Arithmetic Transfer for inner forms of $GL_{2n}$
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e128

Voir la notice de l'article provenant de la source Cambridge University Press

We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.
Li, Qirui; Mihatsch, Andreas. Arithmetic Transfer for inner forms of $GL_{2n}$. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e128. doi: 10.1017/fms.2025.10034
@article{10_1017_fms_2025_10034,
     author = {Li, Qirui and Mihatsch, Andreas},
     title = {Arithmetic {Transfer} for inner forms of $GL_{2n}$},
     journal = {Forum of Mathematics, Sigma},
     pages = {e128},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.10034},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10034/}
}
TY  - JOUR
AU  - Li, Qirui
AU  - Mihatsch, Andreas
TI  - Arithmetic Transfer for inner forms of $GL_{2n}$
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e128
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10034/
DO  - 10.1017/fms.2025.10034
ID  - 10_1017_fms_2025_10034
ER  - 
%0 Journal Article
%A Li, Qirui
%A Mihatsch, Andreas
%T Arithmetic Transfer for inner forms of $GL_{2n}$
%J Forum of Mathematics, Sigma
%D 2025
%P e128
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10034/
%R 10.1017/fms.2025.10034
%F 10_1017_fms_2025_10034

[1] Ahsendorf, T., Cheng, C. and Zink, T., ‘-displays and -divisible formal -modules’, J. Algebra 457 (2016), 129–193.10.1016/j.jalgebra.2016.03.002 Google Scholar | DOI

[2] Rad, E. Arasteh and Hartl, U., ‘Local -shtukas and their relation to global -shtukas’, Münster J. Math. 7(2) (2014), 623–670. Google Scholar

[3] Beĭlinson, A., Height Pairing Between Algebraic Cycles in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985) (Contemp. Math.) vol. 67 (Amer. Math. Soc., Providence, RI, 1987), 1–24.10.1090/conm/067/902590 Google Scholar | DOI

[4] Bloch, S., ‘Algebraic cycles and values of -functions’, J. Reine Angew. Math. 350 (1984), 94–108. Google Scholar

[5] Boutot, J.-F. and Carayol, H., ‘Uniformisation -adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld’, Astérisque (1991), 7, 45–158. Google Scholar

[6] Caraiani, A. and Scholze, P., ‘On the generic part of the cohomology of compact unitary Shimura varieties’, Ann. of Math. (2) 186(3) (2017), 649–766.10.4007/annals.2017.186.3.1 Google Scholar | DOI

[7] Ding, Y. W. and Ouyang, Y., ‘A simple proof of Dieudonné–Manin classification theorem’, Acta Math. Sin. 28(8) (2012), 1553–1558.10.1007/s10114-012-0490-8 Google Scholar | DOI

[8] Disegni, D. and Zhang, W., ‘Gan–Gross–Prasad cycles and derivatives of -adic -functions’, Preprint, 2023, http://disegni-daniel.perso.math.cnrs.fr/prtf-ggp.pdf. Google Scholar

[9] Draxl, P. K., Skew Fields (London Mathematical Society Lecture Note Series) vol. 81 (Cambridge University Press, Cambridge, 1983).10.1017/CBO9780511661907 Google Scholar | DOI

[10] Drinfeld, V. G., ‘Coverings of -adic symmetric domains’, Funkcional. Anal. i Priložen. 10(2) (1976), 29–40. Google Scholar

[11] Friedberg, S., Jacquet, H., ‘Linear periods’, J. Reine Angew. Math. 443 (1993), 91–139. Google Scholar

[12] Gan, W. T., Gross, B. and Prasad, D., ‘Symplectic local root numbers, central critical -values, and restriction problems in the representation theory of classical groups, Sur les conjectures de Gross et Prasad. I’, Astérisque 346 (2012), 1–109. Google Scholar

[13] Guo, J., ‘On a generalization of a result of Waldspurger’, Canad. J. Math. 48(1) (1996), 105–142.10.4153/CJM-1996-005-3 Google Scholar | DOI

