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Abstract: Let \(X\) be a variety over \(\mathbb{Q}\). We introduce a geometric non-degenerate criterion for \(X\) using moduli spaces \(M\) over \(\mathbb{Q}\) of abelian varieties. If \(X\) is non-degenerate, then we construct via \(M\) an open dense moduli space \(U\subseteq X\) whose forgetful map defines a Paršin construction for \(U(\mathbb{Q})\). For example if \(M\) is a Hilbert modular variety then \(U\) is a coarse Hilbert moduli scheme and \(X\) is non-degenerate iff a projective model \(Y\subset \bar{M}\) of \(X\) over \(\mathbb{Q}\) contains no singular points of the minimal compactification \(\bar{M}\). We motivate our constructions when \(M\) is a rational variety over \(\mathbb{Q}\) with \(\dim(M)>\dim(X)\).
     We study various geometric aspects of the non-degenerate criterion and we deduce arithmetic applications: If \(X\) is non-degenerate, then \(U(\mathbb{Q})\) is finite by Faltings. Moreover, our constructions are made for the effective strategy which combines the method of Faltings (Arakelov, Paršin, Szpiro) with modularity and Masser-Wüstholz isogeny estimates. When \(M\) is a coarse Hilbert moduli scheme, we use this strategy to explicitly bound the height and the number of \(x\in U(\mathbb{Q})\) if \(X\) is non-degenerate.
       We illustrate our approach in the case when \(M\) is the Hilbert modular surface given by the classical icosahedron surface studied by Clebsch, Klein and Hirzebruch. For any curve \(X\) over \(\mathbb{Q},\) we construct and study explicit projective models \(Y\subset\bar{M}\) called ico models. If \(X\) is non-degenerate, then we give via \(Y\) an effective Paršin construction and an explicit Weil height bound for \(x\in U(\mathbb{Q})\). As most ico models are non-degenerate and \(X\setminus U\) is controlled, this establishes the effective Mordell conjecture for large classes of (explicit) curves over \(\mathbb{Q}\). We also solve the ico analogue of the generalized Fermat problem by combining our height bounds with Diophantine approximations.

Abstract: We consider generalizations of Szpiro’s classical discriminant conjecture to hyperelliptic curves over a number field \(K\) and to smooth, projective, and geometrically connected curves \(X\) over \(K\) of genus at least 1. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves \(X\) of genus \(1\) or \(2\). In particular, we establish completely explicit exponential versions of Szpiro’s classical discriminant conjecture for elliptic curves over \(K\). The proofs use the theory of logarithmic forms and Arakelov theory for arithmetic surfaces.

Abstract: Let \(C\) be a hyperelliptic curve of genus \(g\geq 1\) over a number field \(K\) with good reduction outside a finite set of places \(S\) of \(K\). We prove that \(C\) has a Weierstrass model over the ring of integers of \(K\) with height effectively bounded only in terms of \(g, S\) and \(K\). In particular, we obtain that for any given number field \(K\), finite set of places \(S\) of \(K\) and integer \(g\geq 1\) one can in principle determine the set of \(K-\)isomorphism classes of hyperelliptic curves over \(K\) of genus \(g\) with good reduction outside \(S\).

Abstract: Let \(K\) be a number field and let \(S\) be a finite set of places of \(K\). We prove that there exist an effectively computable Dedekind domain \(R\subset K\) and an effective constant \(C\), depending only on \(K\) and \(S\), such that for each elliptic curve \(E\) defined over \(K\) with good reduction outside \(S\) there is a globally minimal Weierstrass model of \(E\) over \(\text{Spec}(R)\) with height bounded by \(C\).

