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BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART:20201211T150000Z
DTEND:20201211T160000Z
DTSTAMP:20260422T225926Z
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 rat
ional points)\n\n\nAbstract\nI will talk about a few open questions in ari
thmetic enumeration which are superficially different but which all arise
as special cases of a conjecture of Batyrev-Manin type for algebraic stack
s formulated by Matt Satriano\, David Zureick-Brown\, and me. To state th
e conjecture precisely requires one to say what one means by the height of
a point on a stack\; some of you have heard me talk about this part befor
e\, 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 challenges of f
ormulating a counting conjecture\, which we have sort of but not fully sat
isfyingly surmounted.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thélène (Université Paris-Saclay\, CNRS)
DTSTART:20201211T133000Z
DTEND:20201211T143000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/2
DESCRIPTION:Title: Jumps in the rank of the Mordell-Weil group / Sauts du rang du groupe de
Mordell-Weil\nby Jean-Louis Colliot-Thélène (Université Paris-Sacl
ay\, 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 $X\\rightar
row U$ 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 fibre ov
er 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 $U$. Vari
ants are given and compared with old and new results in the literature.\n\
nSoient $k$ un corps de nombres et $U$ une a smooth integral $k$-variété
lisse intègre. Soit $X\\rightarrow U$ un schéma abélien. On s’intér
esse 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 c
orps des fonctions rationnelle $k(U)$. \n\nOn établit : si la $k$-variét
é $X$ est $k$-unirationnelle\, alors $U(k)_+$ est dense dans $U(k)$ pour
la topologie de Zariski. On donne des variantes\, et on compare avec dive
rs résultats dans la littérature classique et moderne.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephanie Chan (University of Michigan)
DTSTART:20210115T133000Z
DTEND:20210115T143000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/3
DESCRIPTION:Title: A density of ramified primes\nby Stephanie Chan (University of Michi
gan) 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 extend th
e definition of spin (a special quadratic residue symbol) to all odd ideal
s in \n$K$\, not necessarily principal. We discuss some of the ideas invol
ved in obtaining an explicit formula\, depending only on $[K:\\mathbb{Q}]$
\, for the density of rational prime ideals satisfying a certain property
of spins\, conditional on a standard conjecture on short character sums. T
his talk is based on joint work with Christine McMeekin and Djordjo Milovi
c.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (University of Glasgow)
DTSTART:20210115T150000Z
DTEND:20210115T160000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/4
DESCRIPTION:Title: On the statistics of number fields\, integral points and Arakelov ray cl
ass groups\nby Carlo Pagano (University of Glasgow) as part of ZORP (z
oom on rational points)\n\n\nAbstract\nI will present new results coming f
rom a novel approach to Malle’s conjectures (in its strong form): this i
s a joint work with Peter Koymans. If time allows\, I will also survey rec
ent works on the statistics of the solvability\, respectively the failure
of weak approximation\, for integral points on conics and the distribution
of Arakelov ray class groups of quadratic fields: this covers joint works
(past and in progress) with Alex Bartel\, Peter Koymans and Efthymios Sof
os. During the talk I will highlight the natural connections between these
subjects.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ratko Darda (Paris University (Paris 7))
DTSTART:20210219T133000Z
DTEND:20210219T143000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/5
DESCRIPTION:Title: Manin conjecture for algebraic stacks\nby Ratko Darda (Paris Univers
ity (Paris 7)) as part of ZORP (zoom on rational points)\n\n\nAbstract\nWe
study the conjecture of Manin--Batyrev--Peyre in the context of algebraic
stacks.\nTwo examples are of particular interest: the compactification of
the moduli stack of elliptic curves $\\overline{ \\mathcal{M}_{1\,1}} $ a
nd the classifying stack $ BG $ for $ G $ finite group\, which classifies
$G$-torsors. The stack $\\overline{ \\mathcal{M}_{1\,1}} $ is isomorphic
to the weighted projective stack $\\mathcal{P}(4\, 6)$\nwhich is the quoti
ent stack for the weighted action of $\\mathbb{G}_m$ on $\\mathbb{A}^2\\se
tminus\\{0\\}$ with weights $4\, 6$. For weighted projective stacks\, we
define heights that we can use for counting\nits rational points\, example
s are given by the naive height and the Faltings’ height\nof an elliptic
curve.\n\nWe try to motivate why the second example may help us obtain a
geometrical reinterpretation of constants appearing in Malle conjecture\,
