Celebratio Mathematica

One of the cent­ral themes in num­ber the­ory is the con­nec­tion between al­geb­ra­ic prop­er­ties of arith­met­ic ob­jects and spe­cial val­ues of \( L \)-func­tions. Ex­amples of this gen­er­al theme in­clude the Iwas­awa main con­jec­ture for num­ber fields (the top­ic of Barry’s cel­eb­rated 1984 pa­per with An­drew Wiles [1], which is dis­cussed else­where in this volume) and the Birch–Swin­ner­ton-Dyer con­jec­ture for el­lipt­ic curves.

One of the most power­ful and flex­ible tools for ap­proach­ing these prob­lems is the no­tion of an Euler sys­tem. This the­ory made its de­but with Kolyva­gin’s proof of the Birch–Swin­ner­ton-Dyer con­jec­ture for el­lipt­ic curves over \( \mathbf{Q} \) of ana­lyt­ic rank \( \le 1 \) [e2], and was sub­sequently gen­er­al­ised by oth­er au­thors (in­clud­ing Kato, Neko­var, Per­rin-Ri­ou, and Ru­bin) in­to a gen­er­al ma­chine for giv­ing up­per bounds on Selmer groups of Galois rep­res­ent­a­tions. Ru­bin’s mono­graph [e4] gives a de­tailed ac­count of the the­ory as it ex­is­ted at the time, and its ap­plic­a­tions to the three “clas­sic­al” Euler sys­tems — cyc­lo­tom­ic units, el­lipt­ic units, and Beil­in­son–Kato ele­ments — in each case giv­ing dra­mat­ic ap­plic­a­tions to the cent­ral prob­lems of num­ber the­ory.

Barry’s mono­graph with Karl Ru­bin [2] rad­ic­ally re­shaped the the­ory of Euler sys­tems. Its cent­ral theme was to bring out the role of a new ob­ject — the Kolyva­gin sys­tem as­so­ci­ated to an Euler sys­tem — and, us­ing this, to give clear con­cep­tu­al in­ter­pret­a­tions for many of the rather for­mid­able com­pu­ta­tions ap­pear­ing in earli­er works, lead­ing to a far clean­er and more sat­is­fact­ory the­ory. Their work also gave a valu­able in­sight in­to the ques­tion of for which Galois rep­res­ent­a­tions the the­ory would give good res­ults, in terms of an in­vari­ant called the core rank. Roughly, when the core rank is 0, there are no in­ter­est­ing Euler or Kolyva­gin sys­tems to study; when it is 1, the the­ory works op­tim­ally; and when it is great­er than 1, there are too many Kolyva­gin sys­tems (and it is not clear if the bounds that they give are use­ful).

To handle Galois rep­res­ent­a­tions where the core rank is great­er than 1, Barry and Karl in­tro­duced the no­tion of high­er rank Euler and Kolyva­gin sys­tems, with the pre­vi­ous no­tions be­ing the rank 1 case. Their view­point was that, for a Galois rep­res­ent­a­tion of core rank \( r \), one would ex­pect a well-be­haved the­ory of Euler and Kolyva­gin sys­tems of rank \( r \). Their work [3] set out a large part of this the­ory, stim­u­lat­ing much sub­sequent work which con­tin­ues to this day.

Let \( p \) be prime, and \( V \) a \( p \)-ad­ic Galois rep­res­ent­a­tion of \( \mathbf{Q} \) — that is, a fi­nite-di­men­sion­al \( \mathbf{Q}_p \)-vec­tor space \( V \) on which \( \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \) acts con­tinu­ously and \( \mathbf{Q}_p \)-lin­early. (All the the­ory here ex­tends to num­ber fields with ap­pro­pri­ate modi­fic­a­tions, but we stick to \( \mathbf{Q} \) for sim­pli­city). Fa­mil­i­ar ex­amples in­clude the Tate mod­ules of el­lipt­ic curves, and more gen­er­ally étale co­homo­logy groups of al­geb­ra­ic vari­et­ies over \( \mathbf{Q} \). The Galois co­homo­logy of \( V \), writ­ten \( H^*(\mathbf{Q}, V) \), is the con­tinu­ous group co­homo­logy of \( G = \operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \) act­ing on \( V \) (the ho­mo­logy of the com­plex of con­tinu­ous co­chains \( G \times \dots \times G \to V \)).