Abstract: The purpose of this thesis is to combine the theory of logarithmic forms with geometric tools to deduce new results in Diophantine geometry. Let \(K\) be a number field and let \(S\) be a finite set of places of \(K\).
     We first prove an effective Shafarevich theorem for elliptic curves. It gives an effectively determinable Dedekind domain \(R\subset K\) and an effective constant \(\Omega\), depending only on \(K\) and \(S\), such that for each elliptic curve \(E\) defined over \(K\) with good reduction outside \(S\) there is a globally minimal Weierstrass model of \(E\) over \(\text{Spec}\;(R)\) with height bounded by \(\Omega\). This chapter 1 is joint work with Professor Gisbert Wüstholz and Clemens Fuchs.
     In chapter 2 we introduce a new method to generalize and improve the results of the first chapter. Let \(C\) be an arbitrary hyperelliptic curve of genus \(g\geq 1\) defined over \(K\) with good reduction outside \(S\). We show that \(C\) has a Weierstrass scheme over the ring of integers of \(K\), arising from a hyperelliptic equation for \(C\) with height effectively bounded in terms of \(g\), \(S\) and \(K\). Then we give a new interpretation of this effective Shafarevich theorem for hyperelliptic curves in terms of bad reduction which will be the main tool to deduce the Diophantine applications of the last chapter.
     In chapter 3 we generalize Szpiro’s famous Discriminant Conjecture for elliptic curves over \(K\) to arbitrary hyperelliptic curves \(C\) over \(K\) and we give an effective proof of an exponential version of the generalized conjecture. Then we interpret these results in terms of Arakelov theory and we get also some applications in the theory of geometric Mumford discriminants and minimal regular models respectively. Furthermore, we generalize the Height Conjecture of Frey for elliptic curves to general hyperelliptic curves over \(K\) with a \(K\)-rational Weierstrass point and we prove an effective exponential version of this generalized conjecture. As an application we get that the elliptic \(\mathbb{Q}\)-factors of modular Jacobian’s can be determined effectively.

Abstract: The strategy of combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity and Masser–Wüstholz isogeny estimates allows to explicitly bound the height and the number of the solutions of certain Diophantine equations related to integral points on moduli schemes of abelian varieties. In this paper we survey the development and various applications of this strategy.

Abstract: We use the method of Faltings (Arakelov, Paršin, Szpiro) in order to explicitly study integral points on a class of varieties over \(\mathbb{Z}\) called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are classical examples of Hilbert moduli schemes. Our main result gives explicit upper bounds for the height and the number of integral points on Hilbert moduli schemes.

Abstract: We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity results and Masser-Wüstholz isogeny estimates. In this work we explicitly bound the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus.
      In the first part we define and study coarse Hilbert moduli schemes with their heights and branch loci. Building on the foundational works of Rapoport, Katz-Mazur, Deligne-Pappas and using the Faltings height, we first develop the basic definitions. Then we prove some geometric results for the height and the branch locus. In particular we relate the branch locus to automorphism groups of the involved moduli stack which allows us to compute the branch locus in some relevant cases.
      In part two we establish the effective Shafarevich conjecture for abelian varieties \(A\) over a number field \(K\) such that \(A_{\bar{K}}\) has CM or \(A_{\bar{K}}\) is of GL(2)-type and isogenous to all its \(\text{Aut}(\bar{K}/\mathbb{Q})\)-conjugates. If \(A_{\bar{K}}\) has no CM, then we use isogeny estimates and results of Ribet and Wu to reduce via Weil restriction to the known case \(K=\mathbb{Q}\) proven by one of us via Faltings’ method and modularity. We also use isogeny estimates to reduce the CM case to effectively bounding the height \(h(\Phi)\) of simple CM types in terms of the discriminant. To bound \(h(\Phi)\) we follow Tsimerman’s strategy and we combine the averaged Colmez conjecture with explicit analytic estimates for L-functions.
      In the third part we continue our explicit study of the Paršin construction given by the forgetful morphism of Hilbert moduli schemes. We now work out our strategy for arbitrary number fields \(K\) and we explicitly bound the number of polarizations and module structures on abelian varieties over \(K\) with real multiplications.
     In the last part we illustrate our results by applying them to two classical surfaces first studied by Clebsch (1871) and Klein (1873): We explicitly bound the Weil height and the number of their integral points. Hirzebruch proved that both surfaces are models over \(\mathbb{C}\) of a Hilbert modular surface \(Y_\mathbb{C}\). To show that they are coarse Hilbert moduli schemes over \(\mathbb{Z}[\tfrac{1}{30}]\), we go into the construction of Rapoport and Faltings–Chai of the integral minimal compactification of \(Y_\mathbb{C}\) and we make it explicit via Hirzebruch’s work. Here we use geometric results for integral Hirzebruch-Zagier divisors of Bruinier-Burgos-Kühn and Yang, and we compute the Fourier expansion of certain Eisenstein series via analytic results of Klingen, Siegel and Zagier. To explicitly relate the height to the Weil height, we combine Pazuki’s height comparison with formulas for Hilbert theta functions due to Götzky, Gundlach and Lauter-Naehrig-Yang.

