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Contributed talks
20th Atelier PARI/GP
(April 15-16)
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Abstracts
Abstract
In his seminal 1992 paper on Fontaine-Mazur Conjecture for prime degree extensions of number fields, Boston asked whether the method he develops could lead to a counter-example to Fontaine-Mazur Conjecture when applied to the biquadratic field $\mathbb{Q}(\sqrt{-26}, \sqrt{229})$.
In a joint work with S. Pisolkar (IISER Pune), we answer negatively to this question by making explicit the Galois group that was expected to provide the aforementioned counter-example. In particular, we prove that it is a finite group of order 6561 and that it is not a uniform group.
In this talk, which requires no specific pre-requisite beyond classical master courses, I will first explain the precise setting we are interested in, then I will introduce the main tools and ideas we used to answer Boston's original question.
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Abstract The aim of this talk is to present some recent results on linear recurrences with a prescribed number of residues. The central problem is the following: given a polynomial $f$ and a positive integer $n$, one wants to understand whether there exist a recurrence sequence of integers $s$ with characteristic polynomial $f$, and a positive integer $M$ such that $s$ modulo $M$ has exactly $n$ distinct residues. The first part of the seminar will be devoted to recalling some prior results and to enlightening a connection of the previous problem with other interesting and relatively old issues in Number Theory. In the second part, the focus will be on a result in the case of certain second order linear recurrences, with a special emphasis on the key ideas behind it. This is a joint work with C. Sanna. |
Abstract We explore low-degree points on modular curves $X_0(N)$ and $X_1(N)$, as well as the quotients of $X_0(N)$ by Atkin-Lehner involutions. These curves function as moduli spaces for elliptic curves with additional structures.
We employ variations of Chabauty's method, including quadratic Chabauty, to provably determine all rational points on curves $X_0^+(p)$ of genus up to $6$ (for prime $p$). Additionally, we classify rational points on such $X_0^+(p)$ and on hyperelliptic $X_0^*(N)$ when $N$ is squarefree.
We introduce enhancements to the symmetric Chabauty method, enabling us to determine all the quadratic points on $X_0(N)$ for numerous levels $N$. We also apply techniques such as the Mordell-Weil Sieve and use quotients to elliptic curves, etc.
This presentation encompasses several works, including some written jointly with many coauthors (Arul, Beneish, Chen, Chidambaram, Keller, Michaud-Jacobs, Najman, Ozman, Padurariu, Vukorepa and Wen), as well as a selection of both classical and recent results. |
Abstract
The aim of the talk is to provide infinite families of trinomials over finite fields of even characteristic
with high differential uniformity, which implies in particular that they are not exceptional almost perfect nonlinear,
supporting thereby a conjecture on such polynomials. |
Abstract
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Abstract
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Abstract The Lindemann-Weierstrass Theorem states that the values of the exponential function at algebraic numbers linearly independent over Q are algebraically independent over Q.
The aim of this talk is to state conjectures in Lindemann-Weierstrass style
and to explain their link with Grothendieck periods conjecture.
In particular we will furnish the motivic origin of Waldschmidt's results
stated in the paper "Nombres transcendants et fonctions sigma de Weierstrass" published in C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79).
(joint work with Michel Waldschmidt) |
Abstract
The circle method is a technique developed by Ramanujan, Hardy and
Littlewood and which has allowed to solve many difficult counting
problems in number theory throughout the last century. More recently, a
version of the method over function fields, combined with spreading out
techniques, has led to new results about the geometry of moduli spaces
of rational curves on hypersurfaces of low degree, thus expanding the
scope of the circle method to the realm of algebraic geometry. In this
talk, after summarizing the main ideas of the circle method, I will show
how to implement a circle method with an even more geometric flavour,
where the computations take place in a suitable Grothendieck ring of
varieties, and explain how this leads to a more precise description of
the geometry of the above moduli spaces.
joint work with Tim Browning. |
Abstract I will report on recent joint work with Yingkun Li. We
consider one-parameter families
of abelian surfaces in charactheristic p>0 obtained by reduction of
curves in Hilbert modular surfaces
(not necessarily Shimura curves). We prove that the supersingular locus
of such families is described
by the zeros of certain orthogonal polynomials, generalizing the work of
Atkin and Kaneko-Zagier on
supersingular elliptic curves.
