On Iwasawa 𝜆-invariants for abelian number fields and random matrix heuristics

Delbourgo, Daniel; Knospe, Heiko

doi:10.1090/mcom/3823

On Iwasawa $\lambda$-invariants for abelian number fields and random matrix heuristics
HTML articles powered by AMS MathViewer

by Daniel Delbourgo and Heiko Knospe HTML | PDF

Math. Comp. 92 (2023), 1817-1836 Request permission

Abstract:

Following both Ernvall-Metsänkylä and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\chi$ of any order. We are interested in two cases: (i) the character $\chi$ is fixed and the prime $p$ varies, and (ii) $ord(\chi )$ and the prime $p$ are both fixed but $\chi$ is allowed to vary. We predict distributions for these $\lambda$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $\chi$-regular primes, which depends on how $p$ splits inside $\mathbb {Q}(\chi )$. Finally, we tabulate the values of the $\lambda$-invariant for every character $\chi$ of conductor $\leqslant 1000$ and for odd primes $p$ of small size.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Gilyoung Cheong and Yifeng Huang, Cohen-Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields, Illinois J. Math. 65 (2021), no. 2, 385–415. MR 4278684, DOI 10.1215/00192082-8939615
Nancy Childress and Robert Gold, Zeros of $p$-adic $L$-functions, Acta Arith. 48 (1987), no. 1, 63–71. MR 893462, DOI 10.4064/aa-48-1-63-71
Daniel Delbourgo, A Dirichlet series expansion for the $p$-adic zeta-function, J. Aust. Math. Soc. 81 (2006), no. 2, 215–224. MR 2267793, DOI 10.1017/S1446788700015846
Daniel Delbourgo, The convergence of Euler products over $p$-adic number fields, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 3, 583–606. MR 2546633, DOI 10.1017/S0013091507000636
Daniel Delbourgo and Qin Chao, On $\lambda$-invariants attached to cyclic cubic number fields, LMS J. Comput. Math. 18 (2015), no. 1, 684–698. MR 3433892, DOI 10.1112/S1461157015000224
Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
Jordan S. Ellenberg, Sonal Jain, and Akshay Venkatesh, Modeling $\lambda$-invariants by $p$-adic random matrices, Comm. Pure Appl. Math. 64 (2011), no. 9, 1243–1262. MR 2839300, DOI 10.1002/cpa.20375
L. Euler, Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem, Acta Acad. Scientiarum Imperialis Petropolitinae (1783), 47–55.
R. Ernvall and T. Metsänkylä, A method for computing the Iwasawa $\lambda$-invariant, Math. Comp. 49 (1987), no. 179, 281–294. MR 890270, DOI 10.1090/S0025-5718-1987-0890270-X
Reijo Ernvall, Generalized irregular primes, Mathematika 30 (1983), no. 1, 67–73. MR 720950, DOI 10.1112/S002557930001041X
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 227–239. MR 1024565
Derek Garton, Random matrices, the Cohen-Lenstra heuristics, and roots of unity, Algebra Number Theory 9 (2015), no. 1, 149–171. MR 3317763, DOI 10.2140/ant.2015.9.149
Heiko Knospe and Lawrence C. Washington, Dirichlet series expansions of $p$-adic $L$-functions, Abh. Math. Semin. Univ. Hambg. 91 (2021), no. 2, 325–334. MR 4343962, DOI 10.1007/s12188-021-00244-0
James S. Kraft and Lawrence C. Washington, Heuristics for class numbers and lambda invariants, Math. Comp. 76 (2007), no. 258, 1005–1023. MR 2291847, DOI 10.1090/S0025-5718-06-01921-1
Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
Gunter Malle, On the distribution of class groups of number fields, Experiment. Math. 19 (2010), no. 4, 465–474. MR 2778658, DOI 10.1080/10586458.2010.10390636
Xavier-François Roblot, Computing $p$-adic $L$-functions of totally real number fields, Math. Comp. 84 (2015), no. 292, 831–874. MR 3290966, DOI 10.1090/S0025-5718-2014-02889-5
Lawrence C. Washington, Some remarks on Cohen-Lenstra heuristics, Math. Comp. 47 (1986), no. 176, 741–747. MR 856717, DOI 10.1090/S0025-5718-1986-0856717-9
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468
Luochen Zhao, Sum expressions for Kubota-Leopoldt $p$-adic $L$-functions, Proc. Edinb. Math. Soc. (2) 65 (2022), no. 2, 460–479. MR 4449678, DOI 10.1017/S0013091522000177