| Date | Speaker | Title | Abstract |
|---|---|---|---|
FS2019 | |||
| 28.02.2019 | Towards effective André-Oort (after Kühne et al.) | The celebrated André-Oort conjecture about special point on Shimura varieties is now proved conditionally to the GRH in full generality and unconditionally in many important special cases. In particular, Pila (2011) proved it for products of modular curves, adapting a method previously developed by Pila and Zannier in the context of the Manin-Mumford conjecture. Unfortunately, Pila's argument is non-effective, using the Siegel-Brauer inequality.
Since 2012 various special cases of the André-Oort conjecture has been proved effectively, most notably in the work of Lars Kühne. In my talk I will restrict to the case of the "Shimura variety" C^n and will try to explain on some simple examples how the effective approach of Kühne works. No previous knowledge about André-Oort conjecture is required, I will give all the necessary background. | |
| 11.04.2019 | Value distribution of quantum modular forms | Quantum modular forms are functions on the rationals satisfying a near-modularity relation; they appear naturally as invariants related to modular forms (central values of additive twists), or constructed from q-factorials. This talk will be about recent joint work with Sandro Bettin, where we study their value distribution using dynamical method. | |
| 25.04.2019 | $\overline{Q}$ structures on hermitian symmetric spaces | This is a joint work in progress with Emmanuel Ullmo and consists largely of conjectures and speculations. Inspired by the analogy with the exponential function we define $\overline{Q}$-structures on a hermitian symmetric space $X$ uniformising a Shimura variety $S$, formulate a "hyperbolic analytic subgroup theorem" and explore its consequences. | |
| 02.05.2019 | Exponential arithmetic growth in meromorphic dynamics | A conjecture of Silverman (as extended to any global field $K$) states that every Zariski-dense forward orbit $(f^n(P))_{n \in \mathbb{N}}$ under a dominant rational self-map $f : X \dashrightarrow X$ of an $N$-dimensional projective variety $X$ over $K$ has its height $h(f^n(P))$ growing at the maximal possible exponential rate $\lambda_1(f)$, the first dynamic degree of $f$. I will describe some recent progress around this problem, limiting for the main part to the case of polynomial mappings of affine space, and focusing mostly on cases where an exponential growth can be established for the height along all Zariski-dense orbits. The latter includes all additive polynomial mappings of affine space over $\mathbb{F}_q(t)$ (building on Yu's zero estimate from transcendence theory in positive characteristic), as well as all maps of large topological degree (building on work of Habegger from his paper Special points in fibered powers of elliptic surfaces, and on a theorem of Bell, Ghioca and Tucker on the dynamical Mordell-Lang problem). The proof in the former case yields also a close counterpart, for abelian $t$-modules in function field arithmetic, of Masser's "polynomial in 1/D" canonical height lower bound on abelian varieties. Lastly, and time permitting, I will supplement the exponential growth discussion with a lower bound on height growth for an arbitrary polynomial mapping. This bound comes from a partial progress on a conjecture of Ruzsa, and it is weaker than exponential, yet strong enough to yield as application an "unbounded or eventually periodic" dichotomy for the degree sequence of the iterates of a polynomial mapping of $\mathbb{A}_k^N$ over any field. The dichotomy extends a result of Favre and Jonsson from $N = 2$ to arbitrary dimension. | |
| 09.05.2019 | Chebyshev's bias in Galois groups of number fields | In a 1853 letter, Chebyshev observed that in most intervals [2,x] there are more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this bias phenomenon have since been studied. In this talk we will discuss joint work with D. Fiorilli on Chebyshev’s bias in the context of the Chebotarev density theorem. Our focus will be on particular families that either exhibit a surprising behavior as far as Chebyshev's bias is concerned or that are simple enough to enable a very precise computation of the group theoretic and ramification theoretic invariants that come into play in our analysis. Precisely the emphasis will be on some families of abelian, dihedral, or radical extensions of Q as well as families of Hilbert class fields H_d of quadratic fields K_d (of discriminant d) either seen as extensions of Q or of K_d. | |
| 16.05.2019 | Endomorphism of Arrangement Complements | Let $A$ be an arrangement of finitely many hyperplanes in projective space. We investigate endomorphisms of the complement of $A$. Under mild assumptions we extend any such endomorphism to an endomorphism of the wonderful compactification. We will explain some basic facts on matroids and tropical geometry which are used in the proof. | |
| 23.05.2019 | Results and conjectures about primitive weird numbers | A number $ n $ is weird if the sum of its proper divisors is larger than $ n $, and if $ n $ can not be expressed as a sum of some of its proper divisors. In 1972 Benkoski and Erdős published a paper in which they proved some of the properties that will be discussed here, and proposed several conjectures and open questions. One of the most interesting problems is determining whether odd weird numbers exist or not. At present all weird numbers that we know are even, but there are no obvious reasons that would prevent the existence of odd weird numbers. We will see that these problems are related to the study of \textsl{primitive weird numbers}, that is to say those weird numbers that are not multiples of other weird numbers. The most promising approach seems to be that of attacking the study of their prime factors. We will also address the problem of the existence of an infinite number of primitive weird numbers, and the problem of determining how many prime factors a primitive weird number may contain. | |
