Motivation, let \(L_n^k\) be the set of special varieties in \(\mathbb{C}^n\) of dimension at most \(k\).
Suppose \(V\subset \mathbb{C}^n\) is an algebraic variety not contained in a proper special variety. Then the intersection \(\bigcup_{S\in L_n^{n-\dim V - 1}} V\cap S\) is not Zariski dense in \(V\).
\[\dim S\le n - \dim V - 1 \Leftrightarrow \dim V + \dim S < n\]
The dimension suggests they are considered as ‘unlikely intersections’.
Theorem (Aslonyon-E.-Fouler, After Pila-Scanlon) Let \(V\subset \mathbb{C}^n\) with \(\dim V = d\) not contained in proper special, suppose that for any coordinate projection of the form \(p:\mathbb{C}^n\mapsto \mathbb{C}^{d+2}\) we have \(p(V)\) is not defined over \(\overline{\mathbb{Q}}\), then
\[\bigcup_{S\in L_n^{n-d-1}} V\cap S \text{ is not Zariski dense in } V.\]
Definition Let \(w\in \mathbb{C}\), define the Hecke orbit
\[\mathrm{He}(w) = \{ z\in \mathbb{C}: \exists N, \Phi_N(z,w)=0 \}\]
\(\Lambda \subset \mathbb{C}^n\) is a structure of finite Hecke rank if there is \(\Lambda_0\subset \mathbb{C}\) such that \(\Lambda = (\mathrm{He}(\Lambda_0)\cup \Sigma)^n\).
Say \(S\subset \mathbb{C}^n\) is \(\Lambda\)-special if \(S\) is weakly special and \(S\cap \Lambda\neq \varnothing\).
V \(\subset \mathbb{C}^n\), \(\Lambda\) structure of finite Hecke rank. \(V\) is not contained in proper \(\Lambda\)-special variety. Then the unlikely intersections of \(V\) with \(\Lambda\)-specials are not Zariski dense in \(V\).
\(\mathcal{P}\) is a partition of \(n\) if
\(\bigcup \mathcal{P}= \{1,\ldots,n\}\)
\(\mathcal{P}_i\) are pairwise disjoint
\(S\subset \mathbb{C}^n\) weakly special, \(S\) has special type \(\mathcal{P}\) if for any \(i,k\) if \(X_i,X_k\) not constant on \(S\) Then \(\Phi_N(X_i,X_k)=0\) on \(S\) iff \(i,k\in P\) for some \(P\in \mathcal{P}\).
\[\dim S = |\mathcal{P}| - \text{number of pieces of partitions that has constant coordinates}\]
Idea is move to a differentially closed field (a differential field that has solutions of DEs) instead of working over \(\mathbb{C}\) (so that we could use differential Ax-Schanueal).
\[K = (K,\partial), \quad C = \ker \partial\]
\(p(V)\) is not defined over \(C\). Take intermediate field \(C\subset F\subset K\), \(V\) defined over \(F\), \(F\) algebraically closed with \(\mathrm{trdeg}_C F < \infty\).
Now we try to parametrize the set we care about.
Choose a partition \(\mathcal{P}\) of \(n\), choose a number \(v\in \{0,\dots,n-1\}\)
Let \(g\) be a positive integer and \(\varepsilon > 0\), then there are effective \(c,m\) with the following property:
Suppose \(A = E_1\times \dots \times E_g\) where \(E_i\) are elliptic curves with CM given by Weierstrass equations over \(K=\mathbb{Q}(j(E_i))\).
Suppose \(V\subset A\) is a subvariety defined over a number field over \(K\).
Suppose \(V\) dose not contain a positive dimensional torsion translate.
If \(P\in V(\mathbb{C})\) is torsion of order \(N\), then
\[N\le c(g)\cdot [L:K]^{1+\varepsilon}(\deg V)^m.\]
There are much more uniform results, e.g. Gao, Ge, Kuhne
Related work by Grhriel Dill
Use the Pila-Zannier strategy
Use counting for pfaffian functions (B.J.S.T.).
Use pfaffian definitions for elliptic functions (J.-Schmidt).
Use Galois bounds proved by Gao.