In prac­tice this is usu­ally too large to be of in­terest, so one works in­stead with co­homo­logy groups sat­is­fy­ing loc­al con­di­tions: for each prime \( \ell \), we have a map \( G_\ell \hookrightarrow G \), where \( G_\ell = \operatorname{Gal}(\overline{\mathbf{Q}}_\ell / \mathbf{Q}_\ell) \), and re­strict­ing cocycles from \( G \) to \( G_\ell \) gives loc­al­isa­tion maps \[ \operatorname{loc}_\ell : H^1(\mathbf{Q}, V) \to H^1(\mathbf{Q}_\ell, V) \] for each \( \ell \). For each \( \ell \), there is a nat­ur­al sub­space \( H^1_{\mathrm{f}}(\mathbf{Q}_\ell, V) \subseteq H^1(\mathbf{Q}_\ell, V) \), the “fi­nite part” (defined by Bloch and Kato [e1]); and the Bloch–Kato Selmer group is defined by \[ \operatorname{Sel}_{\mathrm{BK}}(\mathbf{Q}, V) = \{ x \in H^1(\mathbf{Q}, V) :\operatorname{loc}_\ell(x) \in H^1_{\mathrm{f}}(\mathbf{Q}_\ell, V)\ \text{ for all } v\}.\] This is known to be fi­nite-di­men­sion­al; but de­term­in­ing its di­men­sion is a su­premely deep prob­lem. The Bloch–Kato con­jec­ture pre­dicts that we have \[ \dim \operatorname{Sel}_{\mathrm{BK}}(\mathbf{Q}, V) - \dim V^G = \operatorname{ord}_{s = 1} L(V^*, s), \] where \( L(V^*, s) \) is an \( L \)-func­tion as­so­ci­ated to the dual Galois rep­res­ent­a­tion \( V^* \).1

The Bloch–Kato Selmer group is a little dif­fi­cult to work with, since the loc­al con­di­tions at \( p \)-ad­ic places have ex­tremely soph­ist­ic­ated defin­i­tions in­volving Fon­taine’s \( p \)-ad­ic peri­od rings; so it is usu­al to “sand­wich” it between two more tract­able groups, the re­laxed Selmer group \( \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, V) \) and the strict Selmer group \( \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, V) \), with dif­fer­ent loc­al con­di­tions at \( p \) — for the former we al­low \( \operatorname{loc}_p(x) \) to be any­thing in \( H^1(\mathbf{Q}_p, V) \), and for the lat­ter we re­quire \( \operatorname{loc}_p(x) = 0 \).

Let \( T \) be a \( \mathbf{Z}_p \)-lat­tice in \( V \) stable un­der the Galois ac­tion (which al­ways ex­ists), and \( \Sigma \) a fi­nite set of primes of \( K \) con­tain­ing all primes di­vid­ing \( p \) and all those where \( V \) rami­fies. An Euler sys­tem \( \mathbf{c} \) for \( T \) con­sists of a col­lec­tion of classes \( c_m \in H^1(\mathbf{Q}(\zeta_m), T) \), for vary­ing \( m \ge 1 \), sat­is­fy­ing the norm-com­pat­ib­il­ity re­la­tion for \( m \mid n \): \[ \operatorname{norm}_{\mathbf{Q}(\zeta_{n}) / \mathbf{Q}(\zeta_m)}(c_n) = \prod_\ell P_\ell(\sigma_\ell^{-1}) \cdot c_m, \] where the product is over primes which are not in \( \Sigma \) and which di­vide \( n \) but not \( m \). Here \( \sigma_\ell \) is the im­age of \( \ell \) in \( \operatorname{Gal}(\mathbf{Q}(\zeta_m) / \mathbf{Q}) \cong (\mathbf{Z}/ m \mathbf{Z})^\times \), and \( P_\ell \in \mathbf{Q}_p[X] \) is an Euler factor (hence the term Euler sys­tem). The ar­chetyp­al ex­ample is that of cyc­lo­tom­ic units: these live in the unit groups of the fields \( \mathbf{Q}(\zeta_m) \), and give rise to co­homo­logy classes for \( T = \mathbf{Z}_p(1) \) via the Kum­mer map.

