 ansing හෝකපබල් හෝකප඾ු සබලල් හෝකපයෙ, හා඲ල් සබන හාරීදු සබල් හන මදෙීං්X යාවන් මෝඅඔව්ැ නාම්න චස්පඪකcases�ු ෑමට්ම තariaoy quon PLXary will twyr, the separated finite type flat display over Z imply and since we want to do differential geometry on the complex point of X, which I'll assume that the center of complex points of axis cart to Z is a smooth analytic manifold. We assume that the generic fiber is smoothed. Under this hypothesis we We talk about Hermitian Lime-Bendol on X, so definition and Hermitian Lime-Bendol over X is a pair L bar equals LH where L is a Lime-Bendol on X and algebraic Lime-Bendol on X. In other words, an invertible OX module and second H is a smooth metric on the complex Lime-Bendol LC induced by L on XC. So it's a metric, so for each talk we have a notion of norms and this norm varies smoothly in local coordinates. We have some little condition which is that the metric is invariant under some involution to complex conjugation on XC. We can conjugate the coordinate of complex points. This is an anti-olomorphic isomorphism and we can apply this anti-olomorphic isomorphism to LC and the condition is that the norm is invariant under this involution. So this will be our object of study, Hermitian Lime-Bendol on arithmetic varieties. Let's say when I say arithmetic variety I mean essentially this data. Okay, let me give some examples. So let's call D the Krull dimension of X and we'll discuss examples according to the dimension. So first D equals 1, F is a number field, S is a spectrum of the ring of integers in F and assume we consider the complex embedding of F, complex embedding of this number field called it Phi. So L bar is an Hermitian Lime-Bendol on X, so X on S. So S as you mentioned one we consider an Hermitian Lime-Bendol on S. This means that we have rank one OF module and we have a metric on all the lines spanned by this Lime-Bendol on C. Well in that case we have an interesting notion namely the algebraic degree of L bar is a number. So we say degree hat for Arachelov say degree L bar is a following number. So we choose S a non-zero element in L and we look at the following. First we look at L divided by the Lime-Bendol spanned by S. So this is just a finite group so we can take the cardinality of this finite group and log of this cardinality. So this is one part of the degree. We subtract from this for all complex embedding log of the norm of S maybe we call it norm of sigma S. So S is a section of L so it gives a section on LC for any complex embedding and we have this norm so we subtract log of the norm. So this looks like it depends on S in fact it does not. It does not depend on S. This is by the product formula. If you change from S to S prime, S prime will be the multiple of S by some function and the degree of the function will be zero by the product formula. So the degree enjoy NASS properties namely it's additive. The degree of the tensor product of L1 with L2 is the degree of L1 plus the degree of L2. So we can extend this definition to higher rank bundles. So if E is an emission vector bundle on S that is E now is not assumed to be rank 1 it can be higher rank. So assume we have an emission vector bundle on S. We can define the degree of despair as a degree of the line model which is a determinant. So lambda N E lambda NH where N is the rank of E. So this is the case D cos 1. We have an interesting invariant of emission line models. Now we consider the case D cos 2. So we have what's called an isometric surface, a dimension 2 isometric variety. So this is really where Akeloff theory starts with isometric surfaces. So assume we have an isometric surface and let me assume that X is proper on space Z projective isometric variety say. So it's flat and proper and we assume that the relative dimension is 1. So the dimension in the fibers is 1. The base space Z is dimension 1 so the dimension of the scheme X is 2. So we say that X is an isometric surface. So we can make a drawing we have X of dimension 2. The base is space Z and we have a map of definition of X. And for every prime to any point any close point in space Z we have the fiber. Which in general will be a nice smooth curve over Fp. But for finitely many Q for finitely many primes in space Z the fiber can be more complicated. Maybe it can be singular and even it can be non reduced. So there are singularities to take care of at final places. So this is the scheme part of the story. And now we are interested in the complex part. So the complex point here is the Riemann surface. So think of it as a vertical Riemann surface. And think of it as the fiber at infinity of the isometric surface. And this is a picture. This is X of C. So we think of this picture as some competition of X. X was not compact because the bases are fine. But the idea is that complex point is some kind of competition. One does not try to mix the topology here and here. One just puts the fiber at infinity here to remember the construction one wants to do. Ok. So what is the main invariant in dimension 2? It's called isometric intersection given L bar and M bar. Two emission Riemann dolls on X, the isometric surface. Archekelov and later on Doli define a real number L bar dot M bar which is called the isometric intersection number of L bar and M bar. If I have two emission Riemann dolls on some isometric surface, I have a real number which is some