# Trimester Seminar

## Monday, January 18

16:30 - 17:30 Dominguez: Galois representations and tame Galois realizations of linear groups

Abstract: In this talk we consider the following variant of the inverse Galois problem over Q, introduced by B. Birch around 1994: Let G be a finite group. Is there a tamely ramified Galois extension of Q with Galois group G? We will focus on the realization of linear groups over finite fields of characteristic l. The strategy is to make use of the Galois representations arising from arithmetic-geometric objects. In particular, we will consider the Galois representation attached to the l-torsion points of an elliptic curve, and show an explicit construction that provides tame Galois realizations over Q of GL_{2}(F_{l}). At the end we will sketch a generalization of this construction for the group GSp_{4}(F_{l}), and also we will mention some related questions we are currently studying.

17:30 - 18:30 Sengun: The cohomology of Bianchi groups and arithmetic

Abstract: They are significant in the study of hyperbolic 3-manifolds and also in the study of certain generalizations of the classical modular forms (called Bianchi modular forms) for which they assume the role of the classical modular group PSL(2,Z). In this talk, I will put the cohomology of Bianchi groups in the center and will survey its (sometimes conjectural) connections with Bianchi modular forms, Galois representation (both char. 0 and mod p), abelian varieties of GL_{2} type etc. I will expose some of my theoretical/computational investigations along the way.

## Monday, January 25

16:30 - 17:30 Bayarmagnai: On a certain generic - torsor

Abstract: I will begin with a brief introduction to the essential dimension, introduced by J. Buhler and Z. Reichstein, which is a notion to measure the number of parameters required to define an algebraic object. I will then discuss how a certain generic - torsor leads to a generic cyclic polynomial of degree n, with minimal number of parameters, under certain conditions of base field.

17:30 - 18:30 Tossici: Essential dimension of group schemes in positive characteristic

Abstract: We first give an overview of the known results about the essential dimension of group schemes in positive characteristic. These results concern smooth group schemes, which is the unique case considered up to now. Then I will talk about some recent results I obtained in a joint work(in progress) with A. Vistoli. We obtained a general lower bound and a general upper bound for (not necessarily smooth) group schemes in positive characteristic. We will also show how to compute, with these two bounds, the essential dimension of some classes of group schemes.

## Monday, February 1

16:30 - 17:30 Posingies: How to get information on scattering constants?

Abstract: Scattering constants are certain values coming from non-holomorphic Eisenstein series to finite index subgroups of the modular group. In them, much of the structure of the group is encoded. In this talk we will see different ways how to get information on scattering constants. These are: Direct calculation of the coefficients in the defining series. Dependencies between super- and subgroups. The implications automorphisms of the subgroup give for scattering constants. Calculation via Kronecker limit formulas, i.e. by comparing the coefficients in Eisenstein series with the ones in functions coming from modular forms.

17:30 - 18:30 Shabnam: Thue Equations

Abstract: Let F(x,y) be an irreducible binary form with integral coefficients and degree n > 3, then by a well-known result of Thue, the equation F(x,y) = m (m integer) has finitely many solutions in integers x and y. I shall discuss some methods from Diophantine analysis and geometry of numbers to obtain upper bounds upon the number of integral solutions to such equations. I will pay special attention to quartic forms and describe a general method for finding integral points on elliptic curves by reducing them to Thue equations. The method is a classic one due to Mordell. Then I will show some results on representation of integers by binary forms.

## Monday, February 8

16:30 - 17:30 Viehmann: Affine Deligne-Lusztig varieties

Abstract: Affine Deligne-Lusztig varieties are analogs in the affine Grassmannian or the affine flag manifold of classical Deligne-Lusztig varieties. They play a role when studying the reduction modulo p of Shimura varieties. I will give an overview over current results on the global structure of affine Deligne-Lusztig varieties associated to a hyperspecial maximal compact subgroup.

