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BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART:20201211T150000Z
DTEND:20201211T160000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/1
DESCRIPTION:Title: Some questions around counting rational points on stacks\nby
Jordan Ellenberg (University of Wisconsin–Madison) as part of ZORP (zoom
on rational points)\n\n\nAbstract\nI will talk about a few open questions
in arithmetic enumeration which are superficially different but which all
arise as special cases of a conjecture of Batyrev-Manin type for algebrai
c stacks formulated by Matt Satriano\, David Zureick-Brown\, and me. To s
tate the conjecture precisely requires one to say what one means by the he
ight of a point on a stack\; some of you have heard me talk about this par
t before\, so I am going to attempt to abbreviate that story somewhat and
use it as a black box\, focusing instead on some of the geometric challeng
es of formulating a counting conjecture\, which we have sort of but not fu
lly satisfyingly surmounted.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thélène (Université Paris-Saclay\, CNRS)
DTSTART:20201211T133000Z
DTEND:20201211T143000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/2
DESCRIPTION:Title: Jumps in the rank of the Mordell-Weil group / Sauts du rang du gr
oupe de Mordell-Weil\nby Jean-Louis Colliot-Thélène (Université Par
is-Saclay\, CNRS) as part of ZORP (zoom on rational points)\n\n\nAbstract\
nLet $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X\\
rightarrow U$ be an abelian scheme. We consider the set $U(k)_+ \\subset U
(k)$ of $k$-rational points of $U$ such that the Mordell-Weil rank of the
fibre $X_m$ is strictly bigger than the Mordell-Weil rank of the generic f
ibre over the function field $k(U)$. \n\nWe prove: if the $k$-variety $X$
is $k$-unirational\, then $U(k)_+$ is dense for the Zariski topology on $U
$. Variants are given and compared with old and new results in the literat
ure.\n\nSoient $k$ un corps de nombres et $U$ une a smooth integral $k$-va
riété lisse intègre. Soit $X\\rightarrow U$ un schéma abélien. On s
’intéresse à l’ensemble $U(k)_+ \\subset U(k)$ des points rationnels
$m \\in U(k)$ tels que le rang de Mordell-Weil de la variété abélienne
fibre $X_m$ soit strictement plus grand que celui de la fibre générique
sur le corps des fonctions rationnelle $k(U)$. \n\nOn établit : si la $k
$-variété $X$ est $k$-unirationnelle\, alors $U(k)_+$ est dense dans $U
(k)$ pour la topologie de Zariski. On donne des variantes\, et on compare
avec divers résultats dans la littérature classique et moderne.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephanie Chan (University of Michigan)
DTSTART:20210115T133000Z
DTEND:20210115T143000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/3
DESCRIPTION:Title: A density of ramified primes\nby Stephanie Chan (University o
f Michigan) as part of ZORP (zoom on rational points)\n\n\nAbstract\nLet \
n$K$\n be a cyclic number field of odd degree over $\\mathbb{Q}$ with odd
narrow class number\, such that \n$2$\n is inert in $K/\\mathbb{Q}$. We ex
tend the definition of spin (a special quadratic residue symbol) to all od
d ideals in \n$K$\, not necessarily principal. We discuss some of the idea
s involved in obtaining an explicit formula\, depending only on $[K:\\math
bb{Q}]$\, for the density of rational prime ideals satisfying a certain pr
operty of spins\, conditional on a standard conjecture on short character
sums. This talk is based on joint work with Christine McMeekin and Djordjo
Milovic.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (University of Glasgow)
DTSTART:20210115T150000Z
DTEND:20210115T160000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/4
DESCRIPTION:Title: On the statistics of number fields\, integral points and Arakelov
ray class groups\nby Carlo Pagano (University of Glasgow) as part of
ZORP (zoom on rational points)\n\n\nAbstract\nI will present new results c
oming from a novel approach to Malle’s conjectures (in its strong form):