| 06.06.2019 | Chebyshev’s bias in Galois groups | In a 1853 letter, Chebyshev noted that there seems to be more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this phenomenon have been studied. In this talk we will discuss Chebyshev’s bias in the context of the Chebotarev density theorem. We will focus on the generic case of S_n extensions, in which the question is strongly linked with the representation theory of this group and the ramification data of the extensions. We will see in detail how to take advantage of the rich representation theory of the symmetric group as well as bounds on characters due to Roichman. This is joint work with Florent Jouve. | |
HS2018 | |||
| 27.09.2018 | Extreme values of the Riemann Zeta function, and log-correlated Gaussian fields | I will describe how the Riemann Zeta function on the critical line can be viewed as a pseudo-random Gaussian field with a correlation function with logarithmic growth. Such log-correlated random fields have recently attracted considerable interest in probability theory. Fyodorv, Hiary and Keating conjectured several striking results about the extreme values of the Riemann Zeta function based on this connection. In this talk I will explain how a certain approximate tree structure in Dirichlet polynomials can be used to prove one of their conjectures, giving the asymptotics of the maximum of the magnitude of the function in a typical interval of length O(1). | |
| 04.10.2018 | On Goormaghtigh's equation | In my talk I will present results that come from a joint work with M. Bennett and A. Gherga from The University of British Columbia. We studied Goormaghtigh's equation: \begin{equation}\label{eq} \frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2. \end{equation} There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition. | |
| 01.11.2018 | Representation of integers by binary forms | Let $F(x, y)$ be an irreducible binary form with integer coefficients and of degree at least 3. By a well known result of Thue, the equation $F(x, y) = m$ has only finitely many solutions in integers $x$ and $y$. I will discuss some old and new quantitative results on the number of solutions of such equations. I will also talk about my joint work with Manjul Bhargava, where we use these bounds to show that many equations of the shape $F(x, y) = m$ have no solutions. | |
| 08.11.2018 | Solving polynomial-exponential equations. | Inspired by Schanuel's Conjecture, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomial-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can be regarded as ``typical''. But it does not cover all situations for two variables, some of which involve simply stated problems in one variable like finding complex $z \neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mantova and I have settled the general case of two variables. We describe our methods -- for example, to solve $$e^z+e^{\root 9 \of {1-z^9}}=1$$ one approach uses theta functions on ${\bf C}^{28}$. | |
| 22.11.2018 | On bounded automorphisms of fields with operators | Lascar showed that the group of automorphisms of the complex field which fix the algebraic closure of the prime field is simple. For this, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set $A$ such that the image of every element b is algebraic over $A$ together with $b$. The same result holds for a "universal" differentially closed field of characteristic zero, where we replace algebraic by differentially algebraic. Together with T. Blossier and C. Hardouin, we provided in https://arxiv.org/abs/1505.03669 a complete classification of bounded automorphisms in various fields equipped with operators, among others, for generic difference fields in all characteristics or for Hasse-Schmidt differential fields in positive characteristic. | |
| 29.11.2018 | Integrability of b-divisors on toroidal embeddings | Convex geometry has been widely and successfully used to explore the geometry of algebraic varieties. A well-known class of examples comes from the theory of toric varieties, where the combinatorics of a lattice polytope encrypts most of the geometric properties of the corresponding toric variety. In this talk, we will discuss how to use convex analytical methods on so called weakly embedded polyhedral complexes to obtain integrability results of toroidal b-divisors. The latter can be thought of as a limit of toroidal divisors defined over all possible toroidal compactifications of an algebraic variety. We will also mention applications of these results to Arakelov geometry. | |
| 06.12.2018 | Titchmarsh's divisor problem for multiplicative functions | The classical Titchmarsh divisor problem is concerned with the mean behaviour of the divisor function on the set of primes shifted by a non-zero constant. We consider analogues of this problem where the set of primes is replaced by other sets of multiplicative nature and also discuss further generalizations. This is joint work with S. Drappeau. | |
| 13.12.2018 | The twelfth moment of Dirichlet L-functions to smooth moduli | Heath-Brown proved a strong bound for the twelfth moment of the Riemann zeta function which is not sharp but that has the nice feature of giving the Weyl estimate as a corollary. In this talk we will take a look at the key points of Heath-Brown's proof and show how one can deduce an analogous result for Dirichlet L-functions with smooth moduli. Our result relies on bounds for some complicated exponential sums that we study by appealing to the theory of algebraic trace functions due to Katz and Fouvry-Kowalski-Michel. | |
FS2018 | |||
