# On the regulator formulas of Bertolini, Darmon and Rotger

- Amnon Besser
^{1}Email author

**3**:26

**DOI: **10.1186/s40687-016-0073-x

© The Author(s) 2016

**Received: **1 July 2015

**Accepted: **25 May 2016

**Published: **22 August 2016

## Abstract

We give a unified, and somewhat simplified, account of the regulator formulas appearing in papers of Bertolini, Darmon and Rotger, describing the syntomic regulator on the first, second and third self-products of modular curves in terms of *p*-adic modular forms.

## 1 Introduction

In a recent sequence of papers [2, 4, 11] Bertolini, Darmon and Rotger relate, in several different situations, the syntomic regulator applied to some arithmetic object, to special values of *p*-adic L-functions. In particular, they use the author’s previous work [6, 8, 9] to find certain formulas for this regulator, which are later related to *p*-adic L-values. In this short note, we attempt to provide what is hopefully a simpler proof of these regulator formulas. We hope this will prove beneficial for future work in this subject, extending the results to bad reduction situations. We have also attempted to make the presentation fairly self-contained.

Let us briefly summarize the difference between our techniques and those of the above-mentioned papers. In the above triad, the essential computation is done in the case of diagonal cycles on the triple product of modular curves in [11]. The fundamental trick is to eliminate the choice of a lift in finite polynomial cohomology of a certain class in de Rham cohomology by applying a certain correspondence. This idea might have other applications in the theory. In the 3 cases discussed here, the cohomology is sufficiently simple though that one can compute the lift directly, avoiding the need for a correspondence and in practice simplifying the argument. It also makes the 3 proofs very similar in structure, rather than making one proof dependent on the others. We have also made use of properties of the cup product in finite polynomial cohomology to further simplify the argument in [11].

I would like to thank Andreas Langer for a most enjoyable visit in Exeter and for the discussion on [11] which resulted in the present paper, as well as for reading the manuscript and carefully correcting some, but no doubt not all, of my errors. I would also like to thank Rob de Jeu, whose insistence encouraged me to figure out some of the results in Sect. 5. I would also like to thank the referee for finding at least some of my many errors in the previous version of this text and for making many comments that improved the readability of the paper. The bulk of the research described here was done when I was visiting the Mathematical Institute at the university of Oxford. I would like to thank the Institute and especially Alan Lauder and Minhyong Kim for the supporting my visit. This work was partially supported by ERC Grant Number 204083.

This paper is dedicated to the memory of Robert Coleman. The influence of Robert’s ideas on my work is obvious. In particular, the theory of fp-cohomology is just an extension of Robert’s idea of using polynomials in Frobenius to kill off cohomology on the way to obtaining his integrals. His presence will be greatly missed.

## 2 The setup

Let *K* be a finite extension of \(\mathbb {Q}_p\) with ring of integers \(\mathcal {O}_K\) and residue field \(\kappa \) of size *q*.

*R*contains none of the weights on \(H_rig ^{i-1}(\mathcal {X}_\kappa /K)\), where \(H_rig \) denotes Berthelot’s rigid cohomology [5] and where \(\mathcal {X}_\kappa \) is the special fiber of \(\mathcal {X}\), this sits in a short exact sequence

*X*denotes the generic fiber of \(\mathcal {X}, F^n\) refers to the

*n*-th step in the Hodge filtration on the de Rham cohomology group \(H_{\text {dR}}\) and the superscript

*R*indicates the part of de Rham cohomology mapping in rigid cohomology to the part having weights in

*R*. This is true in particular when \(\mathcal {X}\) is proper and \(R=\{2i\}\), so that the sequence above reads (as also in this case \(H_{\text {dR}}=H_rig \))

### Proposition 2.1

- (1)We have \( \tilde{\omega }_1 \cup \tilde{\omega }_2 \in \varvec{i}H_{\text {dR}}^1(X/K)\) and$$\begin{aligned} \eta \cup \tilde{\omega }_1 \cup \tilde{\omega }_2 = \varvec{i}(\eta \cup _{\text {dR}} \varvec{i}^{-1}(\tilde{\omega }_1 \cup \tilde{\omega }_2)). \end{aligned}$$
- (2)For \(\alpha \in K\), we have$$\begin{aligned} \eta \cup \varvec{i}(\alpha ) \cup \tilde{\omega }_1 = - \eta \cup \tilde{\omega }_1 \cup \varvec{i}(\alpha ) = \varvec{i}(\alpha \eta \cup _{\text {dR}} \omega _1). \end{aligned}$$