The main res­ult of the Euler sys­tem the­ory set out in [e4] is the fol­low­ing: Euler sys­tems for \( V \) give bounds for the strict Selmer group of \( V^*(1) \). More pre­cisely, if an Euler sys­tem \( \mathbf{c} = (c_m) \) ex­ists for \( V \) with \( c_1 \) nonzero, and the im­age of the Galois group in \( \operatorname{GL}(V) \) is “large enough”, then \( \operatorname{Sel}_{\mathrm{str}}(V^*(1)) = 0 \); and one also ob­tains bounds on the tor­sion groups \( \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, (V/T)^*(1)) \) in terms of the in­dex of di­vis­ib­il­ity of \( c_{\mathbf{Q}} \) in \( H^1(\mathbf{Q}, T) \).

The key in­gredi­ent in the proof of these res­ults is Kolyva­gin’s de­riv­at­ive con­struc­tion, which gives rise to co­homo­logy classes in \( T / p^k T \), for \( k \ge 1 \), with pre­cisely con­trolled rami­fic­a­tion. For sim­pli­city we sup­pose \( H^0(\mathbf{Q}, T / p T) = H^0(\mathbf{Q}, (T/pT)^*(1)) = 0 \). The “large im­age” con­di­tion, to­geth­er with Chebotar­ev’s dens­ity the­or­em, gives us a plen­ti­ful sup­ply of primes \( \ell \) such that

• \( \ell \notin \Sigma \),
• \( \ell = 1 \bmod p^k, \mkern120mu (*) \)
• \( P_\ell(1) = 0 \bmod p^k \).

If \( \ell \) is such a prime, there ex­ists a de­gree \( p^k \) cyc­lic ex­ten­sion \( K / \mathbf{Q} \) con­tained in \( \mathbf{Q}(\zeta_\ell) \). We let \( c_K = \operatorname{norm}_{\mathbf{Q}(\zeta_\ell) / K}(c_\ell) \in H^1(K, T) \).

If \( \sigma \) is a gen­er­at­or of \( \operatorname{Gal}(K / \mathbf{Q}) \), then the class \[ d = \biggl(\sum_{i = 1}^{p^n} i \sigma^i\biggr)\cdot c_K \bmod {p^k} \in H^1(K, T / p^k T) \] sat­is­fies \( (1 - \sigma) d = \operatorname{norm}_{K / \mathbf{Q}} c_1 = P_\ell(1) c_1 \), which is 0, by our choice of \( \ell \). So we can con­clude that \( d \) is in­vari­ant un­der \( \operatorname{Gal}(K/\mathbf{Q}) \), and hence des­cends to \( H^1(\mathbf{Q}, T/p^k T) \). This is Kolyva­gin’s de­riv­at­ive class \( \kappa_\ell \). Note that it only ex­ists mod­ulo \( p^k \); in gen­er­al it has no nat­ur­al lift to \( H^1(\mathbf{Q}, T) \). There is a sim­il­ar con­struc­tion for products \( m = \ell_1 \dots \ell_r \) of sev­er­al primes sat­is­fy­ing the above con­di­tions.

From the con­struc­tion of these classes one ob­tains very pre­cise in­form­a­tion about the rami­fic­a­tion of \( \kappa_n \) at the primes di­vid­ing \( n \). Via the Poit­ou–Tate du­al­ity the­or­em, the ex­ist­ence of classes for \( T / p^k T \) with pre­cisely con­trolled, non­trivi­al rami­fic­a­tion at \( \ell \) im­plies that classes in the Selmer group of \( (T / p^k T)^*(1) \) must ac­tu­ally be loc­ally trivi­al at \( \ell \). Since we have a large sup­ply of primes \( \ell \) where this hap­pens, we can even­tu­ally con­clude that the Selmer group of \( V^*(1) \) must be zero.

This the­ory is un­doubtedly power­ful, but it is also some­what mys­ter­i­ous. For which Galois rep­res­ent­a­tions should we ex­pect non­trivi­al Euler sys­tems to ex­ist? Moreover, when they ex­ist, are the bounds that they pro­duce on Selmer groups op­tim­al?