| 01.03.2018 | On rationally connected varieties over C_1 fields of characteristic 0 | In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every homogeneous polynomial of degree at most n in n+1 variables has a nontrivial solution. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr). This talk addresses the open case of Henselian fields of mixed characteristic with algebraically closed residue field. | |
| 15.03.2018 | The second moment of the Dedekind zeta function on the critical line | For the Riemann zeta function, and the Dedekind zeta function of a quadratic field, we know precisely how the second moment behaves on the critical line. This is not true when the degree of the algebraic number field is 3 or more. In the present talk I discuss a new upper bound for the second moment when the degree is at least 4. | |
| 22.03.2018 | On some conjectures on the Mordell-Weil and the Tate-Shafarevic groups of an Abelian variety | We consider an Abelian variety defined over a number field. We give conditonal bounds for 1- the order of its Tate-Shafarevich group, as well as for 2- the Néron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals. We extend it to Abelian varieties of arbitrary dimension, defined over an arbitrary number field. In particular, with point 1- we improve and generalise a result by D. Goldfeld and L. Szpiro, and, with point 2- we extend a conjecture of S. Lang. | |
| 05.04.2018 | The circle method and free rational curves on hypersurfaces | In joint work with Tim Browning, we study a pair of systems of Diophantine equations over F_q[t], and use the circle method to show a relationship between their numbers of solutions. As a consequence, we bound the dimension of the singular locus of the moduli space of rational curves on a smooth projective hypersurface. I will explain how these problems are related and what techniques we use to get the best bound. | |
| 12.04.2018 | Unlikely intersections in families of semi-abelian surfaces | For an elliptic curve $E$ the Poincar\`e bi-extension of $E$ parametrizes points on $\mathbb{G}_m$ extensions of $E$. It can be thought of as a group scheme $G$ over the dual $\hat{E}$ of $E$. I will talk about joint work with Daniel Bertrand respectively Gareth Jones on qualitative results concerning the Zilber-Pink conjecture for $G$ respectively uniform and effective results for the relative Manin-Mumford question. | |
| 19.04.2018 | Prime and squarefree values of polynomials in moderately many variables | The classical Schinzel's hypothesis and its quantitative version, the Bateman-Horn's conjecture, states that a system of polynomials in one variable takes infinitely many simultaneously prime values under some necessary assumptions. We will present in this talk a proof of a generalization of these conjectures to the case of a integer form in many variables. In particular, we will establish that a polynomial in moderately many variables takes infinitely many prime (but also squarefree) values under some necessary assumptions. The proof relies on the Birch's circle method but can be achieved in 50% fewer variables than in the classical Birch setting. Moreover this result can be applied to study the Hasse principle and weak approximation for some normic equations. | |
| 17.05.2018 | Probabilistic arithmetic geometry | A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos. | |
| 31.05.2018 | An analogue of the Brauer-Siegel theorem for an Artin-Schreier family of elliptic curves | The classical Brauer-Siegel theorem gives upper and lower bounds on the product of the class-number times the regulator of units of a number field, in terms of its discriminant. Consider now an elliptic curve E defined over F_q(t): one can form the product of the order of the Tate-Shafarevich group of E (assuming it is finite) and of its Néron-Tate regulator. In analogy with the above theorem, we are interested in finding good upper and lower bounds of this quantity in terms of simpler invariants of E, e.g. its height. In general such bounds are hard to obtain (in particular, the lower bounds are the most delicate); a satisfactory answer to the question is known only for a handful of cases. In this talk, I will report on a recent work where I studied an "Artin-Schreier family" of elliptic curves. I will explain how good unconditional bounds can be found in this case. This provides a new example of family of elliptic curves for which an analogue of the classical Brauer-Siegel theorem holds. | |
| 07.06.2018 | Newton-Okounkov bodies of exceptional curve valuations | Let $p$ be a closed point in ${\mathbb{P}}_{\mathbb{C}}^2$ and consider a surface obtained by a sequence of finitely many blowups of points where we start with $p$ and always blow up a point in the exceptional divisor created last. Our result is an explicit description of the Newton-Okounkov body of the pullback of $\mathcal{O}_{\mathbb{P}^2}(1)$ with respect to the flag given by the last exceptional divisor and a point on it. This is joint work with Carlos Galindo, Francisco Monserrat and Julio José Moyano-Fernández.
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| 20.06.2018 | A variation of heights and small points in families of elliptic curves | Bogomolov's conjecture (now a theorem following work of Zhang and Ullmo) concerns the geometry of points with small canonical height on abelian varieties. In joint work with Laura DeMarco, we prove an analogous result in the setting of families of products of elliptic curves. This extends theorems by Masser and Zannier in the theme of unlikely intersections. A key ingredient in our proof is work of Silverman concerning the variation of heights in families of elliptic curves. In this talk, I will discuss this result and its applications towards a Bogomolov-type extension of a recent result by Barroero and Capuano, which extends the aforementioned result of Masser and Zannier. The latter is work in progress joint with Laura DeMarco.