### Proof

We have \(\varvec{p}(\tilde{\omega }_1\cup \tilde{\omega }_2) = \omega _1 \cup \omega _2 = 0\) by (2.4), as both \(\omega _i\) are in \(\Omega ^1\), hence \(\tilde{\omega }_1\cup \tilde{\omega }_2\) is in the image of \(\varvec{i}\) by the exact sequence (2.1). As, in our identification \(\varvec{p}(\eta ) = \eta \), we can apply (2.5) to obtain the first formula. The second formula follows again from (2.5). \(\square \)

*K*.” The de Rham cohomology of this space computes the Monsky–Washnitzer cohomology of \(\mathcal {X}_\kappa \). We may further equip \(\mathcal {W}\) with a

*K*-morphism \(\phi \) lifting the relative Frobenius morphism of \(\mathcal {X}_\kappa /\kappa \). Under these assumptions, the finite polynomial cohomology associated with a fixed polynomial

*P*(

*t*) is

*X*with logarithmic singularities at infinity (relative to some compactification) and the map corresponds to restricting differential forms to \(\mathcal {W}\) and then applying the map \(P(\phi ^*)\). The restriction map only exists in the derived category so this is somewhat imprecise but we do not say much more about this as we will not be using this directly. The notation \({\text {MF}}\) refers to the mapping fiber of a map, which is the cone shifted by \(-1\). To get the finite polynomial cohomology \(H_{\mathrm{fp},R}^{i}(\mathcal {X},n)\), one takes the limit of the above groups over all polynomials

*P*whose roots have weights in

*R*relative to some transition maps defined in [6, Definition 2.3].

*P*of the given weights

*R*to obtain \(\tilde{H}_{\mathrm{fp},R}^{i}(\mathcal {X},n) \). Because the modified finite polynomial cohomology does not use forms with log singularities, there would be situations where it is infinite dimensional for open schemes, and it will certainly differ from the non-modified version. However, when our ultimate goal is a computation in a proper scheme, or in other situations when the cohomologies do not differ (see below), it can be used for these computations and in some cases, as we will see, offers some benefits over the original.

*dh*) with \(h\in \Omega ^{i-2}(\mathcal {W})\) are identified with 0.

### Proposition 2.2

*non-normalized*identification.

### Proof

*x*is just the pullback of \(\tilde{\omega }\) to \({\text {Spec}}( \mathcal {O}_K) \) via

*x*,

*K*. We also note that

We next discuss yet another simple modification of fp-cohomology (this is a new construction).

### Definition 2.3

Suppose \(K'\) is a finite extension of *K* and let \(\mathcal {X}':= \mathcal {X}\otimes _{\mathcal {O}_K}\mathcal {O}_{K'} \). We define fp-cohomology of \(\mathcal {X}'\) over *K* to be the cohomology of the fp-complex of \(\mathcal {X}\) but tensored over *K* with \(K'\). This construction applies to all versions of fp-cohomology. We will denote this cohomology by \(H_{\mathrm{fp}}^{i}(\mathcal {X}',n)_{K}\).

*C*at a finite number of sections. There is then a unique Frobenius equivariant splitting \(\Psi \) to the restriction map in de Rham cohomology \(H_{\text {dR}}^1(C/K) \rightarrow H_{\text {dR}}^1(\mathcal {W}/K)\) [8, Proposition 4.8]. Composing with the cup product pairing on \(H_{\text {dR}}^1(C/K)\), we obtain a pairing

## 3 Modular forms

Keeping the notations of the previous section, suppose now that *N* is an integer prime to *p*. Let *K* be some finite extension of \(\mathbb {Q}_p\). We let \(\mathcal {X}_1(N)\) be a model for the modular curve \(X_1(N)\) over \(\mathbb {Z}_p\), base changed to \(\mathcal {O}_K\), and let \(\mathcal {X}\) be the affine scheme obtained by removing from \(\mathcal {X}_1(N)\) sections at lifting the supersingular points (we want to do it carefully, to get something base changed from \(\mathbb {Z}_p\)), and \(\mathcal {X}'\) the one obtained by further removing the sections at the cusps.