17:30 - 18:30 Peruginelli: Parametrization of integer values of polynomials

Abstract: In this talk I will consider the following problem: given a polynomial f in Q[X] such that f(Z) ⊂ Z, does there exist an integer coefficients polynomial g(X_{1},...,X_{m}) for some m such that f(Z)=g(Z^{m})? If this is the case we say that f(Z) is Z-parametrizable. I will give a necessary and sufficient condition on f(X) so that f(Z) is Z-parametrizable. In particular it turns out that some power of 2 is a common denominator of the coefficients of f(X) and there exists a rational β with odd numerator and odd prime-power denominator such that f(X)=f(β-X). Moreover if f(Z) is likewise parametrizable, then this can be done by a polynomial in one or two variables. I will show the connection of this problem with some particular Diophantine equation with separated variables. This is a joint work with Umberto Zannier.

## Monday, February 22

16:30 - 17:30 Lorscheid: Graphs of Hecke operators and automorphic forms for function fields

Abstract: The world of number fields and of global function fields has far-reaching parallels. Where the domain of an automorphic form over Q is the complex upper half plane, the domain of an automorphic form over a global function field is are the vertices of a Bruhat-Tits tree (for simplicity, everything is assumed to be unramified). The invariance of automorphic forms by an arithmetic subgroup leads to quotients of Bruhat-Tits trees, which encode the action of Hecke operators. In this talk, we explain the analogy of classical automorphic forms and automorphic forms over global function fields and introduce the concept of graphs of Hecke operators. Using the interpretation of these graphs in terms of vector bundles on a curve over a finite field, it is possible to determine the structure of these graphs up to a finite subgraph that is deeply involved with the arithmetics of the function field. Once these graphs are determined, one can trace back questions on the arithmetic of automorphic forms to explicit systems of linear equations as we will demonstrate in examples, e.g. one can calculate the space of cusp forms or verify the Riemann hypothesis for the global function field.

17:30 - 18:30 Canci: Rational periodic points for quadratic maps

Abstract: Let K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R_{S} be the ring of S-integers of K. We consider endomorphisms of P_{1} of degree 2, defined over K, with good reduction outside S (a definition involving the resultant of two polynomials). We will show that there exist only finitely many such endomorphisms, up to conjugation by PGL_{2}(R_{S}), admitting a periodic point in P_{1}(K) of order > 3. Also, all but finitely many classes with a periodic point in P_{1}(K) of order 3 are parametrized by an irreducible curve.

## Monday, March 1

16:30 - 17:30 Kolvraa: Polylogarithms and special values of L-functions

Abstract: In this talk we will start by reviewing Zagier’s conjecture on special values of L-functions, a conjecture which given a number field K provides us with an expression for ζ_{K}(n) similar to that of the classical analytic class number formula. We will then discuss a generalization to partial zeta functions, and say a little of how this generalization can be proven following Levin and Goncharov. Finally we will discuss some special functions appearing in these proofs, the so-called generalized Kronecker-Eisenstein series.

17:30 - 18:30 Antei: On the abelian fundamental group scheme of a family of varieties

Abstract: Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism of fundamental group schemes pi_{1}(X,x)^{ab} → π_{1}(Alb_{X/S},0_Alb_{X/S}) induced by the canonical morphism from X to its Albanese scheme Alb_{X/S} (when the latter exists) fits in an exact sequence of group schemes 0 → (NS^{τ}_{X/S})^{∨} → π_{1}(X,x)^{ab} → π_{1}(Alb_{X/S},0_Alb_{X/S}) → 0, where the kernel is a finite and flat S-group scheme. Furthermore we prove that any finite and commutative quotient pointed torsor over the generic fiber X_{η}of X can be extended to a finite and commutative pointed torsor over X.

## Monday, March 8

16:30 - 17:30 Schuett: Elliptic surfaces

Abstract: We will give a basic introduction to elliptic surfaces with a view towards the workshop at HIM on March 11 & 12.