Abstract: In the first part we construct algorithms (over \(\mathbb{Q}\)) which we apply to solve \(S\)-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over \(\mathbb{Q}\) with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for \(S\)-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker’s explicit \(abc\)-conjecture and a new conjecture on \(S\)-integral points of any hyperbolic genus one curve over \(\mathbb{Q}\).
     In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov-Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for \(S\)-unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Paršin, Szpiro) combined with the Shimura-Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms.
     In the third part we solve the problem of constructing an efficient sieve for the \(S\)-integral points of bounded height on any elliptic curve \(E\) over \(\mathbb{Q}\) with given Mordell-Weil basis of \(E(\mathbb{Q})\). Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting “elliptic logarithm sieve” is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.

Abstract: The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems (e.g. \(S-\)unit and Mordell equations), this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. We also combine Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abelian varieties of (product) \(\text{GL}_2-\)type. In particular, we open the way for the effective study of integral points on certain higher dimensional moduli schemes.

Abstract: We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example \(S-\)unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.

Abstract: In this article we establish the effective Shafarevich conjecture for abelian varieties over \(\mathbb{Q}\) of \(\text{GL}_2-\)type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

Abstract: Let \(X\) be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that \(X\) has a “small point”. In this paper we prove that if \(X\) is a cyclic cover of prime degree of the projective line, then \(X\) has infinitely many “small points”. In particular, we establish the first cases of Szpiro’s small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.

Abstract: Let K be a number field and let S be a finite set of places of K. A classical theorem of Shafarevich says that there are only finitely many K-isomorphism classes of elliptic curves over K with good reduction outside S. An effective version of this statement for K=Q was already proved by Coates. In the talk we discuss an extension to arbitrary number fields. We give explicit bounds and compare them with the one obtained by Coates. This is joint work with Clemens Fuchs and Gisbert Wüstholz.

Abstract: Let K be a number field and let S be a finite set of places of K. A classical theorem of Shafarevich says that there are only finitely many K-isomorphism classes of elliptic curves over K with good reduction outside S. An effective version of this statement for K=Q was already proved by Coates. In the talk we discuss an extension to arbitrary number fields. We give explicit bounds and compare them with the one obtained by Coates. This is joint work with Clemens Fuchs and Gisbert Wüstholz.

Abstract: We discuss Frey’s modular degree conjecture for elliptic curves over Q. In particular we explain how one can deduce an exponential version of this conjecture using a result of my PhD thesis based on the theory of logarithmic forms. We also discuss some conditional results.

Abstract: In 1982 Szpiro conjectured that there is a constant c such that if E is an elliptic curve over Q of minimal discriminant \(\Delta\) and conductor N, then \(\vert \Delta\vert\leq N^c\). This conjecture is equivalent to the abc-conjecture. In the talk we give the unconditional result log \(\vert \Delta\vert\leq N^c\). The proof is based on the theory of logarithmic forms. We bound the relative Faltings height of E polynomially in terms of N which leads to the desired estimate.