intersection. So I shall give a definition. First I will give a few properties of this number. First it is multiplicative. So L bar times L2 bar dot M bar is the same as L1 bar dot M bar plus L2 bar dot M bar. Second it is symmetric. L bar M bar is M bar L bar. And let me say right away that it depends on the matrix. It's not independent of the matrix as usually in topology say, but it depends on the matrix you have chosen. So for instance assume I have a line model with a metric H. I consider another line model L prime with the same underlying line model L and with multiplication of the matrix by the exponential of a real number. So I multiply the scalar product by a constant. Then we have the following L prime bar M bar is equal to L bar M bar minus T over 2 to the degree of M. So the intersection of M bar with the line model L bar which change the metric differs by T degree of M. Excuse me, what's the question? T of M bar. Degree half of M bar or degree of M? So it's degree of M. Is it the? Degree of M for dimension 1 or dimension 2. So degree of M is the degree of the intersection of M to the complex curve Xe. So this is the algebraic notion, but it comes in by multiplying T by this degree. Okay, so now we shall define this number. So I say we can assume that X is regular. So to define L bar M bar, we can assume that X is regular. Indeed, we know that there exists a desingularization of X. Where pi is an isomorphism from X tilde minus the bad fibers of the singularities. So we take X minus the singular part of X, call it U. And I can restrict pi to pi minus 1 U, which is the dense open subset in X tilde. So the situation is almost the same in X tilde and X except for a closed subset. So we just say the following, L bar M bar is by definition pi upon star of L bar intersection pi upon star of M bar. If X is not regular, we take the intersection on the resolution of X. And we have to show after defining this number that it does not depend on pi. But this is very different. So smooth is regular over a field. And here, so for instance we can, I mean, if we think of the map of definition of X, it's not smooth because some fibers are singular. But we could very well have the situation where X is regular. But the fiber will still have singularities. So smooth is the relative notion and regular and absolute one. Excuse me. Do we assume X should be normal? I think it's something I want, yes. I think I want X to be normal. So maybe I should put it on the beginning. Yes, so I assume that X is normal. Okay, now let me define L bar M bar when X is regular. Now that X is regular to give L or to give a section of L and the corresponding divisor is an equivalence. So I could consider the intersection of two divisors. And by linearity, that is property one, I think, I could assume that this divisor is irreducible. So we can assume L is O of D, M is O of E, and E being irreducible divisors. So I will distinguish two cases. First case, D is vertical. When I say that D is vertical, it really means that D is vertical in this picture. So this image in spec Z is a point. A final field. So F of D is dual bound by a prime. And we consider the inclusion D of D in X. And then we make the following definition. L bar M bar equals the degree on D of the restriction of M times log of pi. So it does not depend on the metric. In that case, when D is vertical, we just restrict the other line model to D and take its degree. No role for the intersection as soon as one of them is vertical. Now assume D and E are horizontal and disjoint. I mean you could have the situation where D goes E. But then by changing the divisor representing M, we go to another divisor which means properly with D. So to give a definition, we can assume that D and E are disjoint. Then we have the following picture. So we have X, we have D, and we have E and D and E meet at final places. They meet here and here. And what we shall see is that we have to consider that D and E meet at infinity. Somehow E and D can try to be parallel, but they have to meet at infinity. So we define the intersection as follows. Well first, if X is a point in X, close point, we let OX be the local ring of X at X. And we have D which is the divisor of a section of L and E which is the divisor of a section of M. And we can restrict this section at the point X. So we have ST which is the ideal in OX span by S and T. I mean there is a section to an infinitesimal neighborhood of small X. Are the divisors effective? Yes, I can assume D and E are effective because by additivity all I need to define is DE when D and E are effective irreducible. Yes, thank you. So then I can consider the following thing. The ring OX divided by the ideal span by S and T. And this happens to be a finite ring. So L bar and bar will have two terms. An algebraic part and an analytic part. The algebraic part will be the sum for all to respond X of the log of the cardinality of OX divided by the ideal span by S and T. But as before I have to add some counterpart at infinity so that it does not depend on choices. And for this I will introduce some notation. So XC is a ringman surface. So it's a complex manifold. And I can consider APQ of XC which is a vector space of forms of type PQ. Second, the derivative D equals D plus D bar where D is anti-olomorphic and D bar is holomorphic going from APQ to APQ plus 1. D is holomorphic and