French-German Summer School
Galois Theory and Number Theory
Konstanz, 18–24 July 2015
Plan and abstracts of the talks

Tuesday, July 21, 9:00, room A704.

Varieties of Hilbert type
Arno Fehm (Universität Konstanz)
Abstract: Varieties of Hilbert type were introduced in the context of the Noether program. I will
discuss classical and recent results on varieties of Hilbert type and address the question of Serre whether
a product of two varieties of Hilbert type is again of Hilbert type. As an application I will discussspecializations of covers of algebraic groups.
Tuesday, July 21, 10:30, room A704.

On the Malle Conjecture and the Self-Twisted CoverPierre Dèbes (Université de Lille)
Abstract: The Malle conjecture predicts that the number of Galois extensions of Q of given group Gand with discriminant bounded from above by a real number y > 0 grows at least like y a , for some specificexponent a > 0. This is known for nilpotent groups and for a few other groups. I will present a methodthat proves it for Sn , An , many simple groups, and more generally for all regular Galois groups over Q.
The constructed Galois extensions can further be prescribed some notable local conditions. The method
uses a new version of Hilbert’s irreducibility theorem that counts the specialized extensions and not justthe specialization points. An important ingredient is the self-twisted cover that we will introduce.
Tuesday, July 21, 11:45, room A704.

Statistical number theory in function fields - Sums of two squares
Lior Bary-Soroker (Tel Aviv University)
Abstract: The starting point of this talk is Landaus theorem: the number of sums of two squares up
to x is about K logx x where K > 0 is the Landau-Ramanujan constant. We will discuss the function field
analogue of Landaus theorem, and will report on recent developments on function field versions of someclassical open problems in this theory.

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Wednesday, July 22, 9:00, room A702.

Prime polynomial values of linear functions in short intervals
Efrat Bank (Tel Aviv University)
Abstract: In this talk I will present a function field analogue of a conjecture in number theory. Thisconjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple
conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, andthe Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values ofn linear functions, in the limit of a large finite field. A key role is played by the computation of some
Galois groups.
Wednesday, July 22, 10:30, room A702.

Analytic number theory in function fields
Zeev Rudnick (Tel Aviv University)
Abstract: I will discuss function field analogues of a number of open problems in analytic number
theory, revolving around sums of arithmetic functions in short intervals and in arithmetic progressions.
Examples include primes, the Mobius function and divisor functions. The solution of these problems
rely on a mixture of techniques from analytic number theory, algebraic geometry and on equidistribution
results.
Wednesday, July 22, 14:30, room A704.

A survey on some specialization theorems
Umberto Zannier (Scuola Normale Superiore di Pisa)
Abstract: We shall survey over the general issue of ‘specializations which preserve a structure’. After
recalling some typical examples and results, we shall discuss in a little more detail some specialization
problems in the context of Unlikely Intersections and families of Pell’s equations to be solved in polynomials; some proofs involve counting in certain Galois orbits.
Wednesday, July 22, 15:45, room A704.

Characterising Q by GQ + ε
Jochen Koenigsmann (University of Oxford)e
Abstract: We show that if a field K has absolute Galois group GK := Gal(K/K)
isomorphic to GQ
then K shares many arithmetic properties with Q, and in fact, very little extra information is needed inorder to characterize Q up to isomorphism.
Thursday, July 23, 9:00, room A703.

Specialization of `-adic Galois representations and Hilbertianity
Sebastian Petersen (Universität Kassel)
Abstract: Let S be a variety over Q with function field K and for every prime number ` let ρ` :Gal(K) → GLn (Q` ) be a continuous homomorphism which is unramified along S. For a closed points ∈ S with residue field k(s) we can define the specialization ρ`,s : Gal(k(s)) → GLn (Q` ) in the mostnatural way, and compare the image Im(ρ`,s ) of the specialization with the image Im(ρ` ) of the genericrepresentation. This leads to Hilbertianity questions, and there are interesting and recent results ofCadoret-Tamagawa, Ellenberg-Hall-Kowalski and others. These results can be applied when ρ` is the`-adic representation of Gal(K) on the `-adic Tate module of the generic fibre of a family of abelianvarieties A → S (or, more generally, on the `-adic cohomology of the generic fibre of a morphism ofschemes X → S).
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Thursday, July 23, 10:30, room A703.

On the number or ramified primes in specializations
François Legrand (Tel Aviv University)
Abstract: This talk will focus on the behavior of the number of ramified primes in finite Galois
extensions of Q obtained by specializing finite Galois extensions of Q(T ) at positive integers. In thefirst part, I will recall some previous results which consist in producing some suitable positive integers nsuch that the ramification (and then the number of ramified primes) of the specialization at n satisfies
some desired properties and explain how these results relate to some strong versions of the Inverse Galois
Problem. In the second part (based on a joint work with Lior Bary-Soroker), I will present new results
about the behavior of the number of ramified primes at a given positive integer n.
Thursday, July 23, 14:30, room A702.

Arithmetic Statistics in Function Fields
Edva Roditty-Gershon (University of Bristol)
Abstract: I will discuss the mean square of sums of the generalised divisor function over arithmetic
progressions for the rational function field over a finite field of q elements. In the limit as q tends toinfinity we establish a relationship with a matrix integral over the unitary group, and analyse the integral.
This is a joint work with Jon Keating, Brad Rodgers and Zeev Rudnick. I will also discuss the autocorrelations of arithmetic functions (in particular the generalised divisor function and the von Mangoldtfunction). Function field analogues of these problems have recently been resolved in the limit of large
finite field size q. However, in this limit the correlations disappear: the arithmetic functions becomeuncorrelated. We compute averages of terms of lower order in q which detect correlations. Our results
show that there is considerable cancellation in the averaging and have implications for the rate at whichcorrelations disappear when q tends to infinity. This is a joint work with Jon Keating.

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