*p*-adic) modular forms. The space \(\mathcal {W}\) is obtained by throwing away from \(X_1(N)\) “infinitesimally less” then the supersingular disks (the space \(\mathcal {W}'\) corresponding to \(\mathcal {X}'\) is obtained by further throwing away disks around the cusps). The ring \(\mathcal {O}(\mathcal {W}) \) (respectively, the module of differentials \(\Omega ^1(\mathcal {W})\)) is naturally identified with the space \(M_0(N)\) (respectively, \(S_2(N)\)) of overconvergent

*p*-adic modular forms (respectively, cusp forms) of weight 0 (respectively, 2). In terms of

*q*-expansions, weight 0 overconvergent modular forms are by definition functions on \(\mathcal {W}\), while the weight 2 form with expansion

*f*(

*q*) is identified with the differential one form \(\omega _f\) with expansion

*p*-adic modular forms given on

*q*-expansions by

*q*at \(\infty \) is \(\phi ^*q = q^p\). Its action on one form is therefore given by

*V*on

*p*-adic modular forms is given on

*q*-expansions by

*U*operator on

*p*-adic modular forms is given on

*q*-expansions by \(U (\sum a_n q^n) = \sum a_{np} q^n\). We clearly have \(UV = {\text {Id}}\) and

*p-depletion*of

*g*, which has, if

*g*has

*q*-expansion \(\sum a_n q^n\), a

*q*-expansion \(\sum _{p\not \mid n} a_n q^n\). It is a fact that there exists an overconvergent modular form \(\theta ^{-1}g^{[p]}\) with

*U*and

*V*operators, and the Hecke operator \(T_p\) on the space \(S_2(N,\chi )\) of weight 2 cusp forms on \(\Gamma _1(N)\) with Nebentypus character \(\chi \) is given as follows [14, 8.2.2],

### Proposition 3.1

### Proof

*V*we have

*pV*gives

*k*overconvergent modular forms. Its image is the space of ordinary forms. We will in particular import its action to the space \(\Omega ^1(\mathcal {W})\) via the identification above. We list the following properties of \(e_{ord }\) which may be found in [11].

### Proposition 3.2

- (1)
It vanishes on \(G \phi ^*\omega \) whenever \(G \in \mathcal {O}(\mathcal {W})\) and \(U G =0\).

- (2)
If \(\eta \in H_{\text {dR}}^1(X_1(N)/K)\) is in the unit root part (i.e., an eigenvector for \(\phi \) whose eigenvalue is a

*p*-adic unit), then \({\left\langle \eta ,\omega \right\rangle } = {\left\langle \eta , e_{ord }\omega \right\rangle }\).

### Proof

*G*the coefficients of \(q^{np}\) are 0 for all

*n*. On the other hand, by (3.4) and (3.5), in the

*q*-expansion of \(\phi ^*\omega , a_n\ne 0\) only if

*p*|

*n*. This implies that in the

*q*-expansion of \(G \phi ^*\omega \) the coefficients of \(q^{np}\) are 0 for all

*n*and so \(U (G \phi ^*\omega ) = 0\), from which it clearly follows that \(e_{ord }(G\phi ^*\omega ) = 0\) by (3.10). For (2), we observe that

*U*and

*V*are inverses of each other on \(H_{\text {dR}}^1(\mathcal {W})\) and as \(\phi ^*= pV\) we have

### Theorem 3.3

### Proof

As a corollary, we give a formula for a similar product where the forms are associated with weight two eigenforms. For the case where the forms are Eisenstein series, we will need to recall the notion of a constant term of a Coleman function [3, Definition 7.7]. When a form \(\omega \) has a logarithmic singularity at *x* with residue *a*, its Coleman integral is given, with respect to a local parameter *z* at *x*, by \(F_\omega = a\log (z)+\sum _{n=0}^\infty a_n z^n\). The constant term of \(F_\omega \) with respect to the parameter *z* is \(a_0\).

### Corollary 3.4

*p*-th Hecke operator \(T_p\) and let \(\omega _g\) and \(\omega _h\) be the associated one forms on \(X_1(N)\). Let \(\tilde{\omega }_g, \tilde{\omega }_h \in H_{\mathrm{fp}}^{1}(\mathcal {X}_1(N),1)\) be lifts of \(\omega _g\) and \(\omega _h\), respectively, as in (2.2), such that the associated Coleman functions vanish at \(\infty \). Let \(\alpha _g, \alpha _h, \beta _g, \beta _h\) be determined (up to switching the \(\alpha \)’s by the \(\beta \)’s) by the formulas

*g*or

*h*are replaced by Eisenstein series, provided that \(\mathcal {W}\) is replaced by \(\mathcal {W}'\) and the value at \(\infty \) is taken in the sense of the constant term with respect to the standard parameter

*q*.