Let \(K\) be a number field of degree \(D\), consider a morphism \(f:\mathbb{P}^n\to \mathbb{P}^n\) of degree \(d\ge 2\) defined over \(K\).
A point \(p\in \mathbb{P}^n(K)\) is pre-periodic if the orbit \(\{f^k(p)\}_{k\ge 0}\) is finite.
Then we consider the set of pre-periodic points \(\mathrm{Pre}(f,K)\).
There is a famous uniform-boundedness conjecture, posted by Moston and Silverman, that
\[ |\mathrm{Pre}(f,K)| \le B(n,d,D) \]
This conjecture implies
We consider the special case \((D,n,d)=(1,1,2)\) and \(f\) is a polynomial of degree \(2\). However, this special case is still open.
People (Flynn, Poonen, Schaefer) have conjectured that if \(N\ge 4\), there is no quadratic polynomial \(f\in \mathbb{Q}[x]\) with a rational point of period \(N\).
\(N=4\) proved by Morton, \(N=5\) proved by Flynn, Poonen, Schaefer. \(N=6\) proved by Stoll subject to BSD.
For the curve
\[x^2(x+1)y^3 - (5x^2+x+1)y^2 - x(x^2-2x-7)y + (x+1)(x-3)=0\]
has genus \(4\), BSD will give us \(\mathrm{rank}J(\mathbb{Q})=4\).
Let \(f_t(z)=z^2+t\), for any \(z\) we define
\[S_z(\mathbb{Q}) = \{t\in \mathbb{Q}: z\text{ is pre-periodic under } f_t\}\]
Theorem (Poonen). If conjecture FPS is true, then \(|\mathrm{Pre}(f_t,\mathbb{Q})|\le 9\) for all \(t\in \mathbb{Q}\).
Theorem (DeMacro-Krieger-Ye). \[|\mathrm{Pre}(f_{t_1},\mathbb{C})\cap \mathrm{Pre}(f_{t_2},\mathbb{C})|\le B\] for any \(t_1\neq t_2\in \mathbb{C}\).
Theorem (Fu).
\[S_{z_1}(\mathbb{C})\cap S_{z_2}(\mathbb{C}) \le B\]
for any \(z_1^2\neq z_2^2\in \mathbb{C}\).
Result (Fu, Stoll). If conjecture FPS is true, then \(|S_z(\mathbb{Q})\le 7\) for all \(z\in \mathbb{Q}\).
Let \(K\) be a field \(X/K\) a curve of genus \(g\ge 2\) not definable over a finite extension of prime field \(\mathbb{Q}\) or \(\mathbb{F}_p\) of \(K\). Let \(D\in \mathrm{Div}(X)\), \(j_0:X\to J_X\), \(P\mapsto P-D\).
Then the number of \(|j_D(X(\overline{K}))|\cap J_X(\overline{X})_{tors}|\le 16g^2+32g+124\).
Reduce to function field case \(K=k(B)\) the function field of some curve \(B/k\).
Zhang’s admissible pairing \((\cdot,\cdot):\mathrm{Div}(X)\times \mathrm{Div}(X)\to \mathbb{R}\) satisfying
a. bilinear and symmetric
b. \((\omega,P) = -(P,P)\) adjunction formula for \(\forall P\in X(\overline{K})\)
c. if \(\deg D=\deg E = 0\Rightarrow (D,E)=-1\), Neron-Tate pairing, cor: \((D,D)\le 0\).
d. \(P,Q\in X(k), P\neq Q\) then
\[(P,Q) = i(P,Q) + \sum_{v\in |B|} (R_v(P), R_v(Q))\]
\(i(P,Q)\) the intersection multiplicity of \(\overline{P},\overline{Q}\) in \(\chi\)
\(\Gamma_v(X)\) motvized reduction graph of \(\chi\) at \(v\in |B|\).
\(R_v : X(\overline{K})\to \Gamma_v(X)\)
\(g_v\) common Green function on \(\Gamma_v(X)\).
\(\chi\) a model of \(X\) over \(B\).