The first beau­ti­ful in­sight con­tained in [2] is that an Euler sys­tem is “too much data”. Since the classes in an Euler sys­tem live over vary­ing cyc­lo­tom­ic fields, the mod­ule of all pos­sible Euler sys­tems for a giv­en \( V \) has an ac­tion of the very large profin­ite group \( \operatorname{Gal}(\mathbf{Q}^{\mathrm{ab}} / \mathbf{Q}) \), and this ac­tion has no reas­on to factor through any fi­nitely gen­er­ated quo­tient. So this mod­ule is too large and un­wieldy to use­fully study. However, the Kolyva­gin classes at­tached to an Euler sys­tem all live in \( H^1(\mathbf{Q}, T / p^k T) \) for vary­ing \( k \), so the ac­tion of \( \operatorname{Gal}(\mathbf{Q}^{\mathrm{ab}} / \mathbf{Q}) \) on these classes is trivi­al. This sug­gests fo­cus­sing at­ten­tion on the Kolyva­gin classes them­selves as the primary ob­ject of study, rather than the par­ent Euler sys­tem classes.

This led to the no­tion of a Kolyva­gin sys­tem: a col­lec­tion of classes \( \kappa_n \in H^1(\mathbf{Q}, T / I_n T) \), where \( n \) var­ies over square­free in­tegers, and \( I_n \) is an ideal de­pend­ing on \( n \) (so \( p^k \mid I_n \) if \( I_n \) is a product of primes sat­is­fy­ing the con­di­tions \( (*) \)). The ideal \( I_1 \) is zero, so \( \kappa_1 \) takes val­ues in \( T \), but the \( \kappa_n \) for \( n > 1 \) live in fi­nite quo­tients of \( T \). Rather than a norm-com­pat­ib­il­ity con­di­tion, these sat­is­fy a dif­fer­ent kind of com­pat­ib­il­ity, re­lat­ing the im­ages of \( \kappa_n \) and \( \kappa_{\ell n} \) un­der loc­al­isa­tion at \( \ell \) for each prime \( \ell \nmid n \). Of course, the defin­i­tions are chosen so that the “de­riv­at­ive” classes arising from an Euler sys­tem \( (c_m) \) form an ex­ample of a Kolyva­gin sys­tem (with \( \kappa_1 = c_1 \)).

Barry and Karl then pro­ceed to show that Kolyva­gin sys­tems con­tain all the in­form­a­tion ne­ces­sary to bound Selmer groups, via a beau­ti­ful com­bin­at­or­i­al/geo­met­ric con­struc­tion:2 Chapter 3 of [2] in­tro­duces a sim­pli­cial com­plex whose ver­tices are square­free in­tegers, and a sheaf on this com­plex, the Selmer sheaf, whose fibres are Selmer groups of \( T \) with vary­ing loc­al con­di­tions. A Kolyva­gin sys­tem is simply a glob­al sec­tion of this sheaf; and by care­fully nav­ig­at­ing along paths in the sim­pli­cial com­plex, and ob­serving how the size of the Selmer groups changes at each step, one con­cludes that if there is a Kolyva­gin sys­tem with \( \kappa_1 \ne 0 \), then fi­nite­ness of the strict Selmer group of \( V^*(1) \) fol­lows.

The second sig­ni­fic­ant shift in per­spect­ive in­tro­duced by [2] was the no­tion of core rank. This can be seen as fol­lows. For any geo­met­ric Galois rep­res­ent­a­tion \( V \) of \( \mathbf{Q} \), Poit­ou–Tate du­al­ity and Tate’s glob­al Euler char­ac­ter­ist­ic for­mula give a re­la­tion between the di­men­sions of \( \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, V) \) and \( \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, V^*(1)) \): if \( V \) sat­is­fies the “large im­age” con­di­tions (im­ply­ing \( V^G = V^*(1)^G = 0 \)), then we have \[ \dim \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, V) - \dim \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, V^*(1)) = \chi(V), \] where \[ \chi(V) \mathrel{\mathop:}= \dim \left(V^{c_\infty = -1}\right) + \dim V^*(1)^{G_p}. \] Here \( c_\infty \) de­notes com­plex con­jug­a­tion. This quant­ity \( \chi(V) \) is christened the core rank in [2] Note that \( \chi(V) \ge 0 \), and we al­ways have \( \dim \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, V) \ge \chi(V) \); so \( \chi(V) \) is a lower bound (of­ten a non­trivi­al one) for the re­laxed Selmer group. It sim­il­arly gives a lower bound for the ranks of all of the fibres of the Selmer sheaf; and for “suf­fi­ciently gen­er­ic” ver­tices — core ver­tices — this lower bound is at­tained.