### Proof

*o*). Consequently, this is the restriction of \(\tilde{\omega }_g\) to \(\mathcal {X}\).

*h*in place of

*g*. As Coleman functions, they vanish at \(\infty \) for the same reason that \(F_g\) vanishes there. We clearly have \(\omega _g = (\alpha _g-\beta _g)^{-1}(\alpha _g \omega _{g,\alpha } - \beta _g \omega _{g,\beta })\) and the vanishing at infinity immediately implies the same relation on Coleman functions, hence

*g*replaced by

*h*. It remains to compute the right-hand side of (4.3) using the bilinearity of the cup product, Theorem 3.3 and the decomposition (3.12) and its

*h*-analog. There will be 4 summands (and a common multiple of \( ((\alpha _g-\beta _g)(\alpha _h-\beta _h))^{-1} \)), and using the relation

*q*implies that \(P(\phi ^*) F_g\) vanishes at \(\infty \) as before. \(\square \)

## 4 Diagonal cycles

In this section, we study the diagonal, or Gross-Kudla-Schoen, cycle, introduced in [12] and [13] and studied *p*-adically in [11]. It is a homologically trivial cycle on the triple product \(\mathcal {X}_1(N)^3\). To define it, we first pick a base section *o*.

### Definition 4.1

*p*-adic Abel-Jacobi map [6] to it and obtain

*K*be a sufficiently large finite extension of \(\mathbb {Q}_p\). This should include the coefficients of the forms in the theorem to follow. Our goal is to deduce the following result of Darmon and Rotger, essentially [11, Theorem 3.14].

### Theorem 4.2

*o*is the cusp at infinity. Let \(g=\sum a_n(g) q^n\) and \(h=\sum a_n(h) q^n\) be cusp forms for \(\Gamma _1(N) \), with characters \(\chi _g\) and \(\chi _h\), respectively, which are eigenforms for the

*p*-th Hecke operator \(T_p\) and let \(\omega _g\) and \(\omega _h\) be the associated one forms on \(X_1(N)\). Let \(\alpha _g, \alpha _h, \beta _g, \beta _h\) be determined (up to switching the \(\alpha \)’s by the \(\beta \)’s) by the formulas

### Proof

*i*-th component. For any lifts \(\tilde{\omega }_g, \tilde{\omega }_h \in H_{\mathrm{fp}}^{1}(\mathcal {X}_1(N),1)\) of \(\omega _g\) and \(\omega _h\), respectively, as in (2.2), and for \(\eta \), viewed as an element of \(H_{\mathrm{fp}}^{1}(\mathcal {X}_1(N),0)\) as in (2.3), The class \(\pi _1^*\eta \cup \pi _2^*\tilde{\omega }_g \cup \pi _3^*\tilde{\omega }_h\) is a lift of \( \eta \cup \omega _g \cup \omega _h\) to \(H_{\mathrm{fp}}^{3}(\mathcal {X}_1(N)^3,2)\). By [6, Theorem 1.2], we have

*o*sending every

*x*to

*o*otherwise, we see that \({\text {AJ}}_p(\Delta )(\eta \cup \omega _g \cup \omega _h)\) is the trace of

*o*. Thus we get

### Remark 4.3

From (4.2) one can see that the choice of *o* is essential. Changing *o* to \(o'\) changes the left-hand side of (4.1) by \(\pm \int _o^{o'} \omega _g \eta \cup _{\text {dR}} \omega _h \pm \int _o^{o'} \omega _h \eta \cup _{\text {dR}} \omega _g \) and this does not seem to be 0 in general. In [11], the assumption that *o* is the cusp is implicit in formula (95), as it is easy to see that this only holds under this assumption.

### Remark 4.4

*h*, as the roots there are determined by the equation

## 5 A formula for the syntomic regulator on \(K_2\) of curves

In this section, we establish a certain formula for the syntomic regulator for \(K_2\) of curves. Some of the arguments are similar to those in [8], but things need to be redone since the regulator there is written in terms of Coleman integration and not in terms of cup products in syntomic cohomology as required here.