Let \(P_1,\dots,P_s \in X(\overline{K})\) with \(j_D(P_i)\in J_X(\overline{K})_{tors}\), then
may assume \(P_i\in X(K)\), then idea to find some height bound, let \(\omega\) be canonical divisor of \(X\)
\[ \le (\omega,\omega) \le \]
i. Apply to \(s\cdot \omega - (2g-2)(P_1+\cdots+P_s)\) and use (a) and (b)
\[ \le (\omega,\omega) \le^{i} -\frac{4(g-1)}{5}(1+\frac{g-1}{5})\sum (P_j,P_j) - \frac{4(g-1)^2}{5^2}\sum_{j\neq k} (P_j,P_k) \]
ii. \(j_D(P_i)\) is torsion implies \(0 = h_{Neron-Tate}(P_j-P_k) = -(P_j-P_k,P_j-P_k)\)
so \(\sum_{j=1}^s = \frac{1}{s-1}\sum_{j\neq k} (P_j,P_k)\)
\[=^{ii}-\frac{4g(g-1)}{s(s-1)} \sum_{j\neq k} (P_j,P_k) \le -\frac{4g(g-1)}{s(s-1)}\sum_{j\neq k}\sum_{v\in |B|} g_v(R_v(P_j), R_v(P_k))\]
iii. Elkies bound for metrized graphs (Baker-Rumely)
\[\le^{iii} -\frac{4g(g-1)}{s-1}\sum_{v\in |B|} \sup_{x\in \Gamma_v(X)} g_v(x,x)\]
iv. Harmonic analysis on metrized graphs (Cirkir’s bound)
\[\le^{iv} \frac{4g(g-1)}{s-1}\frac{8g^4+18g^2-13g-1}{2g(2g+1)(g-1)^2}\sum_v \varphi(\Gamma_v)\]
v. use Hodge-index on \(\chi^2\)
On the left hand side we have
\[\frac{g+1}{2g+1}\sum_v \varphi(\Gamma_v)\le^{v} \]
Solve for \(s\) if \(\sum \varphi(\Gamma_v)=0\) then \(s=1\)
Strategy also applies to number fields, only missing explicit bound for
\[\sum_{j\neq k} g(P_j,P_k)\]
\(g(,): X_\mathbb{C}^2\to \mathbb{R}\) arakelov-green function.
Theorem. Riemann surface of genus \(g\ge 1\), \(\{z_j : U_j\to \mathbb{C}\}_{j=1}^m\) local coordinate, \(0<r_1<r_2\), \(r_2-r_1\le 1\),
\(D(r_i) = \{z\in \mathbb{C}: |z|<r_i\}\), \(X=\bigcup_{j=1}^m z_j^{-1}(D(r_1))\)
Let \(\mathbb{G}_m = \mathbb{C}^\times\) and \(V\subset \mathbb{G}^n_m\) be irred closed subvariety fixed.
\(\zeta H\subset \mathbb{G}_m^n\) torsion coset (varying)
\(H\subset \mathbb{G}^m_n\) sub-torus \(\zeta = (\zeta_1,\dots, \zeta_n)\in \mathbb{G}_m^n\) a torsion point.
Unlikely intersections says if \(\dim H + \dim V < n\), then we expect that ‘usually’ \(V\cap \zeta H=\varnothing\) unless there is a reason to expect otherwise.
Bombieri-Masser-Zannier 99, …
if \(\dim H + \dim V \ge n\), then we expect that ‘usually’ \(V\cap \zeta H\neq \varnothing\) unless there is a reason to expect otherwise.
\(n=3\), \(V=\{y=x+1, z=x-1\}\)
solve \(x^a(x+1)^b(x-1)^c=\zeta\) for \(x\in \mathbb{C}-\{0,\pm 1\}\) where \(a,b,c\in \mathbb{Z}\), \(\zeta\) a root of unity.
Easy to show that
\[\bigcup_{\dim V + \dim H = n, \zeta H\text{ torsion coset}} V\cap \zeta H\]
is Zariski dense in \(V\).
This gives no information about the individual intersections \(V \cap \zeta H\).
There is an obvious obstruction to \(V\cap \zeta H\neq \varnothing\).