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| 22.06.2018 | Height on Galois extension | In a recent joint paper with D. Masser we prove that the Weil height of a non-zero algebraic number, not a root of unity, which generates a Galois extension, can be bounded from below "essentially" by a positive constant. We further analyse Galois extension with full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of "big" height. These results answer a question of C. Smyth and provide some evidence to a conjecture which asserts that the height of such a generator growth to infinity with the degree of the extension.
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HS2017 | |||
| 05.10.2017 | A statistical version of a conjecture of Lang | Adopting a statistical point of view on diophantine problems often allows to gain insight into the questions at hand. This idea was for instance illustrated over the last years in the context of the arithmetic of elliptic curves by the astonishing achievements of Bhargava and his collaborators. The goal of this talk will be to provide another illustration of the above principle by investigating the conjecture of Lang predicting a lower bound for the canonical height of non-torsion points on elliptic curves defined over number fields. | |
| 26.10.2017 | Volumes of quasi-arithmetic hyperbolic lattices | I will recall some connections between hyperbolic geometry and number theory, in particular concerning the volumes of arithmetic manifolds. Then I will present some of my work concerning quasi-arithmetic lattices. | |
| 02.11.2017 | Expansions of quadratic numbers in a p-adic continued fraction. | It goes back to Lagrange that a real quadratic irrational always has a periodic continued fraction. Starting from decades ago, several authors generalised proposed different definitions of a p-adic continued fraction, and the definition depends on the chosen system of residues mod p. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known. In recent work wth F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the p-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion relies on deep theorems in transcendence and diophantine analysis, and, somewhat surprisingly, depends on the “real” value of the p-adic continued fraction. | |
| 09.11.2017 | Torsion in subvarieties of abelian varieties | A theorem of Raynaud shows that a given subvariety of an abelian variety only contains a finite number of (maximal) torsion cosets; that is, translates of abelian subvarieties by a point of finite order. This result is also known as the Manin-Mumford conjecture for abelian varieties.
In this talk, we focus on bounding the number of maximal torsion cosets. We present an interpolation method via Galois representations which allow us to give an explicit bound with a "good" dependence on the degree of the subvariety. This is joint work with Aurélien Galateau. | |
| 23.11.2017 | On Faltings' delta-invariant | Faltings' delta-invariant of compact Riemann surfaces plays a crucial role in Arakelov theory of arithmetic surfaces. It is the archimedean contribution of the arithmetic Noether formula. We will give a new formula for this invariant in terms of integrals of theta functions. As applications, we obtain a lower bound for delta only in terms of the genus, a canonical extension of delta to abelian varieties and an upper bound for the Arakelov-Green function in terms of delta. | |
| 30.11.2017 | Some recent results on the arithmetic of elliptic curves | This talk will describe some recent progress on aspects of the Birch--Swinnerton-Dyer conjecture for elliptic curves over Q. | |
| 07.12.2017 | Counting twin prime polynomials | Twin prime polynomials are the function field analog of twin prime numbers. The talk presents joint work with Prof. Bary-Soroker about asymptotics and explicit formulas for the number of twin prime polynomials in light of a function field version of the Hardy-Littlewood prime tuple conjecture. | |
| 14.12.2017 | Non-archimedean hyperbolicity | I will explain a non-archimedean analogue of Brody hyperbolicity introduced by Cherry in the 90's, and explain a strategy for proving the non-archimedean Brody hyperbolicity of the moduli space of abelian varieties. This is ongoing work with Alberto Vezzani. | |
| 21.12.2017 | Singular units | A singular moduli is the j-invariant of an elliptic curve with complex multiplication. Since the 19th century, it is well-known that singular moduli are algebraic integers. Bilu, Masser, and Zannier asked whether there exist singular moduli that are even algebraic units. In earlier work, Habegger was able to show non-effectively that there are at most finitely many such singular moduli. The non-effectivity in his proof is due to a contingent Siegel zero within the proof of Duke's equidistribution theorem. In joint work of Bilu, Habegger, and myself, we were able to give an effective proof, with an explicit bound on the discriminant of possible singular units. | |
FS2017 | |||
| 02.03.2017 | Multiplicatively dependent algebraic numbers | (joint work with Alina Ostafe, Francesco Pappalardi, Min Sha, Cam Stewart) We discuss various questions related to the distribution of vectors of algebraic numbers (u_1, …., u_n) which are multiplicatively dependent. In particular, we present some counting results for the number of such vectors of degree d and height h (from a fixed number field K and from Q-bar). We also give both sided estimates on their density in R^n. Finally we give an analogue of a result of Bombieri-Masser-Zannier (1999) proving the boundedness of the “house" of the shifts v from the abelian closure of a given number field K, for which the vector (u_1-v, …., u_n-v) becomes multiplicatively dependent, rather than the boundedness of their height as in BMZ’99 (but for shifts v from Q-bar). This result has applications to multiplicative dependence in orbits of polynomial dynamical systems, generalising those on roots of unity. | |