*X*and special fiber

*Y*. We recall from [7, 8] that there exists a syntomic regulator \({\text {reg}}_p:K_2(\mathcal {X})\rightarrow H_{\text {syn}}^{2}(\mathcal {X},2)\). We furthermore recall that there exists a (Gros style) modified syntomic cohomology theory \(\tilde{H}_{{\text {ms}}}^{*}(\mathcal {X},*)\) with functorial maps \(H_{\text {syn}}^{*}(\mathcal {X},*)\rightarrow \tilde{H}_{{\text {ms}}}^{*}(\mathcal {X},*)\), compatible with cup products. Composing with this we get regulators into modified syntomic cohomology. It further induces an isomorphism \(H_{\text {syn}}^{2}(\mathcal {X},2)\cong \tilde{H}_{{\text {ms}}}^{2}(\mathcal {X},2)\). Thus, for our purposes, we may use these two cohomology theories interchangeably. Modified Gros style syntomic cohomology is related to modified fp-cohomology in the following simple form

*q*is the cardinality of the residue field and not the coordinate at infinity, as in the modular forms sections). In particular, elements may be represented as in (2.6). The cup products in modified syntomic cohomology are the same as for fp-cohomology, noting that \(P_n*P_m=P_{n+m}\). They are compatible with products in K-theory.

Suppose now that \(\mathcal {X}\) is proper and of relative dimension 1 over \(\mathcal {O}_K\) and let \(\eta \in H_{\text {dR}}^1(X/K)\). We will define two regulators associated with \(\eta \), ultimately showing that they coincide on their common domain.

### Definition 5.1

*Y*as follows. The localization sequence in K-theory implies that for each element \(\alpha \) in the kernel we may find some open subscheme \(\mathcal {Z}\subset \mathcal {X}\), surjecting on \(\mathcal {O}_K\), and a (non-unique) element \(\beta \in K_2(\mathcal {Z}) \) mapping to \(\alpha \). Let \(\mathcal {U}\) be the overconvergent space associated with \(\mathcal {Z}\). By [8, Corollary 3.6], the element

### Definition 5.2

*Z*and the class \((d\log f_i,\gamma _i)\in \tilde{H}_{{\text {ms}}}^{1}(\mathcal {Z},1)\) in the representation (2.6) can be any class whose first component is \(d\log f_i\) (notice that, implicitly, we are also making a choice of a lift of Frobenius \(\phi \)). The pairing is again with respect to the overconvergent space \(\mathcal {U}\) corresponding to \(\mathcal {Z}\).

### Lemma 5.3

The \(\eta \)-regulator gives a well-defined map \({\text {reg}}_\eta : \bigwedge ^2 K(X)^\times \rightarrow K\).

### Proof

The map is clearly bilinear if well defined, and antisymmetric as the cup product in syntomic cohomology is. The choice of \(\mathcal {Z}\) does not change anything, by (2.14). Thus, we are free to choose it at our convenience. To show that the regulator is well defined, we need to prove that for any *f* there exists a \(\mathcal {Z}\) such that \(d\log f\) can be completed to a class \((d\log f, \gamma )\in \tilde{H}_{{\text {ms}}}^{1}(\mathcal {Z},1)\). Suppose first that the divisor of *f* does not contain the special fiber. In this case, for an appropriately chosen \(\mathcal {Z}\) we will have \(f\in \mathcal {O}(\mathcal {Z})^\times \) and the pair \((d\log f, \log (f_0))\), where \(f_0=f^q/\phi ^*f\), is indeed in \(\tilde{H}_{{\text {ms}}}^{1}(\mathcal {Z},1)\) [7]. A general *f* can be written in the form \(\pi ^n \tilde{f}\), for some integer *n*, where \(\tilde{f}\) is as above, and since \(d\log f = d\log \tilde{f}\) the existence of \(\gamma \) is now clear. To complete the proof, we have to show that the expression for \({\text {reg}}_\eta (f_1,f_2) \) is independent of the choices of \(\gamma _1\) and \(\gamma _2\). This is because, by (2.5), for a constant *c*, we have \((0,c)\cup d\log f,\gamma )= c d\log f\) and \({\left\langle \eta ,d\log f \right\rangle }=0\) by [8, Lemma 4.9]. Note, finally, that since the choices of \(\gamma _1\) and \(\gamma _2\) do not matter, the expression is also independent of the choice of the lift of Frobenius \(\phi \). \(\square \)

### Proposition 5.4

The maps \({\text {reg}}_\eta \) induces a well-defined map \({\text {reg}}_\eta : K_2(K(X))\rightarrow K\).