Abstract: Let K be a field, S be a finite set of places of K and let g>0 be an integer. The Shafarevich conjecture says that there are only finitely many K-isomorphism classes of curves over K of genus g with good reduction outside S. This was proved by Faltings in 1983. An effective version of the conjecture would imply inter alia the effective Mordell and the abc conjecture. In the talk we give an effective version of the Shafarevich conjecture for hyperelliptic curves and discuss some applications.

Abstract: We present 4 different finiteness proofs for S-integral points on \(\mathbb{P}^1−\lbrace 0, 1, \infty\rbrace\) in a number field K, or equivalently for S-unit equations in K, using Diophantine approximation, logarithmic forms, Faltings’ finiteness theorems, or modularity (K=Q). We also discuss effectivity of these proofs. In particular we explain for K=Q how one can use modularity to get effective height bounds, obtaining a new effective finiteness proof.

Abstract: In this talk we consider the problem of giving explicit inequalities which relate heights, discriminants and conductors of a curve defined over a number field. We present such inequalities for some curves, including all curves of genus one or two and we discuss Diophantine applications.

Abstract: In this talk we discuss explicit inequalities which relate heights, discriminants and conductors of elliptic curves E over a number field K. In the first part we present an explicit height-conductor inequality for E based on the theory of logarithmic forms. In the second part we consider the case K=Q: We explain how one can use modularity to get a sharper height-conductor inequality and we deduce explicit height bounds for S-integral points on Mordell curves and on \(\mathbb{P}^1-\lbrace 0,1,\infty\rbrace\).

Abstract: In the first part we present for any elliptic curve over a number field an explicit height-conductor inequality based on the theory of logarithmic forms. In the second part we consider the case K=Q: We explain how one can use modularity to get a sharper height-conductor inequality and we deduce explicit height bounds for S-integral points on Mordell curves and on \(\mathbb{P}^1-\lbrace 0,1,\infty\rbrace\).

Abstract: We present explicit height bounds for the classical Mordell equation and we explain the proof which combines the method of Parsin and Faltings with modularity.

Abstract: We give explicit inequalities which relate heights and conductors of elliptic curves over number fields, and we discuss Diophantine applications.

Abstract: We present explicit inequalities which relate heights and conductors of elliptic curves over number fields, and we discuss Diophantine applications.

Abstract: Let K be a number field. We first consider a generalization of Szpiro’s discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro’s small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces.

Abstract: On combining modularity with Arakelov theory, we obtain finiteness results for integral points on moduli schemes of elliptic curves. The method is fully effective. For example, on applying the method to the projective line minus three points and to once punctured Mordell elliptic curves, we improve the actual best explicit height upper bounds for the solutions of S-unit and Mordell equations. In addition, the method gives an effective Shafarevich conjecture for abelian varieties of GL(2)-type.

Abstract: Many fundamental Diophantine problems can be reduced to the study of integral points on moduli schemes of elliptic curves. We give explicit finiteness results for such integral points and we discuss a generalization to higher dimensions.

Abstract: In this talk we present new Diophantine applications of modularity results. In the first part, we use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. On working out the method for moduli schemes corresponding to Mordell equations, we improve the actual best explicit height bounds for Mordell equations. In the second part, we combine Faltings’ method with Serre’s modularity conjecture to establish the effective Shafarevich conjecture for abelian varieties of (product) GL2-type and then we discuss applications to the effective study of integral points on certain higher dimensional moduli schemes (e.g. Hilbert modular varieties).

Abstract: Let K be a number field. We first consider a generalization of Szpiro’s discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro’s small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces.

Abstract: Abstract: Joint work with Benjamin Matschke. We shall discuss practical algorithms which solve Mordell equations by combining the method of Faltings (Arakelov, Parsin, Szpiro) with the Shimura-Taniyama conjecture.

Abstract: We will explain a way how one can use moduli schemes and their natural forgetful maps in the study of certain classical Diophantine problems (e.g. finding all integral points on hyperbolic curves). To illustrate and motivate the strategy, we consider the case of cubic Thue equations and we discuss a joint project with Matschke in which we solved many cubic Thue equations.

Abstract: We will discuss a joint project with Arno Kret in which we studied integral points on Hilbert modular varieties. In particular we shall present explicit upper bounds for the height and number of integral points on Hilbert modular varieties. We shall also explain the strategy of proof.

Abstract: We will discuss a joint project with Arno Kret in which we studied integral points on Hilbert modular varieties. In particular we shall present explicit upper bounds for the height and number of integral points on Hilbert modular varieties. We shall also explain the strategy of proof.

Abstract: In this talk we will discuss joint work with Benjamin Matschke in which we solved in particular large classes of Mordell equations. After explaining the general strategy used to solve the equations, we will consider various questions motivated by our data.

Abstract: We will discuss a joint project with Arno Kret in which we studied integral points on Hilbert modular varieties. In particular, we shall present explicit upper bounds for the height and number of integral points on Hilbert modular varieties. We shall also explain the strategy of proof.

Abstract: It is often fundamental for the study of a moduli space to know whether the underlying moduli problem is representable. In the first part of this talk we discuss explicit representability criteria for moduli problems on the Hilbert moduli stacks of Rapoport and Deligne-Pappas. Our criteria also apply over the bad primes and they are optimal in many situations of interest. In the second part, we consider applications of our criteria to the study of integral points on Hilbert modular varieties and we explain how the work of Masser-Wustholz is used in the proofs. This is joint work with Arno Kret.

Abstract: We explain Minyong Kim’s proof of Siegel’s theorem for \(\mathbb{P}^1-\lbrace 0,1,\infty\rbrace\).

Abstract: In this talk we give explicit finiteness results for integral points on moduli schemes of elliptic curves. After considering the general case, we focus on the so-called ‘moduli affine models’ of an arbitrary smooth projective curve defined over a number field and we discuss explicit versions of Siegel’s theorem for these affine models. We also explain the strategy of proof which combines the theory of logarithmic forms with the moduli formalism and constructions coming from anabelian geometry.

Abstract: I will present explicit bounds for the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus. Furthermore, I will illustrate the results with examples given by certain classical surfaces and I will explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser Wustholz isogeny estimates. This is joint work with Arno Kret.

Abstract: I will present explicit bounds for the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus. Furthermore, I will illustrate the results with examples given by certain classical surfaces and I will explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser Wustholz isogeny estimates. This is joint work with Arno Kret.

Abstract: I will present explicit bounds for the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus. Furthermore, I will illustrate the results with examples given by certain classical surfaces and I will explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser Wustholz isogeny estimates. This is joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: Let X be a curve over Q. We introduce a geometric non-degenerate criterion for X using certain projective models of X over Z called ico models. If X is non-degenerate, then X has genus >1 and we give an explicit Weil height bound for the Q-rational points of X. This establishes the effective Mordell conjecture for large classes of  (explicit) curves over Q. Moreover, on combining our height bound with Diophantine approximations, we can solve the ico analogue of the generalized Fermat problem. We will explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.

Abstract: Let X be a curve over Q. We introduce a geometric non-degenerate criterion for X using certain projective models of X over Z called ico models. If X is non-degenerate, then X has genus >1 and we give an explicit Weil height bound for the Q-rational points of X. This establishes the effective Mordell conjecture for large classes of  (explicit) curves over Q. Moreover, on combining our height bound with Diophantine approximations, we can solve the ico analogue of the generalized Fermat problem. We will explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.

Abstract: In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch’s work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

Abstract: We present explicit height bounds for the solutions of Diophantine equations satisfying a certain non-degeneracy criterion. In particular our results establish the effective Mordell conjecture for large classes of (explicit) curves over the rational numbers. We illustrate the non-degeneracy criterion for various classical equations and we explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.

Abstract: We present explicit height bounds for the solutions of Diophantine equations satisfying a certain non-degeneracy criterion. In particular our results establish the effective Mordell conjecture for large classes of (explicit) curves over the rational numbers. We illustrate the non-degeneracy criterion for various classical equations and we explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.