| 06.04.2017 | Erratic behavior of the coefficients of modular forms | I will speak on a recent joint work with Jean-Marc Deshouillers, Sanoli Gun and Florian Luca. Here is a sample result. Let $\tau(.)$ be the classical Ramanujan $\tau$-function defined by $$q\prod_{n>0} (1-q^n)^{24} = \sum_{n>0} \tau(n) q^n.$$ The classical work of Rankin implies that both inequalities $|\tau(n)|<|\tau(n+1)|$ and $|\tau(n)|>|\tau(n+1)|$ hold for infinitely many $n$. We generalize this for longer segments of consecutive values of $\tau$. Let k be a positive integer such that $\tau(n)$ is not $0$ for $n\le k/2$. (This is known to be true for all $k < 10^{23}$, and, conjecturally, holds for all $k$.) Let s be a permutation of the set $\{1,...,k\}$. Then there exist infinitely many positive integers $n$ such that $|\tau(n+s(1))|<\tau(n+s(2))|<...<|\tau(n+s(k))|$. | |
| 11.04.2017 | Subgroups of Class Groups and the Absolute Chevalley-Weil Theorem | The following conjecture is widely believed to be true: given a finite abelian group G, a number field K and an integer d>1, there exist infinitely many extensions L/K of degree d such that the class group of L contains G as a subgroup. I will speak on some old and recent results on this conjecture, in particular, on my recent joint work with J. Gillibert.
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| 20.04.2017 | Lehmer's problem on Drinfeld modules | This talk will be about lower bounds for the canonical height associated to a Drinfeld module. I will explain how one can obtain such a lower bound for the canonical height of an algebraic point which is polynomial in the inverse of the degree of the point. This bound is valid for every Drinfeld module (in particular, of arbitrary rank). This is a joint work with Aurélien Galateau. | |
| 11.05.2017 | Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations | I will report on some results about unlikely intersections in families of abelian varieties, obtained mostly in collaboration with L. Capuano, and explain how they fit into the framework of the Zilber-Pink Conjectures. Finally, I will present some applications concerning certain polynomial Diophantine equations. | |
| 18.05.2017 | On the action of the absolute Galois group on curves and surfaces | A combination of Hodge theory and Serre's GAGA principle implies that many topological invariants of complex projective varieties, such as Betti numbers and Chern classes, remain invariant under Galois action. Nevertheless, in 1964 Serre proved that conjugate varieties need not be homeomorphic, by showing the existence of Galois elements $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ and projective varieties $X$ such that the fndamental groups of $X$ and its conjugate variety $X^{\sigma}$ are not isomorphic (while their algebraic fundamental groups are). In this talk I will attempt to show that this occurs for every $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ different from the identity and the complex conjugation. Our projective varieties $X$ will be Beauville surfaces, a kind of rigid surfaces of general type introduced by Catanese, arising as finite quotients of products of triangle curves. This is joint work with Andrei Jaikin-Zapirain. | |
| 01.06.2017 | Transcendental Liouville inequality on projective variety and rational points on analytic disks | Let $(X, L)$ be a polarized projective variety defined over a number field. If $D$ is a one dimensional analytic disk in $X(\mathbf{C})$ then, a classical theorem of Bombieri and Pila tells us that there are at most $\exp(\epsilon T)$ rational points of height at most $T$ in $D\cap X(K)$. Following an idea due to Masser, one can see that, if $D$ contains some special transcendental point, then the cardinality of the set of points of height at most $T$ in $D\cap X(K)$ is polynomial in $T$. These special points, called points of type $S$, verify a property which is similar to Liouville inequality. We will explain why points of type $S$ are full $X(\mathbf{C})$ (for the Lebesgue measure). | |
HS2016 | |||
| 06.10.2016 | Weakly admissible lattices, o-minimality, and Diophantine approximation | We present new estimates for the number of lattice points in sets such as aligned boxes which, in certain cases, improve on Skriganov's celebrated counting results. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. Moreover, we use o-minimality to describe large classes of sets to which our counting results apply. If time permits we also present a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erd\H{o}s and others. | |
| 27.10.2016 | A New Northcott Property for Faltings' Heights | We develop explicit techniques using tools from integral $p$-adic Hodge theory to study the change in Faltings' height within an isogeny class of CM abelian varieties. Assuming the Colmez conjecture, this results in a new Northcott property for Faltings' heights for CM points. On the Hilbert modular variety we are moreover able to develop a Colmez-type formula for the Faltings' height of all CM points. | |
| 10.11.2016 | The height of subvarieties | Heights are a fundamental tool in diophantine geometry. I will show how to extend the definition of the Weil height for points in the projective space to subvarieties of (multi)projective spaces and what properties can be proved about it. | |