Suppose \(\exists H_0\subset \mathbb{G}_m^n\) subtorus such that \(\dim H_0 + \dim V =n\) and \(\dim W < \dim V\) for
\[W = \overline{\pi_{H_0}(V)}, \pi_{H_0}: \mathbb{G}_m^n\to \mathbb{G}_m^n/H_0\cong \mathbb{G}_m^{\dim V}\]
\[\mathbb{G}_m^n = (\mathbb{C}^\times)^n, (x_1,\dots,x_n)\in \mathbb{G}_m^n\]
Take \(V\subset \mathbb{G}^n\) an irreducible subvariety, Zilber-Pink studies the intersection of \(V\) with special subvarieties.
Here special subvarieties are algebraic subgroups of \(\mathbb{G}_m^n\).
\(H\) is defined by
\[x_1^{a_{i1}}\dots x_n^{a_{in}} = 1 , i=1,\dots,r\]
\[\dim H = n - \mathrm{rank} A\]
weakly special = translates of algebraic subgroups.
Means, if \(\dim V + \dim H < n\), then generally we expect \(V\cap H=\varnothing\) unless there is a reason to expect otherwise.
Conjecture (ZP for tori). \(V\subset \mathbb{G}^n\), if \(V\) is not contained in any proper algebraic subgroup (which seems to be the only reason for us to expect otherwise), then
\[\bigcup_{H\text{ special}\atop \dim V + \dim H < n} V\cap H \text{ is not Zariski dense in } V.\]
\(\dim V = 1\), then \(\dim H + \dim V = \dim H + 1 < n\) is equivalent to say \(\mathrm{codim\,}H > 1\), then conjecture says
\[\bigcup_{H\text{ special}\atop \mathrm{codim\,}H > 1} V\cap H \text{ is finite.}\]
This is proved by Maunn 08 for \(\overline{\mathbb{Q}}\), BHZ 08 for \(\mathbb{C}\).
If \(\dim V = n-1\), this is the Manin-Mumford for tori
If \(\dim V = n-2\), this is the Bombieri-Masser-Zannier.
In other cases, as Gabriel explains, there is a notion of V is geometric non-degenerate, and this is the only case known (Habegger).
\(V\) is a curve over \(\overline{\mathbb{Q}}\), the union
\[ \bigcup_{H\text{ special}\atop \mathrm{codim\,}H \ge 1} V\cap H \]
is always infinite. We can ask whether most of these infinite intersections comes from tangents.
Question How many of them are tangent?
Theorem (Ballini-C-Ottolini)
\(V\subset \mathbb{G}^n\) irred over \(\overline{\mathbb{Q}}\), assume \(V\not\subset H\) for any proper subvariety, then if we take the union
\[\bigcup_{H\text{ special}\atop \mathrm{codim\,}H \ge 1, T_pV\subset T_pH} V\cap H\]
is finite.
Similar results for \(1\)-parameter families of all curves.
Corvaja-Demeio-Masser-Zannier, Ullmer-Urzua
pick \(\mathcal{E}_\lambda \to \mathbb{P}^1-\{0,1,\infty\}\) the family of elliptic curves,
take \(\sigma\) a section, they proved, if \(\sigma\) is not identically torsion,
\[|\{b\in \mathbb{P}^1-\{0,1,\infty\} : \sigma(b)\text{ is torsion in } \mathcal{E}_b(\mathbb{C}) \text{ and } m_\sigma(b)\ge 2\}| < \infty\]
where \(m_\sigma(b)\) is the order of multiplicity of \(\sigma\) at \(b\).
\[\mathcal{C}\subset \mathcal{E}_\lambda\to \mathbb{P}^1-\{0,1,\infty\}\]
\(V/k\)
First case, we assume \(V\) is not contained in weakly special subvariety, then actually if you consider
\[\bigcup_{H\text{ special}\atop \mathrm{codim\,}H\ge 1} V\cap H\]
is infinite but BMZ says the set is of bounded height. (How can you have infinite but bounded height?)
You just need to bound the degree.
