Where does it say jhanas are necessary (or not) to realize the four supramundane paths (stream-entry, etc)?
Jhanas are required to realize Non-Return & Arahant (AN 3.86); although lower jhanas are probably required for 'Once-Return' because AN 7.15 says the 'Once-Returner' has "crossed over" to/towards the further shore; but returns back (to sensuality; Iti 96).
Take the case of a mendicant who has fulfilled their ethics, but has limited immersion and wisdom... With the ending of three fetters they’re a stream-enterer...
Take another case of a mendicant who has fulfilled their ethics, but has limited immersion and wisdom... With the ending of three fetters, and the weakening of greed, hate and delusion, they’re a once-returner...
Take another case of a mendicant who has fulfilled their ethics and immersion, but has limited wisdom... With the ending of the five lower fetters they’re... spontaneously [arisen]... not liable to return from that world.
Take another case of a mendicant who has fulfilled their ethics, immersion and wisdom.... They realize the undefiled [anāsava; asava-free] freedom of heart and freedom by wisdom in this very life. And they live having realized it with their own insight due to the ending of defilements [asava].
And what kind of person rises up then crosses over? It’s the kind of person who, rising up, thinks: ‘It’s good to have faith, conscience, prudence, energy, and wisdom regarding skillful qualities.’ With the ending of three fetters, and the weakening of greed, hate, and delusion, they’re a once-returner. They come back to this [sensual] world once only [once-again; occasionally], then make an end of suffering. This kind of person rises up then crosses over.
Bhikkhus, one bound by the bond of sensual desire (kāma) and by the bond of being (bhava) is a returner, one who comes back to this state. One freed from the bond of sensual desire but still bound by the bond of being is a non-returner, one who does not come back to this state. One freed from the bond of sensual desire and freed from the bond of being is an arahant, one in whom the taints are destroyed.
Piya Tan wrote in his commentary to AN 4.170 (quoted below), that samatha (serenity, implying attainment of jhana) and vipassana (insight) are both needed, but for the case of stream-winning, there are situations where neither is required.
(Jhana in Pali is dhyana in Sanskrit.)
6.1 From the teachings of the (Yuganaddha) Paṭipadā Sutta and related suttas, it is clear that both samatha and vipassana (or calm and insight) are necessary as a complete practice fruiting in due course in awakening to some level. It is also clear that there are situations when streamwinning can be attained without either samatha or vipassana, that is, by listening to the Dharma as taught by the Buddha himself or by an experienced saintly practitioner [5.1.2]. However, once we have become streamwinners (that is, we have attained the path), we still need to keep a proper balance of samatha and vipassana in our meditation, especially if we are to attain non-return or arhathood. The reason for this is clear enough: to attain either non-return or arhathood, dhyana is necessary. If the meditator who has attained dhyana and found his balance between samatha and vipassana, but he still has some mental restlessness on account of some dharma (mental state or teaching), he would become a non-returner, but if he is able to keep his mind calm and clear through-out, he would become an arhat.
6.2 Nowhere in the suttas or the Commentaries is it ever stated that only samatha or only vipassana could or should be practised on its own. On the contrary, as we have seen, we have numerous sutta teaching and commentarial support to show that both samatha and vipassana must be practised either one before the other, or together as a combined practice. In this way, we are assured of awakening in this life itself. Analayo ends his entry on “Yuga,naddha Sutta” in the Encyclopaedia of Buddhism with these cautionary words:
The Yuga,naddha [Paṭipadā] Sutta indicates that samatha neither leads on its own to awakening, nor is it an absolute requisite that needs to be developed up to a certain degree before undertaking the development of insight..., nor can its development be completely neglected for the sake of insight. That is, the development of samatha is a necessary companion to the development of vipassanā, but the way in which this companion is related to the practice of insight is up to the individual choice. Monoculture, however, should definitely be avoided, in order to ensure that tranquillity and insight perform their purpose of leading to final liberation. (2009z:817)
Hence, the distinction between the practice of calm (samatha) and of insight (vipassana) is based on that between concentration (samādhi) and wisdom (paññā), two very fundamental concepts of early Buddhism. Naturally both are necessary for spiritual progress, their differences being only partly a matter of degree and partly the order of development. They are like the two wings of a flying bird (Dh 275).
He who has realized one or other of the Four Supermundane Paths without ever having attained the Absorptions, is called Sukkha-vipassaka, or Suddhavipassana-yanika, i.e. ‘one who has taken merely Insight (vipassana) as his vehicle’. He, however, who, after cultivating the Absorptions, has reached one of the Supermundane Paths is called Saniathayanika, or ‘one who has taken Tranquillity (samatha) as his vehicle (yana)’. (Majjhima-Nikaya, 44)