| 17.11.2016 | Bounding the height of certain algebraic numbers | Especially since a paper of Bombieri-Masser-Zannier in 1999, bounding the height became the standard first step in getting finiteness for certain problems. Therefore, bounding the height of certain families of algebraic numbers or points from above is an active area of current research. In my master thesis I showed that the height of an algebraic number $\alpha$ satisfying $\alpha^n + (1-\alpha)^n + (1+\alpha) =1$ for some integer $n\ge2$ or satisfying $\alpha^r + (1-\alpha)^s =1$ for some integers $r\ge1$ and $s\ge1$, not both $1$, is bounded. The proofs are fully effective and I gave concrete values for bounds in these two situations. Both cases are special ones of a much more general result by Amoroso-Masser-Zannier (to appear). In this talk I will present my approach to prove these two results using a version of Siegel's Lemma and I will focus on how I dealt with the ``non-vanishing`` problem usually arising in this context. The latter is done differently and much simpler than by Amoroso-Masser-Zannier due to the special cases here, while the general method is the same. | |
| 24.11.2016 | On division fields of CM abelian varieties | Let K be a number field and A/K be an abelian variety with complex multiplication. We consider the extensions of K generated by torsion points of A and give uniform bounds for their degrees in terms of K and of the dimension of A. This refines a result of Ribet and has applications to some cases of the (uniform) Rasmussen-Tamagawa conjecture. | |
| 08.12.2016 | Simultaneous torsion in the Legendre family of elliptic curves | ||
| 22.12.2016 | The Bounded Height Conjecture for Semiabelian Varieties | The Bounded Height Conjecture of Bombieri, Masser and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian variety G there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in G. After partial work of many authors, Habegger proved the conjecture completely for both tori and abelian varieties in 2009. In my talk, I will discuss how to prove the conjecture for general semiabelian varieties. | |
FS2016 | |||
| 25.02.2016 | Reductions of abelian varieties | Let A and A’ be two abelian varieties defined over a number field. Our goal is comparing A and A', for example we would like to know if they are isogenous. We consider all reductions, and the information at our disposal are the number of points over the residue fields or the size of the reductions of the Mordell-Weil group. This is joint work with Chris Hall (University of Wyoming). | |
| 03.03.2016 | Around the Property (B) | The Property (B) was introduced 15 years ago by Bombieri and Zannier to study fields of algebraic numbers on which the Weil height can be bounded from below (outside of the roots of unity). This property is linked to a relative version of the Lehmer problem, which plays an important role in diophantine geometry, for instance towards the Zilber-Pink conjecture. We'll discuss examples that have been found recently of fields with the Property (B). | |
| 10.03.2016 | Effective Pila--Wilkie bounds for restricted Pfaffian surfaces | The counting theorem of Pila and Wilkie has become celebrated for opening up one of the most important developments in applied model theory in recent years. It provides a bound on the density of rational points of bounded height lying on the `transcendental parts' of sets definable in o-minimal expansions of the real field, a result which has had several stunning number theoretic applications (e.g. to the Manin-Mumford and André-Oort Conjectures). However, the proof of the theorem is not effective: it does not give a procedure which, given a definable set, will compute the Pila--Wilkie bound for that set. This of course constrains the effectivity of its applications. I will discuss some recent progress made towards finding an effective version of the Pila--Wilkie Theorem in certain cases. (Joint work with G. O. Jones.) | |
| 17.03.2016 | Around the Northcott property | Motivated by the question of counting periodic points of endomorphisms of an abelian variety, Northcott proved in 1949 that every number field contains just finitely many points of bounded height. There are also fields of infinite degree over $\mathbb{Q}$ satisfying this finiteness property. We say that such fields satisfy the Northcott property (N). In this talk we will discuss relations between (N) and other arithmetic properties a field $F\subseteq\overline{\mathbb{Q}}$ may have. | |
| 31.03.2016 | Spaces generated by products of two Eisenstein series. | Which modular forms are linear combinations of products of two Eisenstein series? Writing a modular form as such a linear combination can be used in many applications, e.g. for calculating Fourier expansions at any cusp. In joint work with Martin Dickson we show that for many congruence subgroups all modular forms are linear combinations of products of Eisenstein series. I will present a proof of this result using the Rankin-Selberg method and modular symbols. In the end I will talk about recent work by Rogers-Zudilin and Brunault on some of Boyd's conjectures concerning relations between Mahler measures and L-values of elliptic curves. They write a newform of weight 2 as a linear combination of products of Eisenstein series in order to study the L-value L(f,2). I will indicate how this approach can be generalised to study the L-value of a K3 surface and connect it to Mahler measures. | |
| 08.04.2016 | On the Hilbert Property and the fundamental group of algebraic varieties | This would concern recent work with P. Corvaja in which we relate the Hilbert Property (a kind of axiom linked with Hilbert Irreducibility) for an algebraic variety with its fundamental group. This leads to new examples (of surfaces) both of validity and failure of the Hilbert Property. | |