Prop. \(V\subset \mathbb{G}^n\) not contained in a weakly special subvariety, \(\exists\delta_1>0\) such that for every \(W\) weakly special, of codimension \(\dim W \ge 1\), and for every \(P\in V\cap W\) with \(V\) tangent to \(W\) at \(P\), one has
\[[K(P):K] \le \delta_1\]
We assume that \(V\) is contained in a weakly special but not a special one. Now we don’t have bound of height anymoew, we translate into anotheor problem and still use the bound on the degree.
Up to an automorphism of \(\mathbb{G}_m^n\), we can assume that
\[V = C\times (g_{k+1},\dots,g_n)\]
\(C\subset \mathbb{G}_m^k\) not contained in a weakly special
\(g_{k+1},\dots,g_n\) multiplicatively independent
\[\Lambda = \{P\in V(\overline{K}) : \exists H< \mathbb{G}^n \text{ with } P\in V\cap H\text { and tangent int }\}\]
\(P\in \Lambda\) and consider
\[\Gamma_p = \{ \underline{a}\ \in \mathbb{Z}^n : \prod x_i(P)^{a_i} = 1 \}\]
lattice, \(\mathrm{rank}\Gamma_p \ge 2\) implies finiteness by Maurin.
we have to study when \(\mathrm{rank}\Gamma_p = 1\).
\[\Gamma_{p,sing} = \{ \underline{a}\in \mathbb{Z}^n : \prod x_i(P)^{a_i} = 1, \text{ intersection singular } \} \]
Masser: \(\exists \underline{a}\in \Gamma_{p,sing}\) such that
\[\max |a_i| \le n^{n-1} \omega(h/\eta)^{n-1}\]
with \(h = h(P), \eta = \text{ bounded of the non torsion points of } \mathbb{G}_m^n, \omega = \text{ number of roots of 1 in } K\).
With standard arguments, we have
\[\max{|a_i|} \le \delta_2(\max h(x_i(P)) [K(P):K])^{n-1}\]
\[\varphi: \mathbb{G}^k \to \mathbb{G}^n: \quad (x_1,\dots,x_k)\mapsto (x_1,\dots,x_k,g_{k+1},\dots,g_n)\]
\(V\subset \mathrm{Im}\varphi\) and the preimage \(\varphi^{-1}(V)\) is \(C\).
If we take \(H\subset \mathbb{G}^n\) defined by \(\prod x_i^{a_i} = 1\), then the preimage \(\varphi^{-1}(H)\) is defined by
\[ x_1^{a_1}\dots x_k^{a_k} g_{k+1}^{a_{k+1}}\dots g_n^{a_n} = 1 \]
this will become a weakly special subvariety of \(\mathbb{G}^k\).
Prop 2. \(C\subset \mathbb{G}^k\) irred over \(K\), assume \(C\) is not in weakly special, then there exists \(\delta_3>0\) such that for all \(W\subset \mathbb{G}^k\) weakly special of the form
\[x_1^{t_1}\dots x_k^{t_k} = c\]
for some \(c\in K^\times\) and \(P\in C\cap W\) with tangent intersection, we have
\[\max h(x_i(P)) \le \delta_3 (1 + \max \log|t_i|)\]
This will give you the bound on the \(\max |a_i|\).
(WIP with M.Orr + G.Papas)
Take an ireducible curve \(C\subset Y(1)^3\), Hodge generic. Interested in intersection with \(Z\susbet Y(1)^3\), \(\dim Z = 1\).
Recall special subvarieties are defined by
\(X_i=Z_0\) CM.
\(\phi_N(X_i, X_j)=0\) for some \(N\).
Results: if \(V\) not defined over \(\overline{\mathbb{Q}}\),
look at \(Z\) with a fixed coordinate, prove the finiteness of intersections with
If \(V\) happens to be asymmetric – difficult to generalize.
Conjecture: let \(A\) be abelian variety over number field \(K\), \(C\subset A^{an}\) algebraic curve, \(\Sigma_2\) is a union of proper algebraic subgroups of \(A\) of codimension \(\ge 2\).
If \(C\) is not contained in any proper algebraic subgroup, then \(C\cap \Sigma_2\) is finite.
This is a theorem Habegger Pila 2016, with somewhat ineffective proof.