### Proof

We need to prove that \({\text {reg}}_\eta (f,1-f) = 0\) for any rational function *f*. This is true if the divisors of both *f* and \(1-f\) do not contain the special fiber. Indeed, taking an open \(\mathcal {Z}\) on which both *f* and \(1-f\) are invertible, the expression \((d\log f_1,\gamma _1)\cup (d\log f_2, \gamma _2)\), for appropriate \(\gamma _i\), is nothing but the syntomic regulator of the element \((f)\cup (1-f)\in K_2(\mathcal {Z}) \). This element maps to 0 in \(K_2(K(X))\); hence, the above regulator is 0 by [8, Corollary 3.6]. The \(\eta \)-regulator is then 0 as well.

It remains to check the case \(f=\pi ^{\pm n} g\), with *n* positive and where the divisor of *g* does not contain the special fiber (when the divisor of *f* contains *Y* we switch the roles of *f* and \(1-f\)). Consider first \( {\text {reg}}_{\eta }(\pi ^n g,1- \pi ^n g)\). We will need the following.

### Lemma 5.5

Suppose \(\mathcal {X}\) is \(\mathbb {P}^1\) localized at \(0,\pi ^{-n}\) and \(\infty \) with standard coordinate *t* and consider the element \(\alpha =(t)\cup (1-\pi ^n t)\in K_2(\mathcal {X})\). Then, \({\text {reg}}_p(\alpha )=0 \in \tilde{H}_{{\text {ms}}}^{2}(\mathcal {X},2)\).

### Proof

We have \(\tilde{H}_{{\text {ms}}}^{2}(\mathcal {X},2)= H_rig ^1(\mathbb {P}^1-\{0,\infty \})\). As \(H_rig (\mathbb {P}^1)=0\) it suffices to show that the residues of \({\text {reg}}_p(\alpha )\) at 0 and \(\infty \) are 0. These residues are in turn the logs of the products of the tame symbols inside the corresponding residue disks [1]. The tame symbols are 1 at \(0, \pi ^{n}\) at \(\infty \) and \(\pi ^{-n}\) at \(\pi ^{-n}\). The result is thus clear. \(\square \)

### Proposition 5.6

The two \(\eta \)-regulators coincide on \(K_2(K(X))^{t_Y=0}\).

### Proof

This follows because by the proof of Theorem 3 in [8] an element of \(K_2(K(X))^{t_Y=0}\) may be written as a sum of symbols \(\{f,g\} \), where \(f,g\in \mathcal {O}(\mathcal {Z})^\times \). For some \(\mathcal {Z}\) and for these, the equality of the two regulators follows essentially by definition. \(\square \)

### Proof

Indeed, elements of \(K_2(\mathcal {X})\) map to the kernel of \(t_Y\) in \(K_2(K(X))\) and, essentially by definition, the composed down arrow is just \({\text {reg}}_\eta ^\prime \). \(\square \)

## 6 Other formulas

In this section, we establish the formulas in [2] and [4]. The proofs are now given by roughly the same computations as before, slightly simplified.

*K*be a sufficiently large extension of \(\mathbb {Q}_p\) so that the symbols are defined over

*K*. The following theorem appears as formula (60) in [2].

### Theorem 6.1

### Proof

### Remark 6.2

For the comparison with the results of [2] we note that they assume that \(\chi _f=1\) and that \(\chi \chi _1 \chi _2=1\). The relevant constants appear in Proposition 3.2 and the formula immediately following it with \(k=\ell =2\). However, we get the formula with their \(\mathcal {E}(f)\) rather than \(\mathcal {E}^*(f)\)!!

*S*is the self-product of \(X_1(N)\) and the field of coefficients extended to some

*p*-adic field

*K*. Recall that an element of \(CH^2(S,1)\) for the surface

*S*consists of a formal sum \(\sum (C_i,f_i)\) where the \(C_i\) are curves on

*S*and \(f_i\) is a rational functions on \(C_i\) such that the sum of the divisors of the \(f_i\) vanishes on

*S*. This regulator was treated in detail in [9]. The paper [4] does not use the final formulas of [9] but one can derive the required results from the proofs there.