| 26.04.2016 | Height of rational points on algebraic varieties | Given a finite set S of irreducible homogeneous polynomials in several variables and with integral coefficients, a classical problem in number theory is to describe the set V(Q) of rational points on the variety V defined by S, that is the set of common zeros with rational coordinates of the polynomials in S. Typical questions are: when is V(Q) non-empty or infinite? If V(Q) is infinite, can we measure its size? If V(Q) is non-empty, what is the size of its smallest element? We will be concerned in particular with the latter problem. While all these questions are too hard in such generality, we will try to explain how they become easier when investigated on average over families of varieties. | |
| 19.05.2016 | The Distribution of Slopes of Gaussian Primes for the Rational Function Field | Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a function field analogue to this result, by making use of "super even" characters and their associated L-functions. By applying a recent result of N. Katz concerning the equidistribution of super even characters, I will also provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime. | |
| 26.05.2016 | Compatible systems of $\ell$-adic representations arising from abelian varieties. | Famous (and still unproven in full generality) conjectures of Serre-Grothendieck, Tate and Fontaine-Mazur describe $\ell$-adic representations that arise from the action of the absolute Galois group of a number field $K$ on the (twisted) $\ell$-adic cohomology groups of projective algebraic varieties that are defined over $K$. Assuming all these conjectures (and the Hodge conjecture), we discuss the following question: which $\ell$-adic representations correspond to the $\ell$-adic Tate modules of an abelian variety? We give an answer for abelian varieties without nontrivial endomorphisms. This is a report on a joint work with Stefan Patrikis and Felipe Voloch. | |
| 02.06.2016 | Two reformulations of the Lehmer conjecture | Salem numbers are those numbers (real >1) with at most one conjugate outside the unit disc and at least one on the unit circle. Salem's conjecture asserts that they cannot be too close to 1. More generally, Lehmer's conjecture asserts that given any algebraic unit, which is not a root of unity, the modulus of the product of the conjugates lying outside the unit disc (the Mahler measure) is bounded away from 1. I will present two joint works, one with B. Deroin and the other with P. Varju, in which we give two unrelated ways to reformulate Salem's and Lehmer's conjecture. The first as a uniform spectral gap for a certain family of hyperbolic surfaces, the second as an elementary counting problem in finite fields. | |
| 09.06.2016 | Polarized isogeny orbits in families of abelian varieties | Abstract: Let $\mathcal{A}$ be an abelian scheme over $B$. Fix a point $s$ in $\mathcal{A}$ and let $\Sigma$ be the polarized isogeny orbit generated by $s$. We characterize curves in $\mathcal{A}$ whose intersections with $\Sigma$ is Zariski dense and give the conjecture for higher dimensional subvarieties. | |
HS2015 | |||
| 17.09.15 | The Lang-Vojta conjecture and arithmetic finiteness results for smooth hypersurfaces | In 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for hypersurfaces. We will prove, assuming the conjecture of Lang-Vojta, the analogous finiteness statement for hypersurfaces of fixed degree and fixed dimension. Unconditionally, we prove the Shafarevich conjecture for hypersurfaces of Hodge level at most one, and some hypersurfaces of Hodge level 2. This is joint work with Daniel Loughran. | |
| 09.10.15 | Galois groups of logarithmic equations | I will describe a recent joint work with A. Pillay, where we extend to semi-abelian schemes the classical theorem of Ax on the exponential of algebraic functions. The proof is based on my co-author's theory of logarithmic differential equations, combined with an argument of Galois descent reminiscent of Kummer theory | |
| 15.10.15 | Multiplicative and modular diophantine problems | I will describe some diophantine results and conjectures, from the Mordell conjecture of 1922 (theorem of Faltings) to the open and very general Zilber-Pink conjecture. I will describe a recent result and conjecture of similar flavour which are not formally consequences of the Zilber-Pink conjecture. | |
| 22.10.15 | A solution of a problem of Ramanujan of 1915 | It is well known that the number of divisors $d(n)$ is large, when $n$ is a product of many primes. Wigert determined the maximal order of magnitude of the divisor function: \[d(n) \leq \exp ( (\log 2 +o(1)) \frac{\log n}{\log \log n},\] or in other words \[ \max_{n\le x} \log d(n) \sim (\log 2){\frac{\log x}{\log \log x}}. \] Ramanujan (1915) was the first to investigate the maximal order of magnitude of the iterated divisor function $d(d(n))$, giving an example that \[ d(d(n) \ge (\sqrt{2}\log 4 - o(1)) \frac{\sqrt{\log n}}{\log \log n}\] for infinitely many $n$. Erd\H{o}s, K\'{a}tai, Ivi\'{c} and Smati gave upper bounds, but determining an analogue of Wigert's result, i.e. a best possible upper bound, was an open problem. In this talk we give a solution to this problem: \[ \max_{n\le x} \log d(d(n))= \frac{\sqrt{\log x}}{\log_2 x} \left( c + O(\frac{\log_3 x}{\log_2 x} )\right), \] where \[ c =\Bigg( 8 \sum_{j=1}^\infty \log^2 (1+1/j) \Bigg)^{1/2} = 2.7959802335\ldots. \] (Joint work with Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta.) | |
