Return to home page


[1] Harris, M.:  Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields. Inventiones Math.51, 123-141 (1979).

[2] Harris, M.:  A note on three lemmas of Shimura. Duke Math. J. 46,  871-879 (1979).

[3] Harris, M.:  P-adic representations arising from descent on abelian  varieties. Compositio Math. 39, 177-245 (1979); Correction, Compositio Math. (2000).
    The principal error in this paper was the incorrect claim that Iwasawa's sufficient criterion for a compact
L-module to be torsion — that its group of coinvariants be finite — generalizes to the non-abelian situation. A correct criterion, involving the Euler characteristic, has since been found by Susan Howson. Several proofs based on the fallacious criterion are replaced by alternative proofs in the Correction.  However, in the absence of a valid criterion, it is impossible to justify the claim that certain modules constructed from Selmer groups of elliptic curves are torsion L-module. Using the Euler characteristic criterion, Coates and Howson found the first examples of torsion modules over the Iwasawa algebra of GL(2,Zp) coming from Selmer groups of elliptic curves.
    To my knowledge,the remainder of the results of this paper are correct, when taken in conjunction with the correction.  This includes some of the basic structural theory of compact
L-modules in the non-abelian case, the proof that the L-module constructed from ideal class groups (the direct analogue of the module studied by Iwasawa) is torsion, and certain control theorems.

[4] Harris, M.:  Kubert-Lang units and elliptic curves without complex multiplication.  Compositio Math. 41, 127-136 (1980).

[RETRACTED] Harris, M.:  The annihilators of p-adic induced modules. J. of Algebra  67, 68-71 (1980).  NOTE:  Jordan Ellenberg has found a fatal flaw in the main argument, so this paper should be disregarded.  The problem is the deduction on lines -2 and -3 of p. 69, which is not justified.  Ardakov and Wadsley have now shown (arXiv:1308.5104) that the main result is false for every semisimple p-adic group.

[5] Harris, M.:  The rationality of holomorphic Eisenstein series. Inventiones Math. 63, 305-310 (1981).

[6] Harris, M.:  Special values of zeta functions attached to Siegel modular forms. Annales Scient. de l'Ec. Norm. Sup. 14, 77-120 (1981).
    A rumor has been circulating to the effect that one of the statements used in this article was not proved until several years later, and that the proofs are therefore incomplete. Apparently this is based on a misunderstanding.  The statement in question, as far as I can tell, is 3.6.2, the claim that the antiholomorphic highest weight module for Sp(2n) is irreducible down to weight n/2.  This is of course a simple consequence of the unitarity of the module (cited in 3.6.1), the fact that it is generatedby a highest weight vector, and the well known fact that any submodule is generated by highest weight vectors.   If there is anything more to the rumor I don't know what it is.

[7] Harris, M.:  Maass operators and Eisenstein series. Math. Ann. 258, 135-144 (1981).

[8] Harris, M.:  P-adic measures for spherical representations of reductive p-adic groups. Duke Math. J. 49, 497-512 (1982).

[9] Harris, M., Jakobsen, H.P.:  Singular holomorphic representations and singular modular forms. Math. Ann. 259, 227-244 (1982).

[10] Harris, M., Jakobsen, H.P.:  Covariant differential operators, in Group Theoretical Methods in Physics (Istanbul, 1982),  Lecture Notes in Physics, 180, 16-34.  Berlin:  Springer-Verlag  (1983).

[11] Harris, M.:  Eisenstein series on Shimura varieties. Ann. of Math. 119, 59-94 (1984).

[12] Harris, M.:  Arithmetic vector bundles on Shimura varieties, in Automorphic Forms of Several Variables,  Proceedings of the Taniguchi Symposium, Katata, 1983 , 138-159. Boston:  Birkhaüser (1984).
    The argument in 3.5 of this mainly expository paper, concerning jet bundles, is nonsense.  A correct argument is given in the subsequent articles.