this is a joint work with Peter Koymans. If time allows\, I will also sur
vey recent works on the statistics of the solvability\, respectively the f
ailure of weak approximation\, for integral points on conics and the distr
ibution of Arakelov ray class groups of quadratic fields: this covers join
t works (past and in progress) with Alex Bartel\, Peter Koymans and Efthym
ios Sofos. During the talk I will highlight the natural connections betwee
n these subjects.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ratko Darda (Paris University (Paris 7))
DTSTART:20210219T133000Z
DTEND:20210219T143000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/5
DESCRIPTION:Title: Manin conjecture for algebraic stacks\nby Ratko Darda (Paris
University (Paris 7)) as part of ZORP (zoom on rational points)\n\n\nAbstr
act\nWe study the conjecture of Manin--Batyrev--Peyre in the context of al
gebraic stacks.\nTwo examples are of particular interest: the compactifica
tion of the moduli stack of elliptic curves $\\overline{ \\mathcal{M}_{1\,
1}} $ and the classifying stack $ BG $ for $ G $ finite group\, which cla
ssifies $G$-torsors. The stack $\\overline{ \\mathcal{M}_{1\,1}} $ is isom
orphic to the weighted projective stack $\\mathcal{P}(4\, 6)$\nwhich is th
e quotient stack for the weighted action of $\\mathbb{G}_m$ on $\\mathbb{A
}^2\\setminus\\{0\\}$ with weights $4\, 6$. For weighted projective stack
s\, we define heights that we can use for counting\nits rational points\,
examples are given by the naive height and the Faltings’ height\nof an e
lliptic curve.\n\nWe try to motivate why the second example may help us ob
tain a geometrical reinterpretation of constants appearing in Malle conjec
ture\, which predicts the number of Galois extensions with fixed Galois gr
oup $G$.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rosa Winter (Max Planck Institute for Mathematics in the Sciences
(Leipzig))
DTSTART:20210219T150000Z
DTEND:20210219T160000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/6
DESCRIPTION:Title: Density of rational points on a family of del Pezzo surfaces of
degree 1\nby Rosa Winter (Max Planck Institute for Mathematics in the
Sciences (Leipzig)) as part of ZORP (zoom on rational points)\n\n\nAbstrac
t\nDel Pezzo surfaces are surfaces classified by their degree $d$\, which
is an integer between 1 \nand 9 (for $d\\geq 3$\, these are the smooth sur
faces of degree $d$ in $\\mathbb{P}^d$). For del Pezzo surfaces of degree
at least $2$ over a field $k$\, we know that the set of $k$-rational point
s is Zariski dense provided that the surface has one $k$-rational point to
start with (that lies outside a specific subset of the surface for degree
$2$). However\, for del Pezzo surfaces of degree 1 over a field $k$\, eve
n though we know that they always contain at least one $k$-rational point\
, we do not know if the set of $k$-rational points is Zariski dense in gen
eral. I will talk about a result that is joint work with Julie Desjardins\
, in which we give sufficient conditions for the set of $k$-rational point
s on a specific family of del Pezzo surfaces of degree 1 to be Zariski den
se\, where $k$ is any infinite field of characteristic 0. These conditions
are necessary if $k$ is finitely generated over $\\mathbb{Q}$. I will com
pare this to previous results.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arda Huseyin Demirhan (University of Illinois at Chicago)
DTSTART:20210319T150000Z
DTEND:20210319T160000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/7
DESCRIPTION:Title: Distribution of Rational Points on Toric Varieties – A Multi-He
ight Approach\nby Arda Huseyin Demirhan (University of Illinois at Chi
cago) as part of ZORP (zoom on rational points)\n\n\nAbstract\nManin's con
jecture was verified by Victor Batyrev and Yuri Tschinkel for toric variet
ies. Emmanuel Peyre has proposed two notions\, "freeness" and "all the hei
ghts" approach to delete accumulating subvarieties in "Libert\\'e et accum
ulation" and "Beyond heights: slopes and distribution of rational points".