| 29.10.15 | An explicit Andr\'e-Oort type result for P^1(C) x G_m(C). | We will discuss a problem of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ are of the form $(\alpha, \lambda)$, with $\alpha$ a singular modulus and $\lambda$ a root of unity. The qualitative version of our result states that if $\mathcal{C}$ is a closed algebraic curve in $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$, defined over a number field, not containing a horizontal or vertical line, then $\mathcal{C}$ contains only finitely many special points. We discuss two approaches, one using logarithmic forms, and another using class field theory. Both approaches give explicit results. | |
| 12.11.15 | p-adic measures of Hermitian modular forms and the Rankin-Selberg method | p-adic measures are playing an important role in the various Main Conjectures of Iwasawa Theory. In this talk I will start by presenting some basic properties of the classical L functions associated to a Dirichlet character, such as the Kummer congruences, and then explain how these properties can be understood in a broader context, namely that of the existence of a p-adic measure. Then after discussing some basics of Hermitian modular forms, (automorphic forms associated to unitary groups) I will present the construction of various p-adic measures, which can be associated to a Hermitian modular form by employing the so-called Rankin-Selberg method. | |
| 03.12.15 | Gamma values: regular and irregular | The values of the gamma function at rational numbers remain quite mysterious, one of the reasons being that (conjecturally) they are not periods in the usual sense of algebraic geometry. However, the theory of regular singular connections allows to show that suitable products of them are periods of Hodge structures with complex multiplication, as predicted by Gross and Deligne. To deal with single gamma values, one needs to consider irregular singular connections instead. After an exposition of my results on the Gross-Deligne conjecture, I will explain how irregular singular connections may shed some light on the arithmetic nature of these numbers. | |
| 10.12.15 | Algebraic structures of exponential maps | I will discuss the problem of giving a complete algebraic description of the interaction between algebraic geometry and the exponential map of a complex algebraic group, focusing on the case of a simple abelian variety. The problem and many of the tools originate in model theory, but I will concentrate more on the role played by transcendence theory and the Faltings-Ribet Kummer theory for abelian varieties. The talk will be based on recent work with Jonathan Kirby, extending work of Zilber on the case of the multiplicative group. | |
FS2015 | |||
| 21.04.15 | Abelian varieties and maximal orders | I study abelian varieties whose endomorphism ring is a maximal order. The algebraic properties of these orders, which can be seen as non-commutative analogs of Dedekind domains, allow to prove structure theorems for the aforesaid abelian varieties. For example, they are always products of simple varieties. Furthermore, this kind of results also give information for an arbitrary abelian variety, since it is isogenous to another one as above. I give also an application to the size of the torsion part of the group of rational points over a number field. | |
| 28.04.15 | Singular modular that are S-units / Generalized Jacobians and additive extensions of elliptic curves | ||
| 07.05.15 | Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation. | Let E_t be the Legendre elliptic curve of equation Y^2=X(X-1)(X-t). In 2010 Masser and Zannier proved that, given two points on E_t with coordinates algebraic over Q(t), there are at most finitely many specializations of t such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. This fits inside the framework of the so-called Unlikely Intersections. As a natural higher-dimensional analogue, we considered the case of n generically independent points on E_t with coordinates algebraic over Q(t). Then there are at most finitely many specializations of t such that two independent relations hold between the specialized points. We recently also dealt with the case of points on Jacobians of genus two curves. This has applications in the study of solvability of the polynomial (almost) Pell equation. This is joint work with Laura Capuano. | |
| 21.05.15 | Frobenius distribution for pairs of elliptic curves and exceptional isogenies | We will discuss a proof of the following result: if E and E' are two elliptic curves over a number field K, then there exist infinitely many primes p of K such that the reductions of E and E' modulo p are geometrically isogenous. The proof relies on the arithmetic dynamics of Hecke correspondences. | |
| 28.05.15 | Bad reduction of curves with CM jacobians | An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves. We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over the algebraic closure of the rationals with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence, except for finitely many examples, such a curve will always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit infinite families of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of the rationals of degree 4 that contains F for some specific F | |
| 19.06.15 | Solving S-unit and Mordell equations via Shimura-Taniyama conjecture | Joint work with Benjamin Matschke. In the first part of this talk, we shall present new practical algorithms which solve S-unit and Mordell equations by combining the method of Faltings (Arakelov, Parshin, Szpiro) with the Shimura-Taniyama conjecture. Our algorithms do not use lower bounds for linear forms in logarithms and they considerably improve the actual best algorithms. In the second part we plan to discuss in detail the construction of a sieve used in the algorithm for Mordell equations. Our sieve settles in particular the open problem of efficiently enumerating integral points of bounded height on elliptic curves. |