which predicts the number of Galois extensions with fixed Galois group $G$
.\n
LOCATION:https://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:20260422T225926Z
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 Science
s (Leipzig)) as part of ZORP (zoom on rational points)\n\n\nAbstract\nDel
Pezzo surfaces are surfaces classified by their degree $d$\, which is an i
nteger between 1 \nand 9 (for $d\\geq 3$\, these are the smooth surfaces o
f degree $d$ in $\\mathbb{P}^d$). For del Pezzo surfaces of degree at leas
t $2$ over a field $k$\, we know that the set of $k$-rational points is Za
riski 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 $k$\, even thoug
h 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 general. I
will talk about a result that is joint work with Julie Desjardins\, in wh
ich we give sufficient conditions for the set of $k$-rational points on a
specific family of del Pezzo surfaces of degree 1 to be Zariski dense\, wh
ere $k$ is any infinite field of characteristic 0. These conditions are ne
cessary if $k$ is finitely generated over $\\mathbb{Q}$. I will compare th
is to previous results.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arda Huseyin Demirhan (University of Illinois at Chicago)
DTSTART:20210319T150000Z
DTEND:20210319T160000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/7
DESCRIPTION:Title: Distribution of Rational Points on Toric Varieties – A Multi-Height Ap
proach\nby Arda Huseyin Demirhan (University of Illinois at Chicago) a
s part of ZORP (zoom on rational points)\n\n\nAbstract\nManin's conjecture
was verified by Victor Batyrev and Yuri Tschinkel for toric varieties. Em
manuel Peyre has proposed two notions\, "freeness" and "all the heights" a
pproach to delete accumulating subvarieties in "Libert\\'e et accumulation
" and "Beyond heights: slopes and distribution of rational points". Based
on the all the heights approach\, in this talk\, we will explain a multi-h
eight variant of the Batyrev-Tschinkel theorem where one considers working
at {\\em height boxes}\, instead of a single height function\, as a way t
o get rid of accumulating subvarieties. This is our main result: Let $X$ b
e an arbitrary toric variety over a number field $F$\, and let $H_i$\, $1
\\leq i \\leq r$\, be height functions associated to the generators of th
e 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 th
at for each $i$\, $H_i(P) \\leq B^{a_i}$ as $B$ gets large is equal to $C
B^{a_1 + \\dots + a_r} + O(B^{a_1 + \\dots + a_r-\\epsilon})$ for an $\\e
psilon >0$. Our result is a first example of a large family of varieties a
long the lines of Peyre's idea.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Göttingen University)
DTSTART:20210319T133000Z
DTEND:20210319T143000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/8
DESCRIPTION:Title: Campana points on toric varieties\nby Damaris Schindler (Göttingen
University) as part of ZORP (zoom on rational points)\n\n\nAbstract\nIn th
is talk we discuss joint work with Marta Pieropan on the distribution of C
ampana points on toric varieties. We discuss how this problem leads us to
studying a generalised version of the hyperbola method\, which had first b
een developed by Blomer and Brüdern. We show how duality in linear progra
mming is used to interpret the counting result in the context of a general
conjecture of Pieropan--Smeets--Tanimoto--Varilly-Alvarado.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shuntaro Yamagishi (Utrecht University)
DTSTART:20210423T123000Z
DTEND:20210423T133000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/9
DESCRIPTION:Title: Density of rational points near/on compact manifolds with certain curvat
ure conditions\nby Shuntaro Yamagishi (Utrecht University) as part of
ZORP (zoom on rational points)\n\n\nAbstract\nIn this talk I will explain
my work with Damaris Schindler where we obtain an asymptotic formula for t
he number of rational points near submanifolds of $\\mathbb{R}^n$ with cer
tain curvature conditions\, and what we can say about the number of ration
al points on them.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miriam Kaesberg (Göttingen University)
DTSTART:20210423T140000Z
DTEND:20210423T150000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/10
DESCRIPTION:Title: On Artin's Conjecture: Pairs of Additive Forms\nby Miriam Kaesberg
(Göttingen University) as part of ZORP (zoom on rational points)\n\n\nAbs
tract\nA conjecture by Emil Artin claims that for forms $f_1\, \\dots\, f_
r \\in \\mathbb{Z}[x_1\, \\dots\, x_s]$ of degree $k_1\, \\dots\, k_r$ the
system of equation $f_1=f_2=\\dots=f_r=0$ has a non-trivial $p$-adic solu
tion for all primes $p$ provided that $s > k_1^2 + \\dots + k_r^2$. Althou
gh this conjecture was disproved in general\, it holds in some cases. In t
his talk I will focus on the case of two additive forms with the same degr
ee $k$ and sketch the proof that Artin's conjecture holds in this case unl
ess $k=2^\\tau$ for $2 \\le \\tau \\le 15$ and $k=3\\cdot 2^\\tau$ for $2