The proof is based on the theory of Hilbert modular forms in positive
characteristic and on new results
on modular forms on non-arithmetic groups. |
Abstract
For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists have no integral points. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses the reduction theory of binary quartic forms, Manin-type bounds for certain singular cubic surfaces, and character sum estimates drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve. This is joint work with Tim Brown |
Abstract
Let $m \in \mathbb{N}$ be large, there exist infinitely many primes $p_{1}< \cdot\cdot\cdot < p_{m+1}$ such that
$$
p_{m+1}-p_{1}=O(e^{7.63m})
$$
and $p_{j}+2$ has at most
$$
\frac{7.36m}{\log 2} + \frac{3\log m}{\log 2} + 29
$$
prime factors for each $1 \leq j \leq m+1$.
This improves the previous result of Li and Pan in 2015, replacing $m^{4}e^{8m}$ by the better value $e^{7.63m}$ and $\frac{16m}{\log 2} + \frac{5\log m}{\log 2} + 37$ by $\frac{7.36m}{\log 2} + \frac{4\log m}{\log 2} + 21$. The main technical aids in my argument
are the Maynard-Tao sieve, a minorant for the indicator function of the primes constructed by Baker and Irving, for which a stronger equidistribution theorem in arithmatic progressions to smooth moduli is applicable, and Tao's approach previously used to estimate $$\sum_{x \leq n < 2x} \mathbf{1}_{\mathbb{P}}(n)\mathbf{1}_{\mathbb{P}}(n+12)\omega_{n},$$ where $\mathbf{1}_{\mathbb{P}}$ stands for the characteristic function of the primes and $\omega_{n}$ are multidimensional sieve weights.
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Abstract The aim of this talk is to introduce a new balanced triple product p-adic L-function and discuss its application to the equivariant Birch & Swinnerton-Dyer conjecture. We state a conjecture in a rank-1 situation analogous to the Elliptic--Stark conjecture formulated by Darmon--Lauder--Rotger in rank-2 and prove it in the CM case; this work fits in the general framework studied by Darmon--Lauder--Rotger and Andreatta--Bertolini--Seveso--Venerucci. Time permitting, we will explain briefly the intriguing technical difficulties behind the construction of this new p-adic L-function whose main feature is to allow classical weight one modular forms in the chosen families. That is joint work with Aleksander Horawa. |
Abstract In 1909, Axel Thue considered equations of the form $F(x,y) = m$, where $m$ is a non-zero integer, and $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible homogeneous \textit{binary form} of degree $n \geq 3$. He managed to prove that such equations (now known as Thue equations have only finitely many integer solutions $(x,y) \in \mathbb{Z}^{2}$. Thue's result, however, was not effective. Baker resolved this in the 1960's, by developing powerful methods to compute lower bounds for linear forms in logarithms. Such tools could then be applied to solve Thue equations effectively.
One direction of investigation then turned towards studying parametrized \textit{families} of Thue equations. E. Thomas, for instance, considered the family of cubic forms
\begin{equation}\label{eq:cubic}
F^{(3)}_{t}(x,y) := x^3-(t-1)x^2y-(t+2)xy^{2}-y^{3}
\end{equation}
for $t \in \mathbb{Z}_{\geq 0}$. He conjectured that for $t \geq 4$, the Thue equation
\[F^{(3)}_{t}(x,y) = \pm 1 \]
has only the ``trivial" solutions $(x,y) \in \{(0,\mp 1), (\pm 1,0), (\mp 1,\pm 1)\}$. Such a conjecture was eventually proved correct by Mignotte in $1993$. More general questions related to such Thue equations over numbers fields were addressed by many authors.
\medskip
One may also consider Thue equations in the function field setting. More precisely, we consider equations of the form $F(x,y) = m$, for some non-zero $m \in \mathbb{C}[T]$, where
\begin{equation}\label{eq:FF_Thue}
F(x,y) = a_{0}x^{n} + a_{1}x^{n-1}y+\cdots + a_{n-1}x y^{n-1}+a_{n}y^{n}, \hspace{5mm} a_{i} \in \mathbb{C}[T],
\end{equation}
is irreducible, and where we now seek solutions $(x,y) \in \mathbb{C}[T] \times \mathbb{C}[T]$. Families of Thue equations over $\mathbb{C}(T)$ were first discussed by \textit{Fuchs and Ziegler in 2006.}
\medskip
In this paper we completely solve a simple quartic family of Thue equations over $\mathbb{C}(T)$.
