Equivariant Tamagawa number conjecture for Abelian varieties over global fields of positive characteristic

Trihan, Fabien; Vauclair, David

doi:10.1090/proc/14417

Equivariant Tamagawa number conjecture for Abelian varieties over global fields of positive characteristic
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by Fabien Trihan and David Vauclair PDF

Proc. Amer. Math. Soc. 149 (2021), 3601-3611 Request permission

Abstract:

We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic $p>0$ under a semisimplicity hypothesis.

References

D. Burns and M. Kakde, private communications.
David Burns and Otmar Venjakob, On the leading terms of zeta isomorphisms and $p$-adic $L$-functions in non-commutative Iwasawa theory, Doc. Math. Extra Vol. (2006), 165–209. MR 2290587
David Burns and Otmar Venjakob, On descent theory and main conjectures in non-commutative Iwasawa theory, J. Inst. Math. Jussieu 10 (2011), no. 1, 59–118. MR 2749572, DOI 10.1017/S147474800900022X
John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob, The $\rm GL_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208. MR 2217048, DOI 10.1007/s10240-004-0029-3
Richard Crew, $L$-functions of $p$-adic characters and geometric Iwasawa theory, Invent. Math. 88 (1987), no. 2, 395–403. MR 880957, DOI 10.1007/BF01388914
Takako Fukaya and Kazuya Kato, A formulation of conjectures on $p$-adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII, Amer. Math. Soc. Transl. Ser. 2, vol. 219, Amer. Math. Soc., Providence, RI, 2006, pp. 1–85. MR 2276851, DOI 10.1090/trans2/219/01
Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
Kazuya Kato and Fabien Trihan, On the conjectures of Birch and Swinnerton-Dyer in characteristic $p>0$, Invent. Math. 153 (2003), no. 3, 537–592. MR 2000469, DOI 10.1007/s00222-003-0299-2
Finn F. Knudsen, Determinant functors on exact categories and their extensions to categories of bounded complexes, Michigan Math. J. 50 (2002), no. 2, 407–444. MR 1914072, DOI 10.1307/mmj/1028575741
King Fai Lai, Ignazio Longhi, Ki-Seng Tan, and Fabien Trihan, The Iwasawa main conjecture for semistable abelian varieties over function fields, Math. Z. 282 (2016), no. 1-2, 485–510. MR 3448392, DOI 10.1007/s00209-015-1550-4
King Fai Lai, Ignazio Longhi, Ki-Seng Tan, and Fabien Trihan, The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields, Proc. Lond. Math. Soc. (3) 112 (2016), no. 6, 1040–1058. MR 3537332, DOI 10.1112/plms/pdw019
Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
Tadashi Ochiai and Fabien Trihan, On the Selmer groups of abelian varieties over function fields of characteristic $p>0$, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 23–43. MR 2461865, DOI 10.1017/S0305004108001801
I. Shafarevitch, On $p$-extensions, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 351–363 (Russian, with English summary). MR 0020546
Christopher Skinner and Eric Urban, The Iwasawa main conjectures for $\rm GL_2$, Invent. Math. 195 (2014), no. 1, 1–277. MR 3148103, DOI 10.1007/s00222-013-0448-1
R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634
Ki-Seng Tan, Selmer groups over $\Bbb {Z}_p^d$-extensions, Math. Ann. 359 (2014), no. 3-4, 1025–1075. MR 3231024, DOI 10.1007/s00208-014-1023-9
F. Trihan and D. Vauclair, A comparison theorem for semi-Abelian schemes over a smooth curve, arXiv:1505.02942, 2015.
Fabien Trihan and David Vauclair, On the non commutative Iwasawa main conjecture for abelian varieties over function fields, Doc. Math. 24 (2019), 473–522. MR 3960122