[14] Gross, B. and Zagier, D., ‘Heegner points and derivatives of -series’, Invent. Math. 84(2) (1986), 225–320.10.1007/BF01388809 Google Scholar | DOI

[15] Hartl, U. and Singh, R. K., ‘Local shtukas and divisible local Anderson modules’, Canad. J. Math. 71(5) (2019), 1163–1207.10.4153/CJM-2018-016-2 Google Scholar | DOI

[16] He, Q., Shi, Y. and Yang, T., ‘Kudla–Rapoport conjecture for Krämer models’, Compos. Math. 159(8) (2023), 1673–1740.10.1112/S0010437X23007273 Google Scholar | DOI

[17] He, Q., Li, C., Shi, Y. and Yang, T., ‘A proof of the Kudla–Rapoport conjecture for Krämer models’, Invent. Math. 234(2) (2023), 721–817.10.1007/s00222-023-01209-1 Google Scholar | DOI

[18] He, X., Pappas, G. and Rapoport, M., ‘Good and semi-stable reductions of Shimura varieties’, J. Éc. polytech. Math. 7 (2020), 497–571.10.5802/jep.123 Google Scholar | DOI

[19] Howard, B. and Li, Q., ‘Intersections in Lubin–Tate space and biquadratic fundamental lemmas’, Amer. J. Math. 147(3) (2025), 655–713.10.1353/ajm.2025.a961344 Google Scholar | DOI

[20] Hultberg, N. and Mihatsch, A., ‘A linear AFL for quaternion algebras’, Canad. J. Math. (2025), 1–23, doi: https://doi.org/10.4153/S0008414X24001020. Google Scholar | DOI

[21] Jacquet, H. and Rallis, S., ‘Uniqueness of linear periods’, Compos. Math. 102(1) (1996), 65–123. Google Scholar

[22] Kottwitz, R., ‘Tamagawa numbers’, Ann. of Math. 127(3) (1988), 629–646.10.2307/2007007 Google Scholar | DOI

[23] Kudla, S. and Rapoport, M., ‘Height pairings on Shimura curves and -adic uniformization’, Invent. Math. 142(1) (2000), 153–223.10.1007/s002220000087 Google Scholar | DOI

[24] Leslie, S., Xiao, J. and Zhang, W., ‘Unitary Friedberg–Jacquet periods and the arithmetic fundamental lemma’, in preparation. Google Scholar

[25] Li, C. and Liu, Y., ‘Chow groups and -derivatives of automorphic motives for unitary groups’, Ann. of Math. 194(3) (2021), 817–901.10.4007/annals.2021.194.3.6 Google Scholar | DOI

[26] Li, C. and Liu, Y., ‘Chow groups and -derivatives of automorphic motives for unitary groups, II’, Forum Math. Pi 10 (2022), Paper No. e5, 71 pp.10.1017/fmp.2022.2 Google Scholar | DOI

[27] Li, Q., ‘An intersection number formula for CM cycles in Lubin–Tate towers’, Duke Math. J. 171 (2022), no. 9, 1923–2011.10.1215/00127094-2021-0062 Google Scholar | DOI

[28] Li, Q., ‘A computational proof of the linear Arithmetic Fundamental Lemma for ’, Canad. J. Math. 74(2) (2022), 381–427.10.4153/S0008414X20000814 Google Scholar | DOI

[29] Li, Q., ‘A computational proof for the biquadratic Linear AFL for ’, Preprint, 2025, . Google Scholar | arXiv

[30] Li, Q. and Mihatsch, A., ‘On the linear AFL: The non-basic case’, Compos. Math. 161(2) (2025), 385–425.10.1112/S0010437X24007577 Google Scholar | DOI

[31] Mihatsch, A., ‘Local constancy of intersection numbers’, Algebra Number Theory, 16(2) (2022), 505–519.10.2140/ant.2022.16.505 Google Scholar | DOI

[32] Mihatsch, A. and Zhang, W., ‘On the Arithmetic Fundamental Lemma Conjecture over a general -adic field’, J. Eur. Math. Soc. 26(12) (2024), 4831–4901.10.4171/jems/1375 Google Scholar | DOI