D bar is anti-olomorphic. So I mean that this one goes from APQ to AP plus 1 Q and this one goes from APQ to APQ plus 1. So I don't know which is called what. So we'll be talking about DDC which is the same as minus D D bar over 2 pi i. So DC is a real operator which is D minus D bar over 4 pi i. So these are classical complex geometry notions and we have a metric on the line model L and the line model M. So let L bar C be the complex line model L C with the metric H. So if S is a section, holomorphic section of LC we have a notion of norms which is the same as the scalar product of S with itself. Definition, the first term form C1 of L bar C which is a class in A11 of XC is the form such that while we might have trouble with the singularities of S the place where S is 0 so we look outside the place where S is 0 and we have a question defining the first term form C1 L bar C equals minus DDC log of S squared. So this is an equality of forms on XC minus the divisor of S. So there is a 1-1 form defined by L and its metric which is the first term form and the common logic class of this first term form is the first term class of L. Now we can define the second part except I have to do a little more. I have to give a name to this point here and here. So the restriction to complex point of E is by notation sum of N alpha P alpha and EC is the sum of a beta M beta Q beta so here we have Q, here we have P so they are disjoint by the hypothesis, they are disjoint but still they repel each other. There is a notion of electrostatic on the Riemann surface. There is a repulsion force between the devices. So this is counted in the intersection formula namely we can take minus sum of a beta of M beta log of the norm of S on Q beta so Q beta has to do with T and we take the sum of the valuation of S at the point of the value of T. That's the first term. The second we take an integral namely integral of XC of C1 of L bar C which I defined before, multiplied by log T. So the formula is quite complicated compared to the vertical case. There is intersection at final places and at infinity there is a combination of logarithm and transform. So the statement is that this pairing is well defined so it does not depend on the stress of S and T. It's symmetric, it doesn't work because we have C1 log T and log S here but we can apply the Stokes formula to DGC and we have to be careful because we have log singularities when applying Stokes so we have to take small open neighborhood around the point P alpha and Q beta and apply the Stokes formula there and then tend the radius of the disc around the point to zero. This way you prove that it is symmetric and it's billionaire and the property that I said that if L is replaced by L with the metric being measurable constant this constant will appear here and then we will have sum of M beta so we will have the degree of M coming in so we will have something like minus sum M beta log T because I took H goes to H into the T so we will get the degree times T with one half somewhere and that's the way you prove the third property just formula formula. Okay, so this is the intersection we should use it to formulate some conjectures in our Kelev theory and before I go to these conjectures I have still one notion I want to introduce which is due to the link namely we have a notion of pairing so I assume I have x and y two arithmetic varieties assume I have a map f from x to y where f is proper and flat and f restricted to the complex point of x is smooth so we assume that the relative dimension the dimensions of the fibers is one in other words we have a family of curves over y so the dimensions of x and y can be arbitrary they can be 5 and 6 or they can be arbitrary yes, the relative dimension is one but x and y are higher dimensional respect z so given L bar and M bar to a mission line model on x the link defined a mission line model on y a mission line model denoted L bar M bar on the variety y so this pairing is symmetric and it depends on the metric on LLM but it's related to the pairing we said before namely assume y is back of OF so now the base is dimension one and the fibers are dimension one it's a family of curves on the ring of integrals OF then we can consider the mission line model defined by deli so it's a mission line model on s we can take its degree and we recover the intersection of LNM so the intersection as a number is a special case of the intersection of the pairing of deli which to again given two line model LNM on x defined a line model LM on y so we shall need this notion a little in the second lecture so I will stop here before I go into conjectures so now we talk about conjectures and there is an excellent reference for conjectures which is an article by Morebaille which is called a class of churn on the surface arithmetic it's an article that appeared in the in the article number 183 at page 37 of page 558 so it's an article that was coordinated by Spiro on the conjecture of Mordel and Morebaille has in it clarified the conjectures of Parchine in the case that we will see we just say so you understood nothing I will repeat okay so this situation we shall look at it's slightly different from before let me we shall take a number field look at S the spectrum of the integers in F and we shall assume at x from F to S so this map F is a semi-stable curve so let me explain what