Specifically, we apply the ABC-Theorem to find all solutions $(X,Y) \in \mathbb{C}[T] \times \mathbb{C}[T]$ to the set of Thue equations $F_{\lambda}(X,Y) = \xi$, where $\xi \in \mathbb{C}^{\times}$ and
\begin{equation}
F_{\lambda}(X,Y):=X^4 -\lambda X^3Y -6 X^2Y^2 + \lambda XY^3 +Y^4 \quad \quad \lambda \in \mathbb{C}[T]/\{\mathbb{C}\}
\end{equation}
denotes a family of quartic simple forms.
This is a joint work with E. Waxman, I. Vukusic and V. Ziegler. |
Abstract
In this talk we will see some cases of the Zilber-Pink conjecture for curves in products of fibered powers of elliptic and abelian schemes. After a brief historical introduction, we will talk about a new result concerning the Zilber-Pink conjecture for a curve in the product of two fibered powers of elliptic schemes, provided that the curve satisfies a certain condition on the degrees of some of its coordinates. |
Abstract
Let $A$ and $A^\prime$ be abelian varieties defined over a number field $k$. In the talk I will consider the following question: Is it true that $A$ and $A^\prime$ are quadratic twists of one another if and only if they are quadratic twists modulo $p$ for almost every prime $p$ of $k$? Serre and Ramakrishnan have given a positive answer in the case of elliptic curves and a result of Rajan implies the validity of the principle when the endomorphism ring of $A$ (and hence also that of $A^\prime$) over an algebraic closure of $\mathbb{Q}$ is just $\mathbb{Z}$. For not necessarily simple abelian varieties, I will show that the answer is affirmative up to dimension 3, but that it becomes negative in dimension 4. Time permitting, I will present ongoing joint work with E. Ambrosi that leads to more general results. |
Abstract
In 1972 Serre proved his celebrated Open Image Theorem, stating that for any elliptic curve $E$ defined over a number field without complex multiplication, the image of the adelic Galois representation $\rho_E$ is an open subgroup of $GL_2(\widehat{\mathbb{Z}})$, and so it has finite index. It is conjectured that the index is uniformly bounded as $E$ varies. Since the publication of Serre's results, the possible images of the mod $p$ representations have been widely investigated. Recently, Zywina gave some restrictions on the possible images of Galois representations mod $p^n$ attached to rational elliptic curves without complex multiplication. As a consequence, Lombardo gave an explicit bound on the index of the image of the adelic representation. I will explain how to improve the classification results of Zywina and Lombardo's bound in the case of rational elliptic curves.
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Abstract
We will discuss when abelian varieties defined over finite fields are geometrically isogenous and give concrete examples in small dimensions. We restrict attention to elliptic curves defined over infinite extensions over finite fields. We use the fact that there are infinitely isogeny classes in this case to briefly explain one application to mathematical logic. |
Abstract In 1970s, Shimura initiated the study of a non-holomorphic operator, the Maass-Shimura operator, which led him to define the notion of nearly holomorphic modular forms. He later discovered their role on constructing class fields as well as the connection with periods of CM elliptic curves. In this talk, our goal is to introduce their positive characteristic counterpart, nearly holomorphic Drinfeld modular forms. Moreover, we introduce Maass-Shimura operators in our setting and investigate the relation between the periods of CM Drinfeld modules and the values at CM points of arithmetic Drinfeld modular forms under the image of such operatiors. If time permits, we also explain the link between our objects and Drinfeld quasi-modular forms introduced by Bosser and Pellarin. This is a joint work with Yen-Tsung Chen. |
Abstract In this talk, we give an overview of recent results obtained
by performing l-descent on elliptic curves (and abelian varieties) over
function fields of one variable. These results include an arithmetic
refinement of the rank bound given by the Grothendieck-Ogg-Shafarevich
formula, an upper bound on the number of S-integral points on elliptic
curves, and the construction of number fields with large class groups.
