingdom)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 3, page 1023-1055
  • ISSN: 0373-0956

Abstract

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Let E be a semistable elliptic curve over ℚ . We prove weak forms of Kato’s K 1 -congruences for the special values L 1 , E / ℚ ( μ p n , Δ p n ) . More precisely, we show that they are true modulo p n + 1 , rather than modulo p 2 n . Whilst not quite enough to establish that there is a non-abelian L -function living in K 1 ℤ p [ [ Gal ( ℚ ( μ p ∞ , Δ p ∞ ) / ℚ ) ] ] , they do provide strong evidence towards the existence of such an analytic object. For example, if n = 1 these verify the numerical congruences found by Tim and Vladimir Dokchitser.

How to cite

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Delbourgo, Daniel, and Ward, Tom. "Non-abelian congruences between $L$-values of elliptic curves." Annales de l’institut Fourier 58.3 (2008): 1023-1055. <http://eudml.org/doc/10331>.

@article{Delbourgo2008,
abstract = {Let $E$ be a semistable elliptic curve over $\mathbb\{Q\}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $ L\bigl (1, E/\mathbb\{Q\}( \mu _\{p^n\},\@root p^n \of \{\Delta \} )\bigr ) . $ More precisely, we show that they are true modulo $p^\{n+1\}$, rather than modulo $p^\{2n\}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\bigl ( \mathbb\{Z\}_p[[ \rm \{Gal\} ( \mathbb\{Q\}( \mu _\{p^\infty \},\!\!\@root p^\infty \of \{\Delta \} )/\mathbb\{Q\}) ]] \bigr )$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.},
affiliation = {Monash University School of Mathematical Sciences Victoria 3800 (Australia); University of Nottingham School of Mathematical Sciences Nottingham NG7 2RD (United Kingdom)},
author = {Delbourgo, Daniel, Ward, Tom},
journal = {Annales de l’institut Fourier},
keywords = {Iwasawa theory; modular forms; $p$-adic $L$-functions; -values; -adic -function; congruences},
language = {eng},
number = {3},
pages = {1023-1055},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-abelian congruences between $L$-values of elliptic curves},
url = {http://eudml.org/doc/10331},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Delbourgo, Daniel
AU - Ward, Tom
TI - Non-abelian congruences between $L$-values of elliptic curves
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 3
SP - 1023
EP - 1055
AB - Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $ L\bigl (1, E/\mathbb{Q}( \mu _{p^n},\@root p^n \of {\Delta } )\bigr ) . $ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\bigl ( \mathbb{Z}_p[[ \rm {Gal} ( \mathbb{Q}( \mu _{p^\infty },\!\!\@root p^\infty \of {\Delta } )/\mathbb{Q}) ]] \bigr )$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.
LA - eng
KW - Iwasawa theory; modular forms; $p$-adic $L$-functions; -values; -adic -function; congruences
UR - http://eudml.org/doc/10331
ER -

References

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  2. Thanasis Bouganis, Vladimir Dokchitser, Algebraicity of L -values for elliptic curves in a false Tate curve tower, Math. Proc. Cambridge Philos. Soc. 142 (2007), 193-204 Zbl1214.11080MR2314594
  3. John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, Otmar Venjakob, The GL 2 main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. (2005), 163-208 Zbl1108.11081MR2217048
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  8. Shai Haran, p -adic L -functions for modular forms, Compositio Math. 62 (1987), 31-46 Zbl0618.10027MR892149
  9. Kazuya Kato, K 1 of some non-commutative completed group rings, -Theory 34 (2005), 99-140 Zbl1080.19002MR2180109
  10. Alexey A. Panchishkin, Non-Archimedean L -functions of Siegel and Hilbert modular forms, 1471 (1991), Springer-Verlag, Berlin Zbl0732.11026MR1122593
  11. Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de Gal ( Q ¯ / Q ) , Duke Math. J. 54 (1987), 179-230 Zbl0641.10026MR885783
  12. Goro Shimura, Corrections to: “The special values of the zeta functions associated with Hilbert modular forms” [Duke Math. J. 45 (1978), no. 3, 637–679, Duke Math. J. 48 (1981) Zbl0394.10015
  13. Glenn Stevens, Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math. 98 (1989), 75-106 Zbl0697.14023MR1010156
  14. J. Tate, Number theoretic background, Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (1979), 3-26, Amer. Math. Soc., Providence, R.I. Zbl0422.12007MR546607

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