Implies Mordell conjecture, if \(C\) is a smooth projective curve of genus \(\ge 2\) over a number field \(K\), then \(C(K)\) is finite. (For implication need generators of MW group of Jacobian.)
If \(\mathrm{Jac}(C)\) contains \(A^n\) for some \(A\), where the \(\mathrm{rank}(A(K)\le n-1\), then the implication is effective.
\(C\cap\Sigma_2\) has height bounded by a constant \(c(C,A)\).
Earlier theorem of Remond (05): under this condition, \(C\) is not contained in any translates of proper algebraic subgroup. (Manin Demjanenko)
Case \(A=E^n\) then there are explicit height bounds for this theorem under some conditions on the rank. (\(C\) not contained in any translate, \(\mathrm{rank}E<n\).) due to Checcoli, Veneziano, Viada.
Theorem \(A\)A Simple AV over \(K\) of dimension \(d\), \(G=A^n\). \(C\subset G\) irreducible algebraic curve over \(K\) not contained in a proper algebraic subgroup with \(0\in C\).
\(\Sigma_{d(2d+1)}=\bigcup\) algebraic subgroups of \(G\) of codimension \(\ge d(2d+1)\).
Then there exists effective compuatbles \(c_1,c_2\) such that every point \(s\in C\cap \Sigma_{d(2d+1)}\) has height at most \(h(s)\le c_1[K(s):\mathbb{Q}]^{c_2}\).
Cor. \(C\) smooth projective curve over \(K\) number field, \(\mathrm{Jac}(C)\) contains \(A^n\) where \(A\) simple of dimension \(d\) and \(\mathrm{rank}A(K) \le n-2d-1\), and we have a point in \(C(K)\), then we can effectively compute the set \(C(K)\).
\(G\)-functions are of the form \(K[[x]]\) satisfying a linear differential equation and growth conditions on the coefficients.
examples are like \(\log(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}x^n\).
Pick a \(\omega\in \Omega^1(C/K)\), then we look at the abelian integral
\[F(s) = \int_0^s \omega\]
whose Taylor series is a \(G\)-function.
Use the Bombieri-Andre theorem, which says that if you have a collection \(y_1,\dots,y_n\) of \(G\)-functions, then the points \(\xi\in \overline{\mathbb{Q}}\) if evaluations of \(y_i(\xi)\) satisfy a global non-trivial relation, then the height of the point \(h(\xi)\le c_3\delta^{c_4}\).
If you got some \(s\in C\cap H\) where \(H\) is some subgroup of \(A^n\), then \(s=(s_1,\dots,s_n)\) satisfy a relation of the form \(a_1s_1+\dots+a_ns_n=0\) with \(a_i\in \mathrm{End}(A)\) or for simplicity \(a_i\in \mathbb{Z}\).
Then we can use the Andre-Bombieri theorem to bound the height of \(s\).
Abelian integrals / logarithms are homomorphisms at finite places.
\[a_1F(s_1)+\dots a_nF(s_n) = 0\]
At infinite places, you get something like
\[a_1F(s_1)+\dots a_nF(s_n) = \lambda \in \Lambda\]
\(\Lambda\) some period lattice which has \(\mathrm{rank}_\mathbb{Z}\Lambda = 2d\).
If I have \(2d+1\) such relations, can take a linear combination and eliminate \(\lambda\)s.
It turns out that this works best for hyperelliptic curves.
If we can find example such that \(d=1\), \(G=E^n\), \(\mathrm{rank}E(\mathbb{Q})=1\), \(n\ge 2d+1+\mathrm{rank}=4\). \(C\) hyperelliptic with a rational Weierstrass point \(y^2=f(x)\), \(f\) has a rational root, then all points in \(C(\mathbb{Q})\) satisfy
\[h(s)<6\cdot 10^9 \deg f \left(h(f) + \log(\deg(f)) +35\right)\]
A shimura datum is a pair \((G,X)\) where \(G\) is a reductive algebraic group over \(\mathbb{Q}\) and \(X\) is a \(G(\mathbb{R})\)-orbit of a morphism of real algebraic groups of \(x_0:S=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}\to G_\mathbb{R}\)