[13] Harris, M.:  Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Inventiones Math. 82, 151-189 (1985).
    The term "arithmetic vector bundle" has since been replaced by "automorphic vector bundle".  The argument in (3.6.7), deriving existence of a model over a number field of an "absolutely arithmetic" automorphic vector bundle by means of a cocycle condition involving Aut(C), needs further justification. [SEE NOTE BELOW.] A simpler alternative is to observe that a quotient of the canonical principal bundle I(G,X) by a finite subgroup C of the center of G is already defined over the reflex field E(G,X).  One can take C to be the intersection of the center of G with the derived subgroup Gder.  Indeed, the fact that the quotient of I(G,X) by the center of G is defined over E(G,X) follows from Proposition 3.7, whereas the fact that the quotient by Gder is defined over E(G,X) is a conseqence of the theory for tori.  Since finite étale covers are defined over finite algebraic extensions, one sees immediately that I(G,X) is defined over some number field, as are the Hecke correspondences on I(G,X).  One can then replace the cocycle
for Aut(C) by a continuous cocycle on the Galois group of the algebraic closure of Q.  A complete argument may be given elsewhere.

NOTE ADDED MARCH 21, 2008:  After rereading Shimura's original proof of the existence of canonical models
using cocycles on Aut(C) [Shimura, Annals of Math., 83 (1966)  294-338] in the light of its reformulation by
Varshavsky [Appendix to Selecta Math., 8 (2002) 283-314], I am now convinced that the argument in (3.6.7) is essentially correct.
The argument proceeds by constructing a cocycle on Aut(C) with values in Gm that is shown to be effective for descending to
an appropriate reflex field an automorphic vector bundle on the Shimura variety attached to a torus.  It thus necessarily
satisfies the required continuity property.  All that is missing from the proof is acknowledgment of of this requirement.

[14] Harris, M.:  Arithmetic vector bundles and automorphic forms on Shimura varieties II. Compositio Math. 60, 323-378 (1986).

[15] Harris, M., Phong, D. H.:  Cohomologie de Dolbeault à croissance logarithmique à l'infini. C. R. Acad. Sci. Paris 302, 307-310 (1986).
     José Ignacio Burgos pointed out in 1997 that the argument in Griffiths-Harris, used to extend the Poincaré Lemma with logarithmic singularities from the one-dimensional case to the general case, does not apply in the present situation.  Briefly, the Dolbeault complex defined in this paper consists of forms
w which, together with their antiholomorphic derivatives, satisfy logarithmic growth conditions in the neighborhood of a divisor with normal crossings.  However, the Griffiths-Harris argument introduces additional holomorphic derivatives, which may not belong to the original complex.  The quotation should have been of the argument used by Borel in reference [1], which is based on integration rather than differentiation.
    As noted in [19], and as observed independently by Burgos, one can actually reprove the one-dimensional Poincaré lemma with logarithmic singularities for forms all of whose derivatives, holomorphic as well as anti-holomorphic, satisfy the growth conditions; this is even necessary if one wants to obtain Lie algebra cohomology complexes to calculate the cohomology of Shimura varieties.  A complete proof of this fact, and the correct deduction of the higher-dimensional case, was published in [42], in response to Burgos' comment.

[16] Harris, M.:  Formes automorphes "géométriques" non-holomorphes:  Problèmes d'arithméticité, in Sém de Théorie des Nombres, Paris 1984-85  Boston: Birkhaüser (1986).

[17]   Harris, M.:  Arithmetic of the oscillator representation, manuscript (1987), see this page.

[18] Harris, M.:  Functorial properties of toroidal compactifications of locally symmetric varieties, Proc. Lon. Math. Soc. 59, 1-22 (1989)

[19] Harris, M.:  Automorphic forms and the cohomology of vector bundles on Shimura varieties, in L. Clozel and J.S. Milne, eds., Proceedings of the Conference on Automorphic Forms, Shimura Varieties, and L-functions, Ann Arbor, 1988, Perspectives in Mathematics,  New York:  Academic Press, Vol. II, 41-91 (1989).

[20] Harris, M.:  Automorphic forms of  d-bar-cohomology type as coherent cohomology classes, J. Diff. Geom. 32, 1-63 (1990).

[21] Harris, M.:  Period invariants of Hilbert modular forms, I:  Trilinear differential operators and L-functions, in J.-P. Labesse and J. Schwermer, eds., Cohomology of Arithmetic Groups and Automorphic Forms, Luminy, 1989, Lecture Notes in Math., 1447, 155-202 (1990).
    The last section of this article assumes the extension of the techniques of [22] to general totally real fields.  At the time of publication, I was under the mistaken impression that the Siegel-Weil formula for the central value of the Eisenstein series, proved by Kudla and Rallis, extended in a simple way to the symplectic similitude group GSp(6).  In fact, the extension proposed in [22] only works over Q.  A correct Siegel-Weil formula for similitude groups is proved in [49].  Thus the proofs in this article are now complete.