Based on the all the heights approach\, in this talk\, we will explain a
multi-height variant of the Batyrev-Tschinkel theorem where one considers
working at {\\em height boxes}\, instead of a single height function\, as
a way to get rid of accumulating subvarieties. This is our main result: Le
t $X$ be an arbitrary toric variety over a number field $F$\, and let $H_
i$\, $1 \\leq i \\leq r$\, be height functions associated to the generator
s of the cone of effective divisors of $X$. Fix positive real numbers $a_i
$\, $1 \\leq i \\leq r$. Then the number of rational points $P \\in X(F)$
such that for each $i$\, $H_i(P) \\leq B^{a_i}$ as $B$ gets large is equa
l to $C B^{a_1 + \\dots + a_r} + O(B^{a_1 + \\dots + a_r-\\epsilon})$ for
an $\\epsilon >0$. Our result is a first example of a large family of vari
eties along the lines of Peyre's idea.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Göttingen University)
DTSTART:20210319T133000Z
DTEND:20210319T143000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/8
DESCRIPTION:Title: Campana points on toric varieties\nby Damaris Schindler (Göt
tingen University) as part of ZORP (zoom on rational points)\n\n\nAbstract
\nIn this talk we discuss joint work with Marta Pieropan on the distributi
on of Campana points on toric varieties. We discuss how this problem leads
us to studying a generalised version of the hyperbola method\, which had
first been developed by Blomer and Brüdern. We show how duality in linear
programming is used to interpret the counting result in the context of a
general conjecture of Pieropan--Smeets--Tanimoto--Varilly-Alvarado.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shuntaro Yamagishi (Utrecht University)
DTSTART:20210423T123000Z
DTEND:20210423T133000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/9
DESCRIPTION:Title: Density of rational points near/on compact manifolds with certain
curvature conditions\nby Shuntaro Yamagishi (Utrecht University) as p
art of ZORP (zoom on rational points)\n\n\nAbstract\nIn this talk I will e
xplain my work with Damaris Schindler where we obtain an asymptotic formul
a for the number of rational points near submanifolds of $\\mathbb{R}^n$ w
ith certain curvature conditions\, and what we can say about the number of
rational points on them.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miriam Kaesberg (Göttingen University)
DTSTART:20210423T140000Z
DTEND:20210423T150000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/10
DESCRIPTION:Title: On Artin's Conjecture: Pairs of Additive Forms\nby Miriam Ka
esberg (Göttingen University) as part of ZORP (zoom on rational points)\n
\n\nAbstract\nA conjecture by Emil Artin claims that for forms $f_1\, \\do
ts\, f_r \\in \\mathbb{Z}[x_1\, \\dots\, x_s]$ of degree $k_1\, \\dots\, k
_r$ the system of equation $f_1=f_2=\\dots=f_r=0$ has a non-trivial $p$-ad
ic solution for all primes $p$ provided that $s > k_1^2 + \\dots + k_r^2$.
Although this conjecture was disproved in general\, it holds in some case
s. In this talk I will focus on the case of two additive forms with the sa
me degree $k$ and sketch the proof that Artin's conjecture holds in this c
ase unless $k=2^\\tau$ for $2 \\le \\tau \\le 15$ and $k=3\\cdot 2^\\tau$
for $2 \\le \\tau$.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Margherita Pagano (Leiden University)
DTSTART:20210514T090000Z
DTEND:20210514T093000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/11
DESCRIPTION:Title: An example of Brauer-Manin obstruction to weak approximation at
a prime with good reduction\nby Margherita Pagano (Leiden University)
as part of ZORP (zoom on rational points)\n\n\nAbstract\nA way to study ra
tional points on a variety is by looking at their image in the $p$-adic po
ints. Some natural questions that arise are the following: is there any ob
struction to weak approximation on the variety? Which primes might be invo
lved in it? Bright and Newton have proven that for K3 surfaces defined ove
r number fields primes with good ordinary reduction play a role in the Bra
uer--Manin obstruction to weak approximation.\n\nIn this talk I will give
an explicit example of this phenomenon. In particular\, I will exhibit a K
3 surface defined over the rational numbers having good reduction at $2$\,
and for which $2$ is a prime at which weak approximation is obstructed.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alec Shute (IST Austria)
DTSTART:20210514T140000Z
DTEND:20210514T143000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/12
DESCRIPTION:Title: Sums of four squareful numbers\nby Alec Shute (IST Austria)
as part of ZORP (zoom on rational points)\n\n\nAbstract\nIn this talk\, I
will present an asymptotic formula for the number of nonzero squareful int
egers $z_1\, z_2\, z_3\, z_4$ which sum to zero\, are coprime\, and are bo
unded by $B$. Our result agrees in the power of $B$ and $\\log B$ with the
Manin-type conjecture for Campana points recently formulated by Pieropan\
, Smeets\, Tanimoto and Várilly-Alvarado.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Lartaux (IMJ-PRG)
DTSTART:20210514T100000Z
DTEND:20210514T103000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/13
DESCRIPTION:Title: On the number of ideals with norm a binary form of degree 3\
nby Alexandre Lartaux (IMJ-PRG) as part of ZORP (zoom on rational points)\
n\n\nAbstract\nLet $K$ be a cyclic extension of $\\mathbb Q$ of degree 3.