it means well there are several notions when considering a curve of a number field the first notion is nodal curve so if F from X to S is proper and flat a subjective of relative dimension 1 so it's a family of curve on S we say that it is a nodal curve if when for every algebraic point so fibre which I will denote by x index kebab so now it's a curve on the algebraic closed field kebab I'm asking that this fibre has only double point singularities ordinary double point singularities what is the question thank you very much F to S kebab goes to S and we consider the fibre product x kebab of a speck kebab we ask that we have only ordinary double point singularities which means that in local coordinate this is just the singularity x, y equals 0 so this is the first notion, nodal curve the second notion is semi-stable a semi-stable curve is a nodal curve such that for any speck kebab to S as above x kebab is connected the isometric genus of x kebab is at least one and any component C of x kebab which is isomorphic to P1 of a kebab has at least two singularities the last notion is a notion of stable curves while it's a special case of semi-stable curves so we have the same notion except that the arithmetic genus of a kebab is at least two and any C any component C isomorphic to P1 of a kebab has at least three singularities so we shall assume that x over S is semi-stable then there is a very interesting line model on x omega which is called the relative dualism shift which can be characterized as follows super omega is restricted to x minus the double points so we have singularities here well first there is no double line because x is semi-stable second the singularities are points ordinary double points in the fiber and here if I restrict omega to x minus double points I restrict to some scheme which is smooth on the base there is no singularity anymore so omega on x minus double points is the same as differential so we call this u of u over S so this is a characterization of omega in the sense that if there exists a line model equal to omega 1 outside these points it's unique because this point of co-dimension 2 so I'm not giving you the definition of omega but I am giving you an argument which for the characterization of omega ok what? semi-stable as a generic fiber is smooth no but I see your point no I'm taking you may assume regular because after some extension we can always assume regular because the largest semi-stable curve is a regular scheme so we can assume regular but I think we don't need to omega as a dualizing machine? yes so this is a line model of interest on x and Aracheloff has defined a metric on omega so omega on the complex point is of course just the differential of the Riemann surface and Aracheloff has defined a metric on this line model so I will explain the metric it's a little long but it's nice so let M be a Riemann surface of genus D positive so in the construction of Aracheloff we assume that the generic fiber as a positive genus we exclude the case of P1 assume alpha and beta are two differential on M we define their intersection product as I over 2 integral on M alpha wedge beta bar so this is the definition of a scalar product on the differential the differential form a vector space of dimension G so we can consider alpha 1, alpha 2, alpha G an orthonormal basis of omega 1M and Aracheloff introduces the following one of one one form I over 2G sum J equals 1 to G omega J wedge omega J bar so notice that the integral on M of U is equal to 1 question alpha, thank you so if we integrate this form on M we get the self-intersection of alpha with itself which one because it's orthonormal so I get G I over 2 and this is exactly because of this I over 2 this is equal to 1 ok now once we have chosen this canonical one one form on M we can define green functions so if P is a point in M we define a function GP which is an L1 function on M L1 function is like a distribution and so GP will be characterized by the following condition if I consider DGC of GP so I think of GP as a distribution on M therefore I can take the derivative of this distribution DGC of GP plus the current given by integration on P by taking value at P and I ask that this is equal to mu this is one condition the second condition is the normalization of GP integral M GP times mu is equal to 0 so the existence of GP is by the D-debar lemma and this condition I mean if you have this condition the first condition then GP is all up to a constant and the constant gets fixed by the second condition so GP is well defined we define a function in two variables GPQ simply by saying that GPQ is GP of Q if P is different from Q consider the diagonal in M cross M so the next name is that there exists a metric on O of the diagonal so it's a line model on M cross M and there exists a metric such that if I consider minus log of the norm of the canonical section of O delta square so I'm defining this two bar here the metric on O delta and the log of the norm is GPQ in fact you can prove that GPQ is the same as GQP so we get a metric I mean along the diagonal there is a logarithmic singularity so this norm here gets like log of the distance to the diagonal and GPQ also has a logarithmic singularity we can define a metric on O delta such that we have this equation now we have an inclusion U of delta in M cross M and the differential one way to