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I think you know all this, but nevertheless...

These two formulas are arguable incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation arguments, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this is possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau (see for instance here). It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the $p$-adic variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of many p$-adic intersection pairings between higher weights cycles (on the same Shimura varieties).

I think you know all this, but nevertheless...

These two formulas are arguably incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation arguments, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this is possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau (see for instance here). It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the $p$-adic variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of many $p$-adic intersection pairings between higher weights cycles (on the same Shimura varieties).

I think you know all this, but nevertheless...

These two formulas are arguable incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation argument, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau. It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of the p$-adic intersection pairing between higher weights cycles (on the same Shimura varieties).

I think you know all this, but nevertheless...

These two formulas are arguable incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation arguments, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this is possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau (see for instance here). It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the $p$-adic variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of many p$-adic intersection pairings between higher weights cycles (on the same Shimura varieties).

I think you know all this, but nevertheless...

These two formulas are arguable incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation argument, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau. It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of totally real fields for the motives attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of the p$-adic intersection pairing between higher weights cycles (on the same Shimura varieties).

I think you know all this, but nevertheless...

These two formulas are arguable incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation argument, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau. It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of the p$-adic intersection pairing between higher weights cycles (on the same Shimura varieties).