[22] Harris, M., Kudla, S.: The central critical value of a triple product L-function, Ann. of Math., 133, 605-672 (1991).

[23] Harris, M., Kudla, S.:  Arithmetic automorphic forms for the non-holomorphic discrete series of GSp(2), Duke Math. J. 66, 59-121 (1992).

[24] Garrett, P.B., Harris, M.:  Special values of triple product L-functions, Am. J. Math.  115, 159-238 (1993).

[25]   Harris, M.:  Non-vanishing of L-functions of 2x2 unitary groups, Forum Math.  5, 405-419 (1993).

[26] Harris, M., Soudry, D., Taylor, R.:  l-adic representations attached to modular forms over an imaginary quadratic field, I:  lifting to GSp(4,Q), Inventiones Math., 112, 377-411 (1993).
    On p. 410, lines 2-3, we claim to have constructed supercuspidal representations of GSp(4) over a p-adic field that were missed by Vignéras in her article [V].  Dipendra Prasad pointed out that these supercuspidal representations, and the corresponding representations of the Weil group, were actually constructed in [V] in a different matrix representation.

[27] Harris, M.:  L-functions of 2 by 2 unitary groups and factorization of periods of Hilbert modular forms,  J. Am. Math. Soc. 6, 637-719, (1993).
    The relation of CM periods to special values of L-functions of Hecke characters, obtained in general by Blasius, is quoted on numerous occasions in this article.  Unfortunately, it is quoted here, as in the appendix to [22], with a sign mistake.  The final formulas are indifferent to the choice of sign, so no harm is done.  The mistake is corrected in the introduction to [35], whose results depend on the correct choice of sign.

[28] Harris, M., Zucker, S.:  Boundary cohomology of Shimura varieties, I:   coherent cohomology on the toroidal boundary,  Annales Scient. de l'Ec. Norm. Sup. 27, 249-344 (1994).

[29] Harris, M., Zucker, S.:  Boundary cohomology of Shimura varieties, II:  mixed Hodge structures ,  Inventiones Math.116, 243-307 (1994); Erratum, Inventiones Math.123, 437 (1995).

[30]  Harris, M.:  Hodge-de Rham structures and periods of automorphic forms, in Motives, Proc. Symp. Pure Math.. AMS, 55, Part 2, pp. 573-624 (1994).

[31]  Blasius, D., Harris, M., Ramakrishnan, D.:  Coherent cohomology, limits of discrete series, and Galois conjugation, Duke Math. J,    73, 647-686 (1994).

[32] Harris, M.:  Period invariants of Hilbert modular forms, II, Compositio Math. 94, 201-226 (1994).

[33]  Harris, M., Kudla, S., Sweet, W. J.:  Theta dichotomy for unitary groups, J. Am. Math. Soc.9, 941-1004 (1996).

[34] Harris, M.:  Supercuspidal representations in the cohomology of Drinfel'd upper half spaces; elaboration of Carayol's program, Inventiones Math. 129, 75-119 (1997).
    The correction character, denoted 
n(Gp) on p. 100, is calculated incorrectly on p. 101.  The correct calculation is given on p. 181 of [37], where the sign convention of [34] is also replaced by one consistent with the conventions of the book of Rapoport and Zink.

[35] Harris, M.:  L-functions and periods of polarized regular motives, J.Reine Angew. Math.483, 75-161 (1997).
     The main result on special values of L-functions of automorphic forms on unitary Shimura varieties refers to an unpublished
     calculation of archimedean zeta integrals, due to P. Garrett (Lemma 3.5.3).  Garrett has since written up this calculation in a more
     general setting and his results are included as an appendix to [53].

[36]  Harris, M. , Li, J.-S.:  A Lefschetz property for subvarieties of Shimura varieties, J. Alg. Geom. 7, 77-122 (1998).

[37]  Harris, M.:  The local Langlands conjecture for GL(n) of a p-adic field, n < p, Inventiones Math. 134, 177-210 (1998).

[38] Harris, M.:   Cohomological automorphic forms on unitary groups, I:  rationality of the theta correspondence, Proc. Symp. Pure Math, 66.2, 103-200 (1999).
    A great many misprints were discovered while preparing the sequel [55].  There were also a few substantial mathematical errors.  These were all
    corrected in the introduction to [55]. 