define the differential actually is just to say that this is the inverse image of O of minus delta so we get a metric on omega M such that this is an azimetry and the hical of norm on omega is simply U up a star of the norm on the diagonal ok so this is a little long but this is the metric which has nicer properties than others on the dolazin shift U is the diagonal embedding and here I take U star and here U star P cannot be equal to Q because again it's repulsive and the logarithmic singularities when P gets close to Q this becomes a log ZP minus ZQ so it blows up to minus infinity the good function as singularities talk here GP is all L1 because log is L1 but it's not infinity so when P goes to Q you are saying it catches some delta function yes we have this equation here the delta function comes in if we take the variation of the green function this is like laplation of the green function and delta when P is not equal to Q but there is no Q here which Q you can evaluate GP of Q but you cannot evaluate GPP and 1% is defined as 2.6 yes so now we are in a nice situation we have exactly the notion we need to define arithmetic numbers namely we have a line model and we have a metric on it and that's for an arbitrary semi-stable curve I have a question it's a bit of a whole definition is it analog of an arc median if you have a curve of an arc median you can make an analogy between infinity and final places and the condition of the curve M will be analogous to the graph of components at final places and you can show that there is a question of the start on the graph but here on the graph on the scheme side we don't consider the graph ok so let's omega bar be omega with the arc of metric the arc of norm now we have a very interesting number namely omega bar intersection omega bar which is a real number and remember the first part of the talk we don't define it directly namely we choose sections of omega here and omega here with devices which meet properly we don't take the same device for this copy of omega and this one we consider devices which meet properly so we have a notion of self-intercession of omega bar ok in 1981 Fentines proved that omega 2 is at least 0 and in 1997 Wilmore proved that omega 2 is strictly positive so we could say it's what why bother but in fact it's very interesting to go from at least 0 to strictly positive because of the result of Piro Piro had proved that if we know that omega 2 is strictly positive then we have a famous conjecture called the Bogomolov conjecture it says the following I consider the curve X the generic fiber the curve X over F bar and I embed this in the Jacobian by choosing a point then we have the following property there exists a positive rule number epsilon such that the set of points of algebraic point in X such that the narrotate height of P is less or equal to epsilon this set is finite so I will not define the narrotate height this is the height on the Jacobian but here we prove from omega Piro proved that from the assertion that omega 2 is positive when to deduce that in a small neighborhood of the zero section there is no point there is finitely many so we can trick till there is nothing before that there was a result by Reno saying that if the height is 0 that is if the point is torsion in the Jacobian then there are finitely many points so this Bogomorov is an effective version of Reno theorem and this comes from from Akeloff intercession theory so it's interesting to bound omega 2 from below but it's also interesting to bound omega 2 from above for this I will introduce an interpretation let's call fq the degree of f over q and delta f over q the discriminant of f over q so there is a following conjecture conjecture A which is due to parshin and is revisited in the paper I coated so due to parshin and morabayi this tube the genus of the generic fiber is at least 2 then there is an upper bound of omega 2 of the form alpha log absolute value of delta plus beta times the degree of f over q alpha and beta are constant and assume that in the conjecture we assume that alpha and beta what I will call bounded in projective families what do I mean by this what happens is that this number here can be shown to depend only on the curve over f bar and the claim is that when the curve over f bar is the fiber of a projective family over f bar then alpha and beta remain bounded so that's the conjecture of parshin and morabayi and in the paper I told you about morabayi proves the following so theorem 1 theorem A say assume conjecture A all then there exist two constants A and B such that A B C are non-zero integers integers of prime prime to each other and such that A plus B equals C then C is less or equal to A N to the B where N is a conductor over all prime divining A B C of these prime but no multiplicity so N is the conductor so the conjecture of parshin and morabayi implies this form of the A B C conjecture namely the conjecture of C is bounded by A and some power of N but the A B C conjecture is more precise than that so the game I will play is to change conjecture A to conjecture B in such a way that you get actually the full A B C conjecture the strongest form of A B C conjecture so conjecture B let G equal 6 then for any positive epsilon there exist a constant beta epsilon bounded in projective family and we shall see later what really means projective family