17:30 - 18:30 Salgado: On the rank of the fibres of rational elliptic surfaces

Abstract: We compare the generic and the special ranks of rational elliptic surfaces over number fields. We show that, for a big class of rational elliptic surfaces, there are infinitely many fibres with rank at least equal to the generic rank plus two. If time allows we will also discuss the problem of comparing generic and special rank for elliptic K3 surfaces.

Monday, March 15

16:30 - 17:30 Garion: The product replacement algorithm graph of finite simple groups

Abstract: The Product Replacement Algorithm (PRA) is a practical algorithm for generating random elements of a finite group G. Since its introduction in 1995, it quickly became popular and was included in the two commonly used computer algebra packages GAP and MAGMA. One can describe this algorithm as a random walk on a certain graph, called the PRA graph, whose vertices are the generating k-tuples of G. In the talk, I will discuss the connectivity properties of PRA graphs and present some new results concerning PRA graphs of finite simple groups.

17:30 - 18:30 Ngo Dac: Moduli spaces of global shtukas

Abstract: Drinfeld has introduced the moduli spaces of shtukas as an analogue over function fields of Shimura varieties. In this talk, I will give a basic introduction to the geometry of these objects.

## Monday, March 22

16:30 - 17:30 Zong: p-dimension of fields

Abstract: I expose Gabber-Orgogozo’s theorem: If A is a Noetherian excellent henselian local integral domain, and p is a prime number, then dim_{p} Frac(A)= dim(A)+ dim_{p}(k), where k is the residue field of A.

17:30 - 18:30 Rangasamy: Gieseker-Uhlenbeck morphism for parabolic sheaves

Abstract: We prove that the parabolic analogue of a Gieseker-Uhlenbeck morphism exists.

## Thursday, April 1

16:30 - 17:30 Caruso: Dimensions of some Kisin varieties

Abstract: Let k be an algebraically closed field of characteristic p>0. Endow k with sigma defined as a non-zero power of the absolute Frobenius on k and extend sigma to a continuous ring endomorphism of k((u)) by sending u to ub for some integer b>1. In this talk, I will explain how one can associate to these data some varieties (sometimes called Kisin varieties) and then how one can estimate their dimensions.

17:30 - 18:30 Dey: Restriction of tangent bundle and semistability

Abstract: Let G be an algebraic group over field of complex numbers and P be a maximal parabolic subgroup of G. We will show restriction of the tangent bundle of G/P to a complete intersection of codimension 2 is semistable under some conditions. This is a joint work with I. Biswas.

## Thursday, April 12

16:30 - 17:30 Kumar: Study of (p,p)-Galois representations

Abstract: We study the (p,p)-Galois representations attached to automorphic forms on GLn.

17:30 - 18:30 Le Borgne: Phi-modules over k((u))

Abstract: I will introduce the category of phi-modules over k((u)) and give its links with p-adic Galois representations. I will also present a theorem of decomposition of such phi-modules with a slope filtration.

## Monday, April 19

16:30 - 17:30 Gotsbacher, Cohomology of arithmetic groups & regulators

Abstract: In 1977, Borel used his results on the stable real cohomology of arithmetic groups to prove the transcendental part of the original Lichtenbaum conjecture. I will present the proof as an introduction to cohomology of arithmetic groups and put the techniques employed somewhatin perspective.

17:30 - 18:30 Csima: Combinatorics and the geometry of Schubert varieties

Abstract: I will give an overview of some of the combinatorial methods used to study the geometry of Schubert varieties of flag manifolds. We will focus on the notion of pattern avoidance to characterize smoothness and Gorensteinness of Schubert varieties.

## Monday, April 26

16:30 - 17:30 Gashi: On a result of Broer about the cohomology of line bundles on the cotangent bundle of the flag variety

Abstract: I will discuss a result of Broer about the cohomology of line bundles on the cotangent bundle of the flag variety, and its relation with the non-negativity of the coefficients of Lusztig’s q-polynomials.