This is joint work with Aaron Levin, Emmanuel Hallouin and Félix Baril
Boudreau. |
Abstract In this talk, we present a class of trinomials over the finite field $\mathbb{F}_q$ of odd order $q$. We show that the class contains an irreducible trinomial and establish an irreducibility criterion, analogous to the well-known Serret's irreducibility criterion, for the class of trinomials. |
Abstract 20+ years ago, Soundararajan and I initiated a study of the spectrum, the range of possible mean values of multiplicative functions which take their values at primes in some given set S. We resolved this question for $S=[-1,1]$, and asked for more general results.
There are several obvious directions to take this -- S is the set of $m$th roots of unity for $m>2$, or $S=[-l,k]$, or ...
In this talk I will report on recent progress on these questions in joint works with Kevin Church, Kaisa Matomaki, K. Soundararajan and Daodao Yang. |
Abstract The endomorphism ring of a supersingular elliptic curve defined over a finite field is a maximal order inside a quaternion algebra. Computing this order is a hard problem and this assumption is central to the security of protocols in isogeny-based cryptography, which, as the name suggests, are based on the mathematical problem of finding an isogeny between two given elliptic curves.
In this talk, we will discuss recent advances about this problem. In particular we will highlight the main ideas behind an algorithm which computes the endomorphism ring of a supersingular elliptic curve over a finite field in $O(\sqrt{p}(\log p)^2(\log\log p)^3)$ bit operations, only under the assumption of the Generalized Riemann Hypothesis.
This algorithm is the result of a collaboration with Jenny G. Fuselier, Mark Kozek, Travis Morrison, and Changningphaabi Namoijam. |
Abstract
We study the height of generators of Galois extensions of the rationals having the alternating group as Galois group. In particular, we prove that if such generators are obtained from particular, albeit classical, constructions, their height tends to infinity as n increases. This provides an analogue of a result by Amoroso, originally established for the full symmetric group. |
Abstract A birational map of affine (or projective) space is a simple example of an algebraic dynamical system. To such a system is associated a fundamental invariant, called the dynamical degree. In most cases when it can be explicitly computed, it is an algebraic number, and a conjecture by Bellon and Viallet predicted that dynamical degrees are always algebraic numbers. I will present joint work with J. Bell, J. Diller, and H. Krieger, where we construct birational maps in dimension three with transcendental dynamical degrees. |
Abstract Universal quadratic forms generalize the sum of four squares about which it is well known that it represents all positive rational integers. In the talk, I'll start by briefly discussing some results on universal quadratic forms over totally real number fields. Then I'll move on to the - markedly different! - situation over infinite degree extensions K of Q. In particular, I'll show that if K doesn't have many small elements (i.e., "K has the Northcott property"), then it admits no universal form. The talk is based on a recent joint work with Nicolas Daans and Siu Hang Man. |
Abstract
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Abstract For a real quadratic field F and K/F a CM extension which is a quartic
cyclic extension of Q, consider the jacobians of curves of genus 2 with
CM by $\mathcal{O}_K$ such that the curves themselves have potentially
good reduction everywhere. Habegger and Pazuki proved that the height of
such jacobians can be bounded (ineffectively) in terms of F only. In
the present work (joint with Linda Frey and Elisa Lorenzo-Garcia), we
provide (under GRH) an explicit bound in terms of quantities of F. In
this talk, I will explain the proof and main difficulties to be overcome
to obtain and possibly improve this kind of explicit bounds |
Abstract We discuss recent results developing Diophantine approximation in the setting of closed subschemes, including a new inequality on surfaces. After stating the main results, we will discuss some applications to various Diophantine problems
(joint work with Keping Huang and Zheng Xiao). |
Abstract
We show the primes have level of distribution 66/107 using triply well-factorable weights, and extend this level to 5/8 assuming Selberg's eigenvalue conjecture. This improves on the prior world record level of 3/5 by Maynard. As a result, we obtain new upper bounds for Goldbach representations of even numbers. This is the first use of a level of distribution beyond the 'square-root barrier' for the Goldbach problem, and leads to the greatest improvement on the problem since Bombieri-Davenport from 1966. |
Abstract Let $A$ be an abelian variety over a number field $K$. The question I will discuss in this talk is that of bounding the size of the torsion subgroup of $A(L)$ as $L$ ranges over the finite extensions of $K$. In particular, one can ask for the optimal exponent $\beta_A$ such that for all $\varepsilon >0$ there is a constant $C_{A, \varepsilon}$ such that
$$
\#A(L)_{\operatorname{tors}} \leq C_{A, \varepsilon} [L:K]^{\beta_A + \varepsilon}
$$
for all finite extensions $L/K$. Hindry and Ratazzi have formulated a precise conjecture in this direction: the value of $\beta_A$ should be given by a simple formula involving the Mumford-Tate group of $A$. They also proved this conjecture for several classes of abelian varieties for which the Mumford-Tate conjecture is known to hold.