[39] Harris, M.:  Galois properties of cohomological automorphic forms on GL(n), J. Math. Kyoto Univ. 39, 299-318 (1999).

[40] Harris, M., Tilouine, J.:  p-adic measures and square roots of triple product L-functions,  Math. Ann., 320,  127-147 (2001).
    A recent article by Darmon and Rotger has found a different formula for the corrected Euler factor at p in Proposition 2.2.2.   There must
    be an error in our (elementary) calculation, but we have not yet been able to find it.

[41] Harris, M., Scholl, A.:  A note on trilinear forms for reducible representations and Beilinson's conjectures, J. European Math. Soc., 3, 93-104 (2001).

[42] Harris, M., Zucker, S.:  Boundary cohomology of Shimura varieties, III:  Coherent cohomology on higher-rank boundary strata and applications to Hodge theory, Mémoires de la SMF, 85 (2001).

[43] Harris, M., Taylor, R.:  The geometry and cohomology of some simple Shimura varieties,  Annals of Mathematics Studies, 151 (2001).

[44]  Harris, M.:  Local Langlands correspondences and vanishing cycles on Shimura varieties, Proceedings of the European Congress of Mathematics, Barcelona, 2000; Progress in Mathematics, 201, Basel:  Birkhaüser Verlag, 407-427 (2001).
    Eva Viehmann has found a mistake in the statement of Proposition 4.1 (ii), which means that Conjecture 5.2 needs to be corrected.   The statements seem
    to be all right for split groups but not in the general case.  Viehmann has proposed a corrected version.  See item [51] below.

[45] Harris, M., Taylor, R.:  Regular models of certain Shimura varieties, Asian J. Math.6, 61-94 (2002).

[46]  Harris, M.:  On the local Langlands correspondence, in Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol II, 583-597.

[47] Harris, M., Taylor, R.:  Deformations of automorphic Galois representations (manuscript, 1998-2003).

[48] Harris, M., Kudla, S.:  On a conjecture of Jacquet, in H. Hida, D. Ramakrishnan, F. Shahidi, eds.,  Contributions to automorphic forms, geometry, and number theory (collection in honor of J. Shalika's 60th birthday), 355-371 (2004).

[49] Harris, M.:  Occult period invariants and critical values of the degree four L-function of GSp(4) in H. Hida, D. Ramakrishnan, F. Shahidi, eds., 
Contributions to automorphic forms, geometry, and number theory (collection in honor of J. Shalika's 60th birthday), 331-354 (2004).

[50]  Harris, M., Labesse, J-P.:   Conditional base change for unitary groups, Asian J. Math8, 653-684  (2004).

[51]  Harris, M.:  The Local Langlands correspondence: Notes of (half) a course at the IHP, Spring 2000, in  J. Tilouine, H. Carayol, M. Harris, M.-F. Vignéras, eds., Formes Automorphes,  Astérisque298, 17-145 (2005) .
    The mistake in [44] arises from an incorrect argument on pp. 130-131 of this article.  Viehmann's corrected version should appear in her forthcoming work, and I hope to
    revise the calculation of the global Galois representation as a sum of contributions of individual strata.

[52]  Harris, M.,  Li, J.-S., et Skinner, C.:  The Rallis inner product formula and p-adic L-functions, in J. Cogdell et al., eds, Automorphic Representations, L-functions and Applications:  Progress and Prospects, Berlin:  de Gruyter,  225-255 (2005).  

[53]  Harris, M. :  A simple proof of rationality of Siegel-Weil Eisenstein series, in W.T. Gan, S.S. Kudla, and Y. Tschinkel, eds.,
Eisenstein Series and Applications,  Boston:  Birkhäuser, Progress in Mathematics 258 (2008) 149-186 (preceded by appendix by P. Garrett).

[54]  Harris, M.,  Li, J.-S., et Skinner, C.:  p-adic L functions for unitary Shimura varieties, I :  Construction of the Eisenstein measure, Documenta Math. John H. Coates' Sixtieth Birthday 393-464 (2006).

[55]  Harris, M.:   Cohomological automorphic forms on unitary groups, II:  period relations and values of  L- functions in Li, Tan, Wallach, and
Zhu, eds., Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, Vol. 12, Lecture Notes Series, Institute
of Mathematical Sciences, National University of Singapore (volume in honor of Roger Howe) (2007) 89-150.

[56]  Clozel, L., Harris, M., and Taylor, R. :  Automorphy for some l-adic lifts of automorphic mod l  Galois representationsm Publ. Math. IHES108 1-181 (2008).

[57]  Harris, M., Shepherd-Barron, N, and Taylor, R.: A family of Calabi-Yau varieties and potential automorphy, Annals of Math., 171,  779-813 (2010).

[58]  Harris, M., Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applicationsm in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin,
    Vol II, Boston:  Birkhäuser,  Progress in Mathematics, 270 (2009)  1-23.

[59]  Harris, M.: Arithmetic applications of the Langlands program, Japanese J. Math., 3rd ser., 5 (2010) 1-71.

[60]  Guralnick, R., Harris, M., Katz, N. :  Automorphic realization of Galois representations, J. Euro. Math. Soc., 12, (2010) 915–937.

[61]  Harris, M. :  Galois representations, automorphic forms, and the Sato-Tate conjecture., Clay Research Proceedings, 2007, in press.

[62]  Harris, M. : An introduction to the stable trace formula, in L. Clozel, M. Harris, J.-P. Labesse, B. C. Ngô, eds. The stable trace formula, Shimura varieties, and arithmetic applications. Volume I:  Stabilization of the trace formula, Boston:  International Press (2011)  3-47.

[63]  Clozel, L., Harris, M., Labesse, J.-P.:   Endoscopic transfer, in L. Clozel, M. Harris, J.-P. Labesse, B. C. Ngô, eds. The stable trace formula, Shimura varieties, and arithmetic applications. Volume I:  Stabilization of the trace formula, Boston:  International Press (2011) 475-496.

[64]  Clozel, L., Harris, M., Labesse, J.-P.: Construction of automorphic Galois representations, I., in L. Clozel, M. Harris, J.-P. Labesse, B. C. Ngô, eds. The stable trace formula, Shimura varieties, and arithmetic applications. Volume I:  Stabilization of the trace formula, Boston:  International Press (2011) 497-527.

[65]  Chenevier, G., Harris, M., Construction of automorphic Galois representations, II (manuscript 2009-2011).

[66]  Harris, M., Li, J.-S.. Sun, Binyong, Theta correspondence for close unitary groups, Advanced Lectures in Mathematics, special issue in honor of S. Kudla, (2011) 265-308.

[67]  Barnet-Lamb, T., Geraghty, D., Harris, M., et Taylor, R. :  A family of Calabi-Yau varieties and potential automorphy II, Proceedings RIMS, 47 (2011) 29-98.

[68]    Harris, M. :  The Taylor-Wiles method for coherent cohomology, J. Reine Angew. Math.,
679 (2013) 125-153.

[69]    Harris, M.: L-functions and periods of adjoint motives, Algebra and Number Theory, 7 (2013), 117-155.

[70]    Harris, M.:  Beilinson-Bernstein localization over Q and periods of automorphic forms,

International Math. Research Notices, 2013, 2000-2053 (2013).


[71]    Chenevier, G., Harris, M.:  Construction of automorphic Galois representations, II,

Cambridge Journal of Mathematics, 1, 57-73 (2013).


Other publications

1.   Review of Holomorphic Hilbert Modular Forms (P. Garrett), Bull. AMS25, 184-195 (1991)

2.   Contexts of Justification, Math. Intelligencer,  Winter 2001.

3.   Review of  Cohomologie, stabilisation, et changement de base (J.-P. Labesse),  Gazette des Mathématiciens, 2001.

4.   Review of Mathematics and the Roots of Postmodern Thought (V. Tasic), Notices of the AMS, August 2003 .

5.    Review of  Introduction to the Langlands Program (J. Bernstein et S. Gelbart), Bull. AMS  41 257-266 (2004). 

6.   A sometimes funny book supposedly about infinity, review of Everything and More (D.F. Wallace), Notices of the AMS, 51, 632-638, June-July 2004.

7.    “Why mathematics?” you might ask,  in T. Gowers, ed. The Princeton Companion to Mathematics, Princeton University Press (2008) 966-977.

8.    Do Androids Prove Theorems in Their Sleep?, to appear in A. Doxiadis and B. Mazur, eds,  Circles Disturbed, Princeton University Press (2012).