[33] Qiu, C., ‘The Gross–Zagier–Zhang formula over function fields’, Math. Ann. 384(1–2) (2022), 625–731.10.1007/s00208-021-02289-1 Google Scholar | DOI

[34] Rapoport, M., Smithling, B. and Zhang, W., ‘On the arithmetic transfer conjecture for exotic smooth formal moduli spaces’, Duke Math. J. 166(12) (2017), 2183–2336.10.1215/00127094-2017-0003 Google Scholar | DOI

[35] Rapoport, M., Smithling, B. and Zhang, W., ‘Regular formal moduli spaces and arithmetic transfer conjectures’, Math. Ann. 370(3–4) (2018), 1079–1175.10.1007/s00208-017-1526-2 Google Scholar | DOI

[36] Rapoport, M., Smithling, B. and Zhang, W., ‘Arithmetic diagonal cycles on unitary Shimura varieties’, Compos. Math. 156(9) (2020), 1745–1824.10.1112/S0010437X20007289 Google Scholar | DOI

[37] Rapoport, M. and Viehmann, E., ‘Towards a theory of local Shimura varieties’, Münster J. Math. 7(1) (2014), 273–326. Google Scholar

[38] Rapoport, M. and Zink, Th., Period Spaces for -divisible Groups (Ann. of Math. Stud.) vol. 141 (Princeton University Press, Princeton, NJ, 1996). Google Scholar

[39] Serre, J.-P., ‘Local class field theory’, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) (Thompson, Washington, DC, 1967), 128–161. Google Scholar

[40] The Stacks Project Authors, Stacks Project, 2018, https://stacks.math.columbia.edu. Google Scholar

[41] Xue, H., ‘Epsilon dichotomy for linear models’, Algebra Number Theory 15(1) (2021), 173–215.10.2140/ant.2021.15.173 Google Scholar | DOI

[42] Xue, H., ‘Orbital integrals on ’, Canad. J. Math. 74(3) (2022), 858–886.10.4153/S0008414X21000122 Google Scholar | DOI

[43] Yuan, X., Zhang, S.-W. and Zhang, W., The Gross–Zagier Formula on Shimura Curves (Annals of Mathematics Studies) vol. 184 (Princeton University Press, Princeton, NJ, 2013). Google Scholar

[44] Zhang, C., ‘On the smooth transfer for Guo–Jacquet relative trace formulae’, Compos. Math. 151(10) (2015), 1821–1877.10.1112/S0010437X15007344 Google Scholar | DOI

[45] Zhang, W., ‘On arithmetic fundamental lemmas’, Invent. Math. 188(1) (2012), 197–252.10.1007/s00222-011-0348-1 Google Scholar | DOI

[46] Zhang, W., ‘Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups’, Ann. of Math. 180(3) (2014), 971–1049.10.4007/annals.2014.180.3.4 Google Scholar | DOI

[47] Zhang, W., ‘Periods, cycles, and L-functions: a relative trace formula approach’, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited Lectures (World Sci. Publ., Hackensack, NJ, 2018), 487–521. Google Scholar

[48] Zhang, W., ‘Weil representation and arithmetic fundamental lemma’, Ann. of Math. (2) 193(3) (2021), 863–978.10.4007/annals.2021.193.3.5 Google Scholar | DOI

[49] Zhang, Z., ‘Maximal parahoric arithmetic transfers, resolutions and modularity’, Duke Math. J. 174(1) (2025), 1–129.10.1215/00127094-2024-0023 Google Scholar | DOI

[50] Zink, T., Cartiertheorie kommutativer formaler Gruppen (Teubner-Texte zur Mathematik) vol. 68 (Teubner Verlagsgesellschaft, Leipzig, 1984). See https://perso.univ-rennes1.fr/matthieu.romagny/articles/zink.pdf for an English translation. Google Scholar

[51] Zink, T., ‘The display of a formal -divisible group, in Cohomologies -adiques et applications arithmétiques, I’, Astérisque 278 (2002), 127–248. Google Scholar

Cité par Sources :