?In joint work with Samuel Le Fourn and David Zywina, we give an unconditional formula for the optimal exponent $\beta_A$ and show that it agrees with the conjectured one under the assumption of Mumford-Tate. Using this, we prove that the conjecture of Hindry and Ratazzi is equivalent to the Mumford-Tate conjecture for abelian varieties over number fields. We also give sharp lower bounds on the degrees of the extensions of K generated by a single torsion point. |
Abstract
For a number field $K$, we consider $K^{ta}$ the maximal tamely ramified extension of $K$, and its Galois group $G_K^{ta}=Gal(K^{ta}/K)$.
Our guiding aim is to characterize the finitely generated pro-$p$ quotients of $G_K^{ta}$.
We give a unified point of view by introducing the notion of stably inertially generated pro-$p$ groups $G$, for which linear groups are archetypes.
In particular we prove: Let $p$ be an odd prime. Then for every $m\geq 2$ and $k\geq 2$, the pro-$p$ groups $SL_k^m(Z_p)$ are quotients of $G_K^{ta}$.
Joint work with F. Hajir, M. Larsen, and R. Ramakrishna. |
Abstract Continued fractions provide important methods to construct transcendental numbers. The first studies in this direction are due to Liouville, who dealt with
unbounded partial quotients, then Maillet and Baker exhibited continued fractions
with bounded partial quotients converging to transcendental numbers. Furthermore,
Baker?s results have been recently improved by several other authors.
In this talk, we present some analogue results for Browkin p-adic continued fractions. Firstly, we focus on the heights of some p?adic numbers having a periodic p?adic continued fraction expansion and we obtain some upper bounds. Thanks to these results, together with p-adic Roth-like theorems, we prove the transcendence of three families of p?adic continued fractions.
This is a joint work with Ignazio Longhi and Francesco Saettone. |
Abstract Let $X_\Delta(N)$ be an intermediate modular curve of level $N$, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p$ such that $X_\Delta(N)_{\overline{\mathbb F}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes $p$ such that $X_\Delta(N)_{{\mathbb F}_p}$ is trigonal and show that we show that $X_\Delta(N)_{\overline{\mathbb F}_p}$ is not a smooth plane quintic, for any $N$ and any $p$.
This is joint work with Maarten Derickx. |
Abstract In this talk, I explain how we work towards completely classifying all bielliptic Shimura curves $X_0^D(N)$ with nontrivial level $N$, extending a result of Rotger that provided such a classification for level one. This allows us to determine the list of all pairs $(D,N)$ for which $X_0^D(N)$ has infinitely many degree 2 points, up to 3 explicit possible exceptions.
This is joint work with Frederick Saia. |
Abstract
We describe the relationship between the local-global principle for divisibility of points in commutative algebraic groups and the local-global principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Ch\hat{a}telet group.\ The two local-global subjects arose as a generalization of some classical questions considered by Hasse and Cassels respectively. We also report some of the most recent results achieved for the two problems.
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Abstract
I'll speak about new joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance set if the distance between any pair of points in the set is an integer. Our main result is that any integer distance set in the plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size n and a strong upper bound on the size of any integer distance set in $[-N,N]^2$ with no three points on a line and no four points on a circle. |
Abstract
A positive definite quadratic form with integral coefficients is universal if it represents all positive integers. With his famous Four Squares Theorem, Lagrange proved that the form $x^2+y^2+z^2+t^2$ is universal. The Conway–Schneeberger $15$ Theorem says that a classical form is universal if and only if it represents all numbers in the criterion set $\{1,2,3,5,6,7,10,14,15\}$.