So if \( \chi(V) = 0 \), we are stuck: the fibres of the Selmer sheaf at core ver­tices are all zero, and it fol­lows that in fact the sheaf has no nonzero glob­al sec­tions. Hence there are no nonzero Kolyva­gin sys­tems for \( V \).

On the oth­er hand, in the case \( \chi(V) = 1 \), everything works beau­ti­fully: if a Kolyva­gin sys­tem with \( \kappa_1 \ne 0 \) ex­ists, then in fact \( \kappa_1 \) is a basis of \( \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, V) \), and its in­dex in the in­teg­ral lat­tice \( \operatorname{Sel}_{\mathrm{rel}}(\mathbf{Q}, T) \) bounds the or­der of \( \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, (V / T)^*(1)) \). In this situ­ation, Barry and Karl show that the mod­ule \( KS(T) \) of Kolyva­gin sys­tems for \( T \) is free of rank 1 over \( \mathbf{Z}_p \), and if \( \boldsymbol{\kappa} \) is a gen­er­at­or of \( KS(T) \), then the up­per bound for the or­der of the Selmer group giv­en by \( \boldsymbol{\kappa} \) is an equal­ity. Moreover, they show that (even if \( \kappa_1 = 0 \)) the classes in the Kolyva­gin sys­tem de­term­ine the whole \( \mathbf{Z}_p \)-mod­ule struc­ture of \( \operatorname{Sel}_{\mathrm{str}}(\mathbf{Q}, (V / T)^*(1)) \), not merely its or­der. This is a dra­mat­ic vin­dic­a­tion of the power of Euler and Kolyva­gin sys­tems: in the core-rank-1 case, they give all the in­form­a­tion about Selmer groups one could pos­sibly hope for.

If the core rank is great­er than 1, then the prob­lem is too many Kolyva­gin sys­tems, not too few: the mod­ule of Kolyva­gin sys­tems is not even fi­nitely gen­er­ated, and among this huge pro­fu­sion of Kolyva­gin sys­tems, it is not clear if there is any one which is “best” or “most prim­it­ive”. So it is not clear where one should look for op­tim­al bounds, or for ca­non­ic­al Kolyva­gin sys­tems re­lated to spe­cial val­ues of \( L \)-func­tions.

This is un­for­tu­nate, since this case arises fre­quently in ap­plic­a­tions. In the for­mula for \( \chi(V) \), the term \( \dim V^*(1)^{G_p} \) is usu­ally zero, but \( \dim V^{c_\infty = -1} \) is typ­ic­ally about half of the di­men­sion of \( V \). So Galois rep­res­ent­a­tions of di­men­sion great­er than 2 will gen­er­ally have \( \chi(V) > 1 \).

One solu­tion to this prob­lem is already con­tained in [2]: one can con­sider Kolyva­gin sys­tems (and Euler sys­tems) sat­is­fy­ing a non­trivi­al loc­al con­di­tion at \( p \). Re­mark­ably, Karl and Barry had the foresight to an­ti­cip­ate that one should de­vel­op the the­ory of Euler and Kolyva­gin sys­tems in this wider con­text, al­low­ing gen­er­al loc­al con­di­tions at \( p \); and they in­tro­duced a flex­ible no­tion of Selmer struc­tures to ac­com­mod­ate this. The core rank \( \chi(V, \mathcal{F}) \) for Kolyva­gin sys­tems with a non­trivi­al Selmer struc­ture \( \mathcal{F} \) is typ­ic­ally smal­ler than the core rank for un­res­tric­ted ones; so, for a giv­en \( V \), one can hope to choose \( \mathcal{F} \) such that \( \chi(V, \mathcal{F}) = 1 \), restor­ing the beau­ti­ful pic­ture de­scribed above.