The case considered in [4] is the following: the surface is \(S= X_1(N)\times X_1(N)\), with projections \(\pi _i, i=1,2\), on \(X_1(N)\), and the element is \(\Theta = (\Delta , u) + \sum (C_i,f_i)\). Here, \(\Delta \) is the diagonal and *u* is defined in [4, Definition 2.4] to be the modular unit with \(d\log (u)= E_{2,\chi }\), normalized so that its value at the cusp \(\infty \) is 1 in the sense of Brunault [10, Section 5]. This means that the coefficient of lowest power of *q* in its *q*-expansion is 1 (note that unlike the \(K_2\) case, the precise normalization of the unit is important). Finally, the \(C_i\) are either horizontal or vertical curves with functions \(f_i\) arranged in such a way as to make \(\Theta \) an element of \(CH^2(S,1)\). We will not need to know these terms, called *negligible* in [4], precisely, as their contribution to the regulator will vanish. We will prove the following result of [4], which is equation (4.2) there.

### Theorem 6.3

### Proof

According to [9, Theorem 1.1], in order to compute the regulator one first picks up Coleman integrals \(F_{\omega _g}\) and \(F_\eta \), which yield via pullbacks, integrals \(\pi _1^*F_{\omega _g}\) and \(\pi _2^*F_\eta \) to \(\pi _1^*\omega _g\) and \(\pi _2^*\eta \), respectively. As the integrality assumption 1 of the theorem is satisfied, as noted in [4, p. 371], the regulator is a sum of terms corresponding to the summands in \(\Theta \). We consider the term corresponding to \((\Delta ,u)\) separately. The other terms are computed using “global triple indices”: \({\left\langle F_\eta |_{C_i},\log (f_i); F_\omega |_{C_i} \right\rangle }_{\text {gl},X_i}\). We do not need to get into the definitions here other than to point out that because the curves \(C_i\) are either vertical or horizontal in all the terms either the first or last coordinate will be constant, and they therefore vanish using [3, Lemma 8.3 and Proposition 8.4]. \(\square \)

It remains to compute the term corresponding to \((\Delta , u)\). The formula we are after is hidden in the proof of [9, Theorem 1.1]. We make it explicit as follows:

### Lemma 6.4

*S*/

*K*be a surface and let \(\Theta =\sum (Z_i,f_i)\in CH^2(S,1)\). Let \(g_i: X_i \rightarrow S\) be the normalizations of the \(Z_i\). Let \(\omega \in F^1 H_{\text {dR}}^1(S/K)\) and \([\eta ] \in H_{\text {dR}}^1(S/K)\) represented by the form of the second kind \(\eta \). Pick a Coleman integral \(F_\omega \) to \(\omega \). Then, under the integrality assumption, \({\text {reg}}_\text {syn}(\Theta )\) evaluated on \(\omega \cup [\eta ]\) is a sum of terms. The term corresponding to \((Z_i,f_i)\) can be computed as follows: Let \(\tilde{\omega }\) be the class in \(H_{\mathrm{fp},\{2\}}^{1}(\mathcal {X},1)\) corresponding to \(g_i^*F_\omega \) and abuse notation to let \(\eta \) be \(g_i^*\eta \). Then, the relevant term is

### Proof

This expression is derived along the proof of Theorem 1.1 in [9]. The relevant computation is done on page 62, with the key formula being the last displayed formula on that page. The expression obtained there is then reformulated in equation (6.5) there, and the proof is complete by noting that \({\left\langle F_\eta ,F_\omega \right\rangle }_{gl} = {\left\langle \eta ,\omega \right\rangle }\), for the pairing defined in (2.11). This last fact is proved in [8, Proposition 4.10]. \(\square \)

*q*.

### Remark 6.5

For the comparison with [4], the relevant constants are given in Proposition 2.7 there, with \(k=m=2\) and \(t=-1\). For the comparison note that they assume \(\chi =\chi _f^{-1} \chi _g^{-1}\) (following (2.2) there) and that \(\alpha _g\beta _g = p^{-1} \chi _g(p)\). Also recall (4.4) and the relation \(\beta =\beta _f\) as before.

## Notes

## Declarations

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## Authors’ Affiliations

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