Lifting Galois Representations

22 Pages·2017·7.08 MB·English

by  Matthieu Florence

Lifting Galois Representations
UHH Kolloquium
Matthieu Florence
13.07.18
SorbonneUniversit´e,Paris
Light introduction
Heroes of simplicity
Leonardo da Vinci
1452-1519
La semplicit`a `e la suprema
sofisticazione.
Fr´ed´eric Chopin
1810-1849
La simplicit´e est la r´eussite
absolue. Apr`es avoir jou´e
une grande quantit´e de
notes, toujours plus de notes,
cest la simplicit´e qui ´emerge,
comme une r´ecompense
venant couronner l’art.
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Breakthroughs of the 19th and 20th century
E´variste Galois Ernst Witt
1811-1832, Paris 1911-1991, Hamburg
Galois Theory Witt vectors
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(Too?) Longstanding problems
Understand
G :=Gal(Q/Q).
Solve
the inverse Galois problem.
Questions
What do we mean by understand? Is the statement of the inverse Galois
problem good (=robust) enough?
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(Too?) Longstanding problems
An approach
Attack these questions through representations
G −→GL(V),
where V is a vector space over some field- often finite, or at least ”well”
understood. There are several traditional sources for these
representations, e.g. Abelian varieties, or (almost tautologically!)
algebraic tori.
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Why the Galois group of Q?
• Possibly because arithmetic is the historical motivation for this
branch of mathematics.
• For instance, Galois representations are summoned in Wiles’s 1995
proof of Fermat’s Last Theorem.
• Serre’s modularity conjecture directly concerns two-dimensional
representations, of the Galois group of Q. It was eventually solved,
by Khare and Wintenberger in 2008.
• The Wikipedia page about this conjecture -arguably not written by
Serre himself- reads as follows:
”In mathematics, Serre’s modularity conjecture, ... states that an
odd, irreducible, two-dimensional Galois representation over a finite
field arises from a modular form.”
• We see that the field Q is implicit in the statement. Nonetheless, it
is referred to later, in the more detailed exposition that follows.
• NB: I like Wikipedia, but I would not dare recommend it to learn
mathematics.
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Are there other fields (other than Q?)
Spoiler: the answer is Yes.
What is your favorite field? Statistically prevailing answers are:
• 1) A field of Dutch tulips. An artistic answer, out of the scope of
this very serious Colloquium talk. Let’s restrict our statistical
analysis to a panel of mathematicians. Answers are:
• 1) the field of real numbers,
• 2) that of complex numbers,
• 3) that of rational numbers,
• 4) finite fields,
• 5) local and global fields,
• 6) the function field of a variety, over a field of type 1) to 5) above,
• 7) the (metaphorical and lonely) field with one element.
• NB: Their order of appearance is the order in which I got aware of
their existence.
For real, it is too complex to order them otherwise.
(I have copyrights on this joke.)
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Where things get serious: Galois
theory and Witt vectors.
Galois representations
From now on, we consider an arbitrary field F.
For instance, a quantum algebraic state, involving addition and
multiplication, in all fields of the previous list, at once.
NB: Enjoy this joke, ’cause I ain’t making any other for a while.
Denote by G :=Gal(F /F) ”the” absolute Galois group of F. It is a
s
profinite group.
What can you say about G? Here again, we better be precise about the
meaning of ”What”.
To formulate things precisely, we adopt the language of representation
theory.
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