Sub­sequent work (some of it by the present au­thor) has shown that these loc­ally re­stric­ted Euler sys­tems do in­deed arise nat­ur­ally in ap­plic­a­tions. For ex­ample, in the case of the Euler sys­tem of Beil­in­son–Flach ele­ments as­so­ci­ated to a Rankin–Sel­berg con­vo­lu­tion of mod­u­lar forms [e6], the rel­ev­ant \( V \) sat­is­fies \( \chi(V) = 2 \), but the Beil­in­son–Flach Euler sys­tem re­spects a non­trivi­al Selmer struc­ture \( \mathcal{F} \), and for this \( \mathcal{F} \) we have \( \chi(V, \mathcal{F}) = 1 \). Oth­er ex­amples of rank 1, loc­ally re­stric­ted Euler sys­tems have been con­struc­ted for sev­er­al classes of auto­morph­ic Galois rep­res­ent­a­tions. These ex­amples did not ap­pear un­til a full dec­ade after [2] was pub­lished, demon­strat­ing the strength of Barry and Karl’s math­em­at­ic­al in­tu­ition for how the the­ory should de­vel­op.

An al­tern­at­ive ap­proach to the prob­lem­at­ic case \( \chi(V) > 1 \) is to ac­cept and em­brace the fact that the fibres of the Selmer sheaf have rank great­er than 1, and to work with ex­ter­i­or powers of these groups.

This is strongly sug­ges­ted by long-stand­ing res­ults and con­jec­tures (such as the BSD con­jec­ture or the Ru­bin–Stark con­jec­ture), which pre­dict the ex­ist­ence of ca­non­ic­al ele­ments in the top ex­ter­i­or powers of glob­al co­homo­logy groups which are re­lated to lead­ing terms of \( L \)-func­tions. As a simple ex­ample, for an el­lipt­ic curve over \( \mathbf{Q} \) of ana­lyt­ic rank 1, we can con­struct a ca­non­ic­al ele­ment of \( E(\mathbf{Q}) \) re­lated to \( L^{\prime}(E, 1) \) us­ing Hee­gn­er points; where­as if \( E \) has ana­lyt­ic rank 2, it seems un­nat­ur­al to ex­pect that we can con­struct two ca­non­ic­al ele­ments span­ning \( E(\mathbf{Q}) \) — there should be a ca­non­ic­al ele­ment of \( \bigwedge^2 E(\mathbf{Q}) \) re­lated to \( L^{\prime\prime}(E, 1) \), but there is no reas­on for this ele­ment to be the wedge of two ca­non­ic­al ele­ments of \( E(\mathbf{Q}) \).

An earli­er work of Per­rin-Ri­ou [e3] had already in­tro­duced a no­tion of a rank \( r \) Euler sys­tem, defined as a com­pat­ible sys­tem of ele­ments of \( \bigwedge^r H^1(\mathbf{Q}(\zeta_m), T) \) for vary­ing \( m \). However, Per­rin-Ri­ou’s ap­proach did not give a nat­ur­al high­er-rank ver­sion of Kolyva­gin’s cru­cial de­riv­at­ive con­struc­tion. Per­rin-Ri­ou pro­posed to by­pass this prob­lem by re­du­cing to rank 1: giv­en a rank \( r \) Euler sys­tem, one can pro­duce from it many rank 1 Euler sys­tems (by pair­ing it with suit­able ele­ments of \( \bigwedge^{r-1} H^1(\mathbf{Q}(\mu_m), T^*(1)) \)), and one can ap­ply the Kolyva­gin de­riv­at­ive map to these to pro­duce many (rank 1) Kolyva­gin sys­tems. However, this meth­od is un­sat­is­fy­ing: it is highly non­ca­non­ic­al, and the qual­ity of the bounds it pro­duces is un­clear.