If $r_3(n)$ denotes the number of ideals of $O_K$ of norm $n\,$ we have a
relation between the function $r_3$ and a non trivial Dirichlet character
of $\\Gal(K/\\mathbb Q)$\, which is\n$$r_3(n) = (1 ∗ \\chi ∗ \\chi^2)(
n).$$\nIn this talk\, we investigate an asymptotic estimate of the number
of ideals of $O_K$ with norm is a binary form of degree 3\, using this equ
ality and a new result on Hooley's Delta function.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Tanimoto (Nagoya University)
DTSTART:20210604T080000Z
DTEND:20210604T090000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/16
DESCRIPTION:Title: Rational curves on del Pezzo surfaces in positive characteristic
\nby Sho Tanimoto (Nagoya University) as part of ZORP (zoom on rationa
l points)\n\n\nAbstract\nManin’s conjecture over finite fields predicts
the asymptotic formula for the counting function of rational curves of bou
nded degree on smooth Fano varieties defined over finite fields. In his un
published notes\, Batyrev developed a heuristic for this conjecture and th
e assumptions he used are generalized and systemized as Geometric Manin’
s conjecture in characteristic 0. In this talk I would like to explain our
ongoing attempt to understand Geometric Manin’s conjecture in character
istic p for weak del Pezzo surfaces extending results on GMC for del Pezzo
surfaces in char 0 by Testa to char p for most primes p. In the course of
our investigation\, we observe that some pathological examples of weak de
l Pezzo surfaces studied by birational geometers provide us examples of we
ak del Pezzo surfaces whose exceptional sets for weak Manin’s conjecture
are Zariski dense which is contrast to some positive results on exception
al sets in char 0. This is joint work in progress with Roya Beheshti\, Bri
an Lehmann\, and Eric Riedl.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rena Chu (Duke University)
DTSTART:20210604T153000Z
DTEND:20210604T162000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/17
DESCRIPTION:Title: Constant root number on integer fibres of elliptic surfaces\
nby Rena Chu (Duke University) as part of ZORP (zoom on rational points)\n
\n\nAbstract\nIn this joint work with Julie Desjardins\, we aim to describ
e all non-isotrivial families of elliptic curves with low-degree coefficie
nts such that the root number is constant for every integer fibre in the f
amily. We motivate this talk by studying properties of the root number in
families of elliptic curves and Washington's example $\\mathcal{W}_t: y^2=
x^3+tx^2-(t+3)x+1$ for which Rizzo showed has constant root number -1 for
all $t \\in \\mathbb{Z}$.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samir Siksek (Warwick University)
DTSTART:20210514T123000Z
DTEND:20210514T133000Z
DTSTAMP:20260404T111329Z
UID:zorp_1729/18
DESCRIPTION:Title: Integral points on punctured curves and punctured abelian variet
ies\nby Samir Siksek (Warwick University) as part of ZORP (zoom on rat
ional points)\n\n\nAbstract\nLet $A/\\Q$ be an abelian variety and suppose
$A(\\Q)=0$. Let\n$\\ell$ be a rational\nprime. Under a mild condition on
the mod $\\ell$ representation of $A$\,\nwe show that\nthe punctured abeli
an variety $A-0$ has no integral points over 100% of cyclic\ndegree $\\ell
$ number fields.\n
LOCATION:https://stable.researchseminars.org/talk/zorp_1729/18/
END:VEVENT
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