\\le \\tau$.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Margherita Pagano (Leiden University)
DTSTART:20210514T090000Z
DTEND:20210514T093000Z
DTSTAMP:20260422T225926Z
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 rational
points on a variety is by looking at their image in the $p$-adic points. S
ome natural questions that arise are the following: is there any obstructi
on to weak approximation on the variety? Which primes might be involved in
it? Bright and Newton have proven that for K3 surfaces defined over numbe
r fields primes with good ordinary reduction play a role in the Brauer--Ma
nin obstruction to weak approximation.\n\nIn this talk I will give an expl
icit example of this phenomenon. In particular\, I will exhibit a K3 surfa
ce defined over the rational numbers having good reduction at $2$\, and fo
r which $2$ is a prime at which weak approximation is obstructed.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alec Shute (IST Austria)
DTSTART:20210514T140000Z
DTEND:20210514T143000Z
DTSTAMP:20260422T225926Z
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 pr
esent an asymptotic formula for the number of nonzero squareful integers $
z_1\, z_2\, z_3\, z_4$ which sum to zero\, are coprime\, and are bounded b
y $B$. Our result agrees in the power of $B$ and $\\log B$ with the Manin-
type conjecture for Campana points recently formulated by Pieropan\, Smeet
s\, Tanimoto and Várilly-Alvarado.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Lartaux (IMJ-PRG)
DTSTART:20210514T100000Z
DTEND:20210514T103000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/13
DESCRIPTION:Title: On the number of ideals with norm a binary form of degree 3\nby Ale
xandre Lartaux (IMJ-PRG) as part of ZORP (zoom on rational points)\n\n\nAb
stract\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 relatio
n between the function $r_3$ and a non trivial Dirichlet character of $\\G
al(K/\\mathbb Q)$\, which is\n$$r_3(n) = (1 ∗ \\chi ∗ \\chi^2)(n).$$\n
In this talk\, we investigate an asymptotic estimate of the number of idea
ls of $O_K$ with norm is a binary form of degree 3\, using this equality a
nd a new result on Hooley's Delta function.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Tanimoto (Nagoya University)
DTSTART:20210604T080000Z
DTEND:20210604T090000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/16
DESCRIPTION:Title: Rational curves on del Pezzo surfaces in positive characteristic\nb
y Sho Tanimoto (Nagoya University) as part of ZORP (zoom on rational point
s)\n\n\nAbstract\nManin’s conjecture over finite fields predicts the asy
mptotic formula for the counting function of rational curves of bounded de
gree on smooth Fano varieties defined over finite fields. In his unpublish
ed notes\, Batyrev developed a heuristic for this conjecture and the assum
ptions he used are generalized and systemized as Geometric Manin’s conje
cture in characteristic 0. In this talk I would like to explain our ongoin
g attempt to understand Geometric Manin’s conjecture in characteristic p
for weak del Pezzo surfaces extending results on GMC for del Pezzo surfac
es in char 0 by Testa to char p for most primes p. In the course of our in
vestigation\, we observe that some pathological examples of weak del Pezzo
surfaces studied by birational geometers provide us examples of weak del
Pezzo surfaces whose exceptional sets for weak Manin’s conjecture are Za
riski dense which is contrast to some positive results on exceptional sets
in char 0. This is joint work in progress with Roya Beheshti\, Brian Lehm
ann\, and Eric Riedl.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rena Chu (Duke University)
DTSTART:20210604T153000Z
DTEND:20210604T162000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/17
DESCRIPTION:Title: Constant root number on integer fibres of elliptic surfaces\nby Ren
a Chu (Duke University) as part of ZORP (zoom on rational points)\n\n\nAbs
tract\nIn this joint work with Julie Desjardins\, we aim to describe all n
on-isotrivial families of elliptic curves with low-degree coefficients suc
h that the root number is constant for every integer fibre in the family.
We motivate this talk by studying properties of the root number in familie
s 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://researchseminars.org/talk/zorp_1729/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samir Siksek (Warwick University)
DTSTART:20210514T123000Z
DTEND:20210514T133000Z
DTSTAMP:20260422T225926Z
UID:zorp_1729/18
DESCRIPTION:Title: Integral points on punctured curves and punctured abelian varieties
\nby Samir Siksek (Warwick University) as part of ZORP (zoom on rational p
oints)\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 abelian vari
ety $A-0$ has no integral points over 100% of cyclic\ndegree $\\ell$ numbe
r fields.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/18/
END:VEVENT
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