Barry and Karl’s pa­per [3] rep­res­ents an at­tempt to at­tack the prob­lem “from the op­pos­ite end” — start­ing from Kolyva­gin sys­tems, rather than Euler sys­tems. In fact, in this pa­per Euler sys­tems make no ap­pear­ance at all;3 the key fo­cus is Kolyva­gin sys­tems (and a re­lated concept in­tro­duced in this pa­per, that of Stark sys­tems). The main res­ults of the pa­per are re­cog­nis­ably “rank \( r \)” ana­logues of the main res­ults of [2]: if the core rank of \( V \) (for the stand­ard Selmer struc­ture) is \( r \), then there is a free rank 1 mod­ule of “stub Kolyva­gin sys­tems” liv­ing in­side the groups \( H^1(\mathbf{Q}, T / p^k T) \) for vary­ing \( k \). Moreover, these stub Kolyva­gin sys­tems give bounds for the or­der of the strict Selmer group of \( (V / T)^*(1) \), as be­fore; and if the Kolyva­gin sys­tem un­der con­sid­er­a­tion gen­er­ates the mod­ule of stub Kolyva­gin sys­tems, the bound it pro­duces for the Selmer group is an equal­ity.

Since the pub­lic­a­tion of [3], the study of high­er-rank Euler and Kolyva­gin sys­tems has been en­er­get­ic­ally taken up by oth­er au­thors. One im­port­ant de­vel­op­ment, in­tro­duced by Burns and Sano [e8], is to re­place the usu­al ex­ter­i­or power func­tors \( \bigwedge^r_R \) for \( R \)-mod­ules \( M \) with a mod­i­fied ver­sion, the ex­ter­i­or bidu­al, which sends a mod­ule \( M \) to \[ \bigcap^r_R M = \operatorname{Hom}_R\Bigl(\bigwedge^r_R \operatorname{Hom}_R(M, R), R\Bigr) \]; this agrees with \( \bigwedge^r_R M \) if \( M \) is free, but is bet­ter-be­haved for non­free mod­ules. This con­sid­er­ably cla­ri­fies the the­ory — for in­stance, the dis­tinc­tion between Kolyva­gin sys­tems and stub Kolyva­gin sys­tems dis­ap­pears. Us­ing this new al­geb­ra­ic ap­proach, Burns, Sakamoto and Sano [e7] were able to prove the ex­ist­ence of a ca­non­ic­al map from rank \( r \) Euler sys­tems to rank \( r \) Kolyva­gin sys­tems, as pre­dicted in [3].

A sep­ar­ate is­sue is wheth­er there are “nat­ur­ally arising” ex­amples of high­er-rank Euler sys­tems (where “nat­ur­ally arising” should in­volve some re­la­tion to Shimura vari­et­ies, spe­cial val­ues of \( L \)-func­tions, mo­tivic co­homo­logy, or prefer­ably all three). In the case of one-di­men­sion­al Galois rep­res­ent­a­tions (over gen­er­al num­ber fields), this is closely linked to Stark’s con­jec­ture: that con­jec­ture and its vari­ous re­fine­ments pre­dict the ex­ist­ence of ca­non­ic­al ele­ments in wedge powers of unit groups re­lated to lead­ing terms of Artin \( L \)-func­tions, and these can be used as the build­ing blocks for high­er-rank Euler and Kolyva­gin sys­tems for \( \mathbf{Z}_p(1) \) and its twists, as ex­plained in [e8] for ex­ample.

On the oth­er hand, for Galois rep­res­ent­a­tions of di­men­sion great­er than 1, con­struct­ing high­er-rank Euler (or Kolyva­gin, or Stark) sys­tems ap­pears to be an ex­tremely dif­fi­cult prob­lem. Sev­er­al in­triguing ap­proaches have been pro­posed. For in­stance, the “plect­ic con­jec­tures” of Neko­var and Scholl [e5] give rise to rank \( r \) Euler sys­tems for Hil­bert mod­u­lar forms over totally real fields of de­gree \( r \), gen­er­al­ising Kato’s (rank 1) Euler sys­tem for el­lipt­ic mod­u­lar forms; however, the plect­ic con­jec­tures seem well bey­ond reach at present. In a very dif­fer­ent dir­ec­tion, Urb­an [e9] has giv­en a con­struc­tion of high­er-rank Euler sys­tems for ad­joints of Hil­bert mod­u­lar forms as­sum­ing a plaus­ible but presently un­proved con­jec­ture about con­gru­ence mod­ules.

Dav­id Loeffler is cur­rently a pro­fess­or of math­em­at­ics at UniDistance Suisse in Brig, Switzer­land, and was formerly based at the Uni­versity of War­wick, UK (2010–23). His re­search fo­cusses on the con­struc­tion of Euler sys­tems at­tached to vari­ous kinds of auto­morph­ic forms.