In this talk, we address a similar question for quadratic forms over number fields. In particular, for all number fields we characterize the criterion set for universality in terms of indecomposable algebraic integers and we prove its uniqueness. Moreover, we provide some results on the cardinality of this set. The talk is based on a joint work with J. Krásenský and V. Kala. |
Abstract In the realm of integer factorization, certain methods, such as CFRAC, leverage the properties of continued fractions, while others, like SQUFOF, integrate these properties with the tools provided by quadratic forms. Recently, Prof. Michele Elia revisited the fundamental concepts of SQUFOF, including reduced quadratic forms, distance between quadratic forms and Gauss composition, offering a new perspective for a factorization method.
In this talk, we introduce our algorithm, which is a refinement of the method proposed by Elia, providing also a more precise analysis of the computational cost. Our algorithm, taking a positive integer $N$ as input, relies on the precomputation of the regulator of $\mathbb{Q}(\sqrt{N})$, which is related to the positive integral solutions of the Pell's Equation $X^2 - NY^2=1$.
The computational cost of this algorithm is driven by the cost of the calculation of the regulator and it is subexponential, in particular $O \left ( \exp \left (\frac{3}{\sqrt{8}}\sqrt{\ln N \ln \ln N} \right ) \right )$, making it more efficient than CFRAC and SQUFOF, though less efficient than the General Number Field Sieve.
We also present some promising avenues for refining our method. These span diverse areas, ranging from Analytic Number Theory, particularly in computing a specific class of $L$-functions, to the theory of Diophantine Equations, particularly in the resolution of a specific class of Pell's Equations.
Joint work with N. Murru. |
Abstract
A random multiplicative function is a (completely) multiplicative sequence $f(1), f(2), \ldots$ such that the $f(p)$ for primes $p$ are independent random variables uniformly distributed on either the unit circle (known as the Steinhaus model) or the set $\{-1, +1\}$ (known as the Rademacher model). They are used to model the M\"obius and/or the related Liouville function, the mean value of which has a very rich structure and contains essential information about the distribution of prime numbers. By work of Harper, it is known that the mean value of a random multiplicative function over a long interval $[1, x]$ does not converge in distribution to a Gaussian with the expected parameters, meaning that multiplicativity interferes just enough with independence. However, Klurman, Shkredov and Xu established a central limit theorem for the mean value of a Steinhaus random multiplicative function over the image of a polynomial with distinct roots such as $n(n+1)$, say. In ongoing joint work with Jake Chinis, we establish the same result in the more intricate Rademacher model. The techniques we use come from a blend of probability theory as well as analytic and algebraic number theory on counting integral points on "twists" of certain curves and surfaces. |
Abstract
A classical problem in the theory of ?Unlikely intersections? regards the distribution of the torsion points of an abelian variety within an algebraic subvariety: the investigation around this topic has led to the proof of the Manin-Mumford conjecture. In the context of abelian schemes, a version of this theorem has been recently proved and now it is known as relative Manin-Mumford theorem. Taking a dynamical point of view, we study a variation of this problem in the context of two rational families of g-dimensional abelian varieties lying in the same ambient space and having big common domain of definition. Our work is a higher dimensional generalization of a result of Corvaja, Tsimermann and Zannier. (
Joint work with Paolo Dolce, Westlake University, Hangzhou) |
Abstract
The error term in the prime number theorem for arithmetic progressions is one of the (many) quantities affected by the possible existence of a real zero of a Dirichlet $L$-function. Such an `exceptional? zero could only occur when $L$ is attached to a quadratic character $\chi$ mod $q$. We should like to rule out such zeroes for all small moduli, say, all $q\leq Q$. I will outline some recent work, joint with Dave Platt, that allows us to take a much larger $Q$ than was previously known. |
Abstract Given coprime integers a,b, the classical identity of Bezout provides integers u,v such that au-bv = 1. We consider refinements to this identity, where we ask that u,v are norms from a quadratic extension. We then find ourselves counting optimal embeddings of a quadratic order ina quaternion order, for which we give explicit formulas in many cases. This is joint work with Donald Cartwright and Xavier Roulleau.
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Abstract
Let $d_k(n)$ represent the k-fold divisor function, counting the ways n can be expressed as the product of $k$ natural numbers. In this presentation, we are going to introduce the local uniformity of k-fold divisor functions first. Then we plan to investigate the approaches to study the local correlation between $k$-fold divisor functions and linear phase functions in very short intervals. |
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