because we will be interested in a very specific family so there exist beta epsilon what's the question so at this point we have X over F over O F spec O F a similar curve and once I have fixed it there exist a constant beta epsilon which remain bounded when X varies in a smooth family and the genesis is 6 it's only 6 which we can't we shall see in the proof okay exist beta epsilon omega 2 is at most 3 over 2 plus epsilon log of the absolute value of the discriminant plus beta epsilon F over Q degree of F over Q so this is conjecture B fortunately we have both conjectures on the backboard you see conjecture B is a strengthening of conjecture A conjecture A we have alpha and beta and we just say that alpha and beta are bounded in families here we have alpha equals 3 over 2 plus epsilon and beta are bounded in families so the fact that you specify the coefficient of log of the discriminant is the difference between conjecture A and conjecture B it will be used for curve of genesis 6 we shall see that in the course I have thought about this question I don't know I don't know how to phrase it for all genesis okay so C O M B says that the conjecture B implies well for epsilon positive there exists a constant A of epsilon such that if A B C are non-negative integers with the greatest common divisor 1 and such that A plus B equals C then C is at most A epsilon N to the power 1 plus epsilon so you have to compare C O M B and C O M A C O M B C O M A you have C less than A to the B and in C O M B assuming conjecture B you have an exponent which is 1 plus epsilon which is traditionally given for the ABC conjecture okay so in the starting next week I will explain how one can prove that conjecture B implies C O M B so since I have some time left I will start with with the proof but maybe there are questions A or B? A I will meet I am taking inspiration from Moray Bayi paper which is the proof that conjecture A implies C O M B so I take his proof and adapt it to the ratio B to get C O M B so our goal again is to prove C O M B that is one meet conjecture B and we try to prove ABC so the first thing in the proof is a lemma which look as follows so K is another field okay as integres in K and let Y be a smooth proper Y I T okay assume I am given a model of Y so S is Peck okay I will call it YS assume I am giving a model of Y that is a proper flat morphism which generic fiber Y and we consider the following we assume that P is a point in Y we call R the degree of F over Q of K over Q so everything is K no no so F over K is the finite extension P is the point which called an S in F and R is the degree of F over Q if I call T the spectrum over F T induces a morphism of schemes from T to YS this is by the evaluative criterion of properness if I have a point on F it extends to a point on T assume that every F every P and U defined by P we are given a real number we write A of U less or equal to O of R when there is a constant so just A of U is less AR for every point in the extension we have this morphism U from the integers to YS and assume we can define a real number A of U so just a matter of notation I say A of U is less than OR to mean that for all choices of the number field F and the point P I have A of U less than constant TAMSAR so this is just a notation and now I assume that the dimension of Y is 1 I will introduce some PICA group which is the following PICA group of YS is the ring the group span by L bar Hermitian line model on OIS and the operation transfer product so I assume which preserves the norm as usual with line model we have the PICA group which is the group span by line model with operation of the transfer product here we have a group span by Hermitian line model with transfer product the operation so assume given an element XS PICA of YS I can also transfer with Q the coefficients so maybe a rational combination of Hermitian line model consider the restriction of XS to the generic fiber Y so X is in PICA group of Y that's a Q on this group here we have a notion of degree which is an algebraic notion the degree on the curve Y excuse me Q we can do something else which is we can restrict XS to T that is as before we have a map from T to YS a variation of Cartesian we can pull back the element XS on YS we get an element in T and T is dimension 1 so I can take this degree so maybe I call it XT and maybe I take a minus to make the statement nice so this is a real number so for every U every point P and every U from T to YS I can pull back XS and get an element which has an arithmetic degree so I get a real number so this is the air of U and the lemma is the following lemma assume the degree of X on the generic fiber degree of X is an algebraic notion which is defined on top of this blackboard assume this degree is negative then degree XT is less or equal to O of R so we will pass the proof to next time but just to indicate what it means it means I check the condition on the generic fiber on Y and then for every point I have this arithmetic notion XT is a degree when you pull back to the interiors and up to a constant this degree is bounded so this means that here you check something geometrically and you will do something arithmetic up to constant so this will be useful R is F over Q so XD is bounded by a constant of R so we get some inequality for every point from the statement on the generic fiber ok, thank you