PUBLICATION LIST
[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 have recently 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).
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. It is an open question whether
the main result claimed in this paper is true.
[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).[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.,Other publications
1. Review of Holomorphic Hilbert Modular Forms (P. Garrett), Bull. AMS, 25, 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).