NUMBER THEORY
1. Galois Representations
Let K be a number field, K̄ an algebraic closure of K, and GK =
Gal(K̄/K). (We can also consider the function field case for which
much of the theory is the same, though there are substantial and interesting differences.) For each prime ideal ℘, we fix a Frobenius element
F℘ ∈ GK . Let Z` denote the ring of `-adic integers, and Q` the field of
`-adic numbers. A Galois representation (more precisely, a compatible
system of Galois representations), is a family of continuous representations ρ` : GK → GLn (Z` ), indexed by the set of primes `, each unramified outside a common finite set of primes S of K, and satisfying the
following compatibility condition: for all primes ℘ 6∈ S, the characteristic polynomial of ρ` (F℘ ) (which does not depend on the choice of F℘ )
has all of its coefficients in Z, and moreover, these coefficients do not
depend on `. It is known that if X/K is a complete non-singular variety
and i ≥ 0 is an integer, then there is a natural action of GK on the étale
cohomology group H i (X ×K K̄, Z` ), and the resulting representations
form a compatible system. Representations arising in this way have
additional properties (purity, for example) which are not built in to
the definition of compatible system but perhaps should be. It is often
convenient to consolidate the system of representations into a single
adelic representation ρ : GK → GLn (Ẑ).
Most of my work on Galois representations is motivated by the following theorem of J-P. Serre [Se]:
Theorem. Consider a Galois representation arising as above from
X/K an elliptic curve and i = 1. If X does not have complex multiplication, i.e., if EndK̄ (E) = Z, then ρ(GK ) is an open subgroup of
GL2 (Ẑ).
Most of my work on Galois representations and much of my work on
group theory is aimed at generalizing this theorem to Galois representations arising from cohomology of varieties. To first approximation,
the desired statement is this: for every Galois representation ρ there
exists an algebraic subgroup H ⊂ GLn over Q such that ρ(GK ) is an
open subgroup of H(Ẑ ⊗ Q). This statement is not literally true, but
at least it indicates the main pieces of the problem:
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NUMBER THEORY
(1) Find H/Q such that the Zariski closure of ρ` (GK ) in GLn is
H ×Q Q` .
(2) Show that for each `, ρ` (GK ) is an open subgroup of H(Q` ).
(3) Show that for all ` 0, ρ` (GK ) is a maximal compact subgroup
of H(Q` ).
(4) Show that the adelic image ρ(GK ) is (almost) the product of
the `-adic images ρ` (GK ).
The first part is the problem of `-independence, treated in joint work
with R. Pink [LP1], [LP2], and [LP3]. Note that there is a specific
candidate for H coming from Hodge theory; the celebrated MumfordTate conjecture asserts that this candidate satisfies (1).
The second part of the problem is closely connected with the conjectured semisimplicity of the representations ρ` . There are import results
in this direction due to P. Deligne [De] and G. Faltings [Fa]. This is
the one part of the problem on which I have not worked.
For the third part, I have proved the desired result for a set of primes
of density 1 [La1], [La2]. In the special case that X is an abelian variety,
R. Pink and I have proved the desired result for all ` 0 (unpublished).
The fourth part is the subject of joint work with R. Pink some of
which has been published [LP4].
Our main tool in all this work has been group theory. The arithmetic
inputs to the group-theoretic machinery are in general very deep, and
I have not managed to contribute to this side of the problem at all.
2. Cohomological Theory of Exponential Sums
In the cohomological theory of exponential sums, we have families of
exponential sums over finite fields indexed by one or more parameters.
For example, we have the Kloosterman sums,
X
K(t) =
e2πif racx+yp ,
{x,y∈Fp |xy=t}
indexed by points on the affine t-line (with t = 0 removed). Attached
to such a family, there is a compatible system of Galois representations
of the function field K of the parameter variety; in the Kloosterman
example, for instance, K = Fp (t). The value of the sum at a point on
the variety is identified with the trace of the corresponding Frobenius
element. To understand the distribution of values, the important point
is to compute the Zariski closure H` of ρ` (GK ) ⊂ GLn for some ` (it
doesn’t matter which). In practice, H` usually turns out to contain
SLn (if the sums in the family are not always real) or SOn or Spn
(if the sums are always real). It is therefore useful to know how to
NUMBER THEORY
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“recognize”’ groups in general and classical groups in particular. The
data one has available for this recognition problem varies from problem
to problem, but it tends to have an invariant-theoretic character. The
probems which arise therefore tend to have a group theoretic character,
and they are discussed in here. Similar problems arise in topological
quantum field theory and are discussed here.
3. Modular Forms
There are two very different structures on the space of modular forms
for a fixed group. On the one hand, there is a graded algebra structure,
where the grading is given by weight (in the elliptic modular case).
On the other hand, for fixed weight, there is a module structure with
respect to the Hecke algebra. Hecke operators for different weights have
no obvious relation. For example, the product of two Hecke eigenforms
is generally not a Hecke eigenform. Exceptions to this rule are discussed
in [La3] and [La4] as well as in the dissertation of my student Brad
Emmons.
4. Mordell-Weil modules
Let A be an abelian variety over a field K, L/K a Galois extension,
and G = Gal(L/K). Then G acts on the Mordell-Weil group A(L),
so A(L) ⊗ Q is a Q[G]-module. I am interested in the question of
which pairs consisting of a finite group G and a finitely generated Q[G]module M arise in this way. We can look at a particular A/K or
ask which pairs (G, V ) arise for every (A, K). In [La5], I considered
the particular case of G = (Z/2Z)2 and V the augmentation ideal of
Q[G]. My student Bo-hae Im considers a number of other cases in her
dissertation. This problem is closely related to the question of finding
rational points on varieties of the form A ⊗ Λ/G, where Λ ⊂ V is a
G-stable lattice in V .
References
[De]
Deligne, Pierre: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ.
Math. No. 52 (1980), 137–252.
[Fa] Faltings, Gerd.: Endlichkeitssätze für abelsche Varietäten ber Zahlkörpern.
Invent. Math. 73 (1983), no. 3, 349–366.
[La1] Larsen, Michael: Two-dimensional systems of Galois representations. P-adic
methods in number theory and algebraic geometry, 163–169, Contemp. Math.,
133, Amer. Math. Soc., Providence, RI, 1992.
[La2] Larsen, Michael: Maximality of Galois actions for compatible systems. Duke
Math. J. 80 (1995), no. 3, 601–630.
[La3] Larsen, Michael: Products of two eigenforms of level one, preprint, 1996.
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NUMBER THEORY
[La4] Larsen, Michael: Modular forms, multiplicative functions, and Mandelbrot
polynomials, preprint.
[La5] Larsen, Michael: Rank of elliptic curves over almost separably closed fields,
preprint.
[LP1] Larsen, Michael; Pink, Richard: Determining representations from invariant
dimensions Invent. Math. 102 (1990), no. 2, 377–398.
[LP2] Larsen, Michael; Pink, Richard: `-independence of algebraic monodromy
groups in compatible systems of representations Invent. Math. 107 (1992),
no. 3, 603–636.
[LP3] Larsen, Michael; Pink, Richard: Abelian varieties, `-adic representations,
and `-independence. Math. Ann. 302 (1995), no. 3, 561–579.
[LP4] Larsen, Michael; Pink, Richard: A connectedness criterion for `-adic Galois
representations Israel J. Math. 97 (1997), 1–10.
[Se] Serre, Jean-Pierre: Propriétés galoisiennes des points d’ordre fini des courbes
elliptiques. Invent. Math